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b/1tFQT4oBgHgl3EQfEzWQ/content/tmp_files/2301.13239v1.pdf.txt @@ -0,0 +1,1669 @@ +arXiv:2301.13239v1 [math.QA] 30 Jan 2023 +Periodic Y-systems and Nahm sums: the rank 2 case +Yuma Mizuno +Abstract +We classify periodic Y-systems of rank 2 satisfying the symplectic property. We find +that there are six such Y-systems. In all cases, the periodicity follows from the existence +of two reddening sequences associated with the time evolution of the Y-systems in positive +and negative directions, which gives rise to quantum dilogarithm identities associated with +Donaldson-Thomas invariants. We also consider q-series called the Nahm sums associated +with these Y-systems. We see that they are included in Zagier’s list of rank 2 Nahm sums +that are likely to be modular functions. It was recently shown by Wang that they are indeed +modular functions. +1 +Introduction +1.1 +Background +The Y-system is a system of algebraic relations satisfied by coefficients of a cluster algebra, +which has the following form: +Yi(u)Yi(u − ri) = +� +j∈I +ri−1 +� +p=1 +Yj(u − p)[nij;p]+� +1 + Yj(u − p) +�−nij;p +(1.1) +where I is a finite index set, Yi(u) for i ∈ I, u ∈ Z are commuting variables, ri ∈ Z≥1, and +nij;p ∈ Z. We also use the notation [n]+ := max(0, n). +Such equations are first discovered +by Zamolodchikov in the study of thermodynamic Bethe ansats [36], prior to the discovery of +cluster algebras by Fomin and Zelevinsky [7]. The most striking feature of Zamolodchikov’s Y- +systems, as well as their generalizations [22, 30] defined shortly after the Zamolodchikov’s work, +is that they are periodic, which was fully proved by applying the theory of cluster algebras +[9, 10, 15, 16, 21]. +A systematic treatment of the Y-systems in the general setting of cluster algebras, including +the Y-systems arising from the thermodynamic Bethe ansatz as spacial cases, was given by +Nakanishi [27]. This approach was further developed in [24], and it was shown that the algebraic +relation (1.1) arises from a cluster algebra if and only if the data ri, nij;p have a certain symplectic +property. This allows the “axiomatic” study of Y-systems without explicitly referring to cluster +algebras. In this general setting, however, the Y-system is typically not periodic, and so the +study of periodic Y-systems as a generalization of Zamolodchikov’s Y-systems would be further +developed. In particular, the classification problem for periodic Y-systems is a challenging open +problem (see the last comments in [27, Section 3]). +There are several classification results in the literature. Fomin and Zelevinsky [10] showed +that the classification when ri = 2, nij;p ≤ 0, and nii;p = 0 for any i, j, p coincides with the +Cartan-Killing classification. Galashin and Pylyavskyy [12] generalized this result to show that +the classification when ri = 2 and nii;p = 0 for any i, p coincides with the classification of ADE +bigraphs of Stembridge [31]. On the other hand, the situation is more complicated when ri > 2 +for some i, and so far there has been no comprehensive classification results except when |I| = 1 +1 + +where it is not difficult to give a complete classification thanks to the work by Fordy and Marsh +[11] (e.g. see [24, Example 5.6]). +In this paper, we make a first attempt to give a classification result involving the case ri > 2 +for some i. Precisely, we classify the periodic Y-systems of the form (1.1) with |I| = 2 satisfying +the symplectic property. We would like to emphasize that we consider general ri, nij;p in the +classification. The result is given in the next section. +We also discuss the relation to Nahm’s conjecture on q-series [26, 35] in Section 1.3. +1.2 +Main result +Let I be a finite set. +We denote by Y0 the set of pairs (r, n) where r = (ri)i∈I and n = +(nij;p)i,j∈I,p∈N are families of integers satisfying ri ≥ 1 for any i and +n± +ij;p = 0 unless 0 < p < ri +(1.2) +for any i, j, p. +Definition 1.1. Let (r, n) ∈ Y0. Let P be a semifield, and (Yi(u))i∈I,u∈Z be a family of elements +in P. We say that (Yi(u)) satisfies the Y-system associated with the pair (r, n) if the relation +(1.1) holds for any i, u. The equation (1.1) itself is called the Y-system associated with (r, n). +We also say that (Yi(u)) is a solution of the Y-system if it satisfies the Y-system. +It is useful to think a pair (r, n) ∈ Y0 as a triple of matrices with polynomial entries by the +map (r, n) �→ (N0(z), N+(z), N−(z)) : Y0 → (MatI×I N[z])3 defined by +N0(z) := diag(1 + zri)i∈I, +N±(z) := +� � +p∈N +n± +ij;pzp +� +i,j∈I +(1.3) +where we set n± +ij;p := [±nij;p]+. We also define the map (r, n) �→ A±(z) : Y0 → (MatI×I Z[z])2 +by A±(z) := N0(z) − N±(z). Since this map is injective by the condition (1.2), we will identify +Y0 with the image of this map. For example, we will use the term “the Y-system associated +with A±(z) ∈ Y0”. +Definition 1.2. We say that A±(z) ∈ Y0 satisfies the symplectic property if +A+(z)A−(z−1)T = A−(z)A+(z−1)T, +(1.4) +where T is the transpose of a matrix. We denote by Y the subset of Y0 consisting of pairs +satisfying the symplectic property. +The pair A±(z) ∈ Y0 satisfies the simplectic property if and only if the Y-system associated +with A±(z) ∈ Y0 is realized as the exchange relations of coefficients in a cluster algebra [24]. +We review this fact in Section 2.1. +Definition 1.3. We say that a solution of a Y-system is periodic if there is a positive integer +Ω > 0 such that Yi(u + Ω) = Yi(u) for any i, u. +Definition 1.4. We say that a pair A±(z) ∈ Y is of finite type if any solution (in any semifield) +of the Y-system associated with this pair is periodic. In this case, we also say that Y-system +itself is periodic. +The purpose of this paper is to classify periodic Y-systems of rank 2. Before stating the +result, we give a few remarks. We say that A±(z) ∈ YI is decomposable if it is a direct sum +of some A′ +±(z) ∈ YI′ and A′′ +±(z) ∈ YI′′ with nonempty I′ and I′′. We say that A±(z) ∈ YI +is indecomposable if it is not decomposable. It is enough to consider indecomposable pairs in +the classification. We also note that A±(z) is of finite type if and only if A±(z)op := A∓(z) is +of finite type by the correspondence between solutions Yi(u) �→ Yi(u)−1. The main results are +summarized as follows: +2 + +A+(z) +A−(z) +h+ +h− +� +1 + z2 +−z +−z +1 + z2 +� +� +1 + z2 +0 +0 +1 + z2 +� +3 +2 +(1) +� +1 + z2 +−z +−z − z5 +1 + z6 +� +� +1 + z2 +0 +−z3 +1 + z6 +� +8 +6 +(2) +� +1 + z2 +−z +−z − z5 − z9 +1 + z10 +� +� +1 + z2 +0 +−z3 − z7 +1 + z10 +� +18 +10 +(3) +� +1 + z2 +−z +−z +1 + z2 +� +� +1 + z2 − z +0 +0 +1 + z2 − z +� +3 +3 +(4) +� +1 + z2 +−z +−z − z2 +1 + z3 +� +� +1 + z2 − z +0 +0 +1 + z3 +� +5 +3 +(5) +� +1 + z2 +−z +−z +1 + z2 − z +� +� +1 + z2 +0 +0 +1 + z2 +� +5 +2 +(6) +Table 1: Finite type classification for Y-systems of rank 2. The numbers h± are the length of +reddening sequences in positive and negative directions, respectively. +Theorem 1.5. Suppose that I = {1, 2}. +(1) Any pair A±(z) ∈ Y in Table 1 is of finite type. +(2) Any indecomposable pair A±(z) ∈ Y of finite type is reduced to exactly one pair in Table 1 +by permuting the indices, changing sign, and changing slices (see Section 3.1), if necessary. +The claim (1) can be proved by concrete calculation in a suitable universal algebra since +A±(z) in Table 1 is concrete. We, however, give another proof involving cluster algebras. We +give a quiver and a sequence of mutations for each A±(z) in Table 1 that yields the Y-system as +the exchange relation of coefficients in the cluster algebra. See Table 3 for quivers and mutations. +We can verify that some iteration of this sequence of mutations, as well as its inverse, is a +reddening sequence (Theorem 2.8). Thanks to the deep results in the theory of cluster algebras, +this property is enough to imply the periodicity (Proposition 2.7). The number h± in Table +1 are the length of reddening sequences in positive and negative directions, respectively. This +verification of the periodicity is interesting not only because it is computationally more efficient, +but also because it leads to nontrivial dilogarithm identities associated with Donaldson-Thomas +invariants (Corollary 2.10). +The claim (2) is proved in Section 3.2 by the following steps: +Step 1. We recall the result in [24] that asserts that A±(1) satisfies a certain positivity, which +in particular implies that tr A±(1) and det A±(1) are positive. This allows us to significantly +reduce the candidates for finite type A±(z). +Step 2. For a fixed A+(1) in the candidates obtained in Step 1, we search for A−(1) satisfying +the symplectic property (1.4) at z = 1. +Step 3. During the search in Step 2, we discard the pair A±(1) that cannot be endowed with +the parameter z (Lemma 3.3 and 3.4). +Step 4. At this point, we have six candidates up to a permutation of the indices and a change of +sign. For each A±(1) in the six candidates, we try to endow with the parameter z. It turns out +that this is possible for all the six candidates. We give all possible A±(z) in Lemma 3.5–3.8. +3 + +Step 5. We finally check that each remaining candidate reduces to one of A±(z) in Table 1 by +change of slices. +Remark 1.6. Most of the Y-systems obtained from Table 1 are already known in the litera- +ture. (1)op and (6)op are Zamolodchikov’s Y-system of type A2 [36] and T2 (“tadpole”) [30], +respectively. (2)op is the reduced sine-Gordon Y-system associated with the continued fraction +3/4 = [1, 3] = 1/(1+ 1/3), and (5)op with z replaced by z2 is the reduced sine-Gordon Y-system +associated with 3/5 = [1, 1, 2] = 1/(1 + 1/(1 + 1/2)) [32] . (4) is the “half” of the Y-system +associated with the pair (A2, A2) [30]. (3) appears to be new: +Y1(u)Y1(u − 2) = +1 +1 + Y2(u − 1)−1 +Y2(u)Y2(u − 10) = +� +1 + Y1(u − 3) +�� +1 + Y1(u − 7) +� +� +1 + Y1(u − 1)−1�� +1 + Y1(u − 5)−1�� +1 + Y1(u − 9)−1� +although it is implicitly given in the author’s previous work [24, Table 2]. +Remark 1.7. The pair A±(z) ∈ Y is called the T-datum in [24] since it describes the T-systems, +which is a companion to the Y-systems. We do not use this term since we only consider the +Y-systems in this paper. Moreover, the definition of the T-datum in [24] allows to have a non- +diagonal N0 and have a nontrivial symmetrizer D, which is more general than the definition in +this paper. See also Section 1.4 for the Y-systems involving nontrivial symmetrizers. +Remark 1.8. There is another expression of the Y-system using a pair of matrices A±(z) +directly. Let A±(z) ∈ Y0, and define aij;p ∈ Z by +A±(z) = +�� +p∈N +a± +ij;pzp +� +i,j∈I +. +Let (P ± +i (u))i∈I,u∈Z be a family of elements in a multiplicative abelian group P. We say that +(P ± +i (u)) satisfies the multiplicative Y-system associated with A±(z) if +� +j∈I +� +p∈N +P + +j (u − p)a+ +ij;p = +� +j∈I +� +p∈N +P − +j (u − p)a− +ij;p +(1.5) +for any i, u (schematically, “A+(z) · log P + = A−(z) · log P −” under the action z : u �→ u − 1). +The solution (P ± +i (u)) is called normalized if P is endowed with a semifield structure, and +P + +i (u) + P − +i (u) = 1 +for any i, u. We have a one-to-one correspondence between solutions of the Y-system (1.1) and +normalized solutions of the multiplicative Y-system (1.5). The correspondence is given by +Yi(u) �→ P + +i (u) +P − +i (u), +P + +i (u) �→ +Yi(u) +1 + Yi(u), +P − +i (u) �→ +1 +1 + Yi(u). +In the setting of cluster algebras, this correspondence is nothing but the normalization of the +coefficients described by Fomin and Zelevinsky [7, Section 5]. +1.3 +Relation to Nahm sums +Consider the q-series defined by +G(q) = +∞ +� +n=0 +qn2 +(q)n +, +H(q) = +∞ +� +n=0 +qn2+n +(q)n +, +(1.6) +4 + +where (q)n := (1 − q)(1 − q2) · · · (1 − qn) is the q-Pochhammer symbol. The famous Rogers- +Ramanujan identities express these q-series as the following infinite products: +G(q) = +� +n≡±1 mod 5 +1 +1 − qn, +H(q) = +� +n≡±2 mod 5 +1 +1 − qn. +These expressions in particular implies that q−1/60G(q) and q11/60H(q) are modular functions +on some finite index subgroup of SL(2, Z). In fact, it is a rare case that an infinite sum of the +form (1.6) is modular. It is known that the q-series +∞ +� +n=0 +q +1 +2 an2+bn+c +(q)n +(1.7) +with a, b, c ∈ Q is modular only if a = 1/2, 1, or 2 [35]. +Nahm [26] considered higher rank generalization of (1.7), which we call the Nahm sum. Let +I be a finite set, and suppose that A ∈ QI×I is a symmetric positive definite matrix, B ∈ QI is +a vector, and C ∈ Q is a scalar. The Nahm sum is the q-series defined by +fA,B,C(q) := +� +n∈NI +q +1 +2 nTAn+nTB+C +� +a(q)ni +. +When |I| ≥ 2, it is not well understood when fA,B,C(q) is modular. Nahm gave a conjecture +providing a criterion on the modularity of fA,B,C(q) in terms of torsion elements in the Bloch +group [26, 35]. See [3, 33] for the development of this conjecture. +Nahm used Zamolodchikov’s periodicity to provide an evidence of the conjecture. In fact, +there is a natural way to give a candidate of modular Nahm sums from finite type A±(z) ∈ Y +in general. Precisely, the matrix K := A+(1)−1A−(1) is always symmetric and positive definite +for finite type A±(z) ∈ Y, and it is conjectured that it gives a modular Nahm sum fK,0,C(q) for +some C [24]. (This construction is essentially the same as that in [18], except that they did not +prove that K is symmetric and positive definite. A special case can also be found in [23].) We +note that the symplectic property (1.4) at z = 1 plays an important role here since it implies +that K is symmetric. On the other hand, the positive definiteness is related to the periodicity +of the Y-system. +Based on our classification, we find that: +Theorem 1.9. Suppose that I = {1, 2}. The Nahm sum fK,0,C(q) is modular for any finite type +A±(z) ∈ Y, where C is given in Table 2. +In fact, every K from finite type A±(z) is included in the Zagier’s list [35, Table 2] for rank +2 candidates of modular Nahm sums. There are Rogers-Ramanujan type identities that enable +us to write each Nahm sum in the list in terms of theta functions. The proof of the desired +identities was partially given in [1, 4, 6, 33, 35], and was recently completed by Wang [34] except +for one candidate that does not appears in our construction from Y-systems. See Table 2. +Remark 1.10. We can define the refinement f (s) +A±(z)(q) of the Nahm sum fK,0,0(q), which is +parametrized by s ∈ H for an abelian group H of order det A+(1) such that it reduces to the +original one by taking summation [24, Definition 5.12]: +fK,0,0(q) = +� +s∈H +f (s) +A±(z)(q). +It is conjectured that each f (s) +A±(z)(q) is already modular after multiplying qC for some C. We +note that the symplectic property (1.4) at z = 1 again plays an important role in the definition +of the refinement. We will discuss this refinement for rank 2 case in more detail elsewhere. We +remark that similar refinement also appears in the context of 3-dimensional quantum topology +[13, Section 6.3]. +5 + +A±(z) +K +−24C +RR +A±(z) +K +−24C +RR +(1) +� +4/3 +2/3 +2/3 +4/3 +� +4 +5 +[6] +(1)op +� +1 +−1/2 +−1/2 +1 +� +6 +5 +[33] +(2) +�3/2 +1 +1 +2 +� +5 +7 +[34] +(2)op +� +1 +−1/2 +−1/2 +3/4 +� +9 +7 +[34] +(3), (6) +�2 +2 +2 +4 +� +4 +7 +[1] +(3)op, (6)op +� +1 +−1/2 +−1/2 +1/2 +� +10 +7 +[34] +(4) +� +2/3 +1/3 +1/3 +2/3 +� +1 +[35] +(4)op +� 2 +−1 +−1 +2 +� +1 +[35] +(5) +�1 +1 +1 +2 +� +3 +4 +[34] +(5)op +� 2 +−1 +−1 +1 +� +5 +4 +[4] +Table 2: The list of the matrix K = A+(1)−1A−(1). The Nahm sum fK,0,C(q) is modular, which +can be proved by using Rogers-Ramanujan type identities (RR for short) given in the references +in the table. +1.4 +Remarks on higher rank and skew-symmetrizable case +We have seen that the following properties hold for rank 2 case: +(P1) We have reddening sequences in both positive and negative directions. +(P2) The map A±(z) �→ A+(1)−1A−(1) gives modular Nahm sums. +We expect that the properties (P1) and (P2) also hold for any finite type A±(z) ∈ Y of general +rank. The followings are some known examples: +• For the Y-system associated with the untwisted quantum affine algebras Uq(X(1) +r ) with +level ℓ restriction [22], (P1) holds with h+ = tℓ and h− = t · (dual Coxeter number of Xr) +where t = 1, 2, or 3 is the multiplicity in the Dynkin diagram of Xr [15, 16], and (P2) +holds under the assumption [14, Conjecture 5.3] by the result of Kac and Peterson [17]. +• For the Y-system associated with a pair of finite type simply laced Dynkin type (Xr, X′ +r′) +[30], (P1) holds with h+ = (Coxeter number of Xr) and h− = (Coxeter number of X′ +r′) +[19, 21]. +• For the (reduced) sine-Gordon Y-system associated with the continued fraction p/q = +[nF, . . . , n1] = 1/(nF + 1/(· · · + 1/n1)) [29, 32], (P1) appears to hold with h+ = 2p and +h− = 2q. +• For the Y-system associated with an admissible ADE bigraph (Γ, ∆) [12], (P1) appears to +hold with h+ = (Coxeter number of Γ) and h− = (Coxeter number of ∆). +Moreover, we can consider Y-systems associated with skew-symmetrizable cluster algebras +rather than skew-symmetric ones discussed in this paper. In this case, the symplectic property +(1.4) becomes +A+(z)DA−(z−1)T = A−(z)DA+(z−1)T, +where D is a diagonal matrix called symmetrizer [24]. We also expect that the properties (P1) +and (P2) also hold for skew-symmetrizable case. See [24, Definition 5.12] for the definition of +the Nahm sum in skew-symmetrizable case. +Acknowledgment. This work is supported by JSPS KAKENHI Grant Number JP21J00050. +6 + +2 +Y-systems and cluster algebras +2.1 +Preliminaries on cluster algebras +In this paper, a semifield is a multiplicative abelian group equipped with an addition that is +commutative, associative, and distributive with respect to the multiplication. +Definition 2.1. Let I be a set. +The set of all nonzero rational functions in the variables +y = (yi)i∈I with natural number coefficients is a semifield with respect to the usual addition +and multiplication. This semifield is called the universal semifield, and denoted by Q>0(y). We +have a canonical bijection Homsemifield(Q>0(y), P) ∼= Homset(I, P) for any set I and semifield P. +Definition 2.2. Let I be a set. The tropical semifield Trop(y) is the multiplicative free abelian +group generated by the variables y = (yi)i∈I equipped with the addition defined by +� +i +yai +i + +� +i +ybi +i = +� +i +ymin(ai,bi) +i +. +Let I be a finite set and P be a semifield. A Y-seed is a pair (B, y) where B = (Bij)i,j∈I +is a skew-symmetric integer matrix and y = (yi)i∈I is a tuple of elements in P. We sometimes +represent B as the quiver whose signed adjacency matrix is B. For a Y-seed (B, y) and k ∈ I, +the mutation in direction k transforms (B, y) into the new Y-seed µk(B, y) = (B′, y′) given by +B′ +ij := +� +−Bij +if i = k or j = k, +Bij + [−Bik]+Bkj + Bik[Bkj]+ +otherwise, +(2.1) +y′ +i := +� +yk +if i = k, +yiy[Bki]+ +k +(1 + yk)−Bki +otherwise. +(2.2) +A mutation is involutive, that is, µk(B, y) = (B′, y′) implies (B, y) = µk(B′, y′). We have the +commutativity +µiµj = µjµi +if Bij = 0, +(2.3) +which allows us to write µi for a set i ⊆ I such that Bij = 0 for any i, j ∈ i to mean the +successive mutations along arbitrarily chosen order on i. +For a Y-seed (B, y) and a bijection ν : I → I, we define a new Y-seed ν(B, y) = (B′, y′) by +B′ +ν(i)ν(j) := Bij and y′ +ν(i) := yi. +2.2 +Solving Y-systems by cluster algebras +Let A±(z) ∈ Y. We will construct a solution of the Y-system associated with A±(z) based on +[24, Section 3.3]. We first define a subset R ⊆ I × Z by +R := {(i, u) ∈ I × Z | 0 ≤ u < ri}, +(2.4) +and define a skew-symmetric R × R integer matrix B by +B(i,p)(j,q) = −nij;p−q + nji;q−p + +� +k∈I +min(p,q) +� +v=0 +� +n+ +ik;p−vn− +jk;q−v − n− +ik;p−vn+ +jk;q−v +� +, +(2.5) +where we understand nij;p = 0 if p < 0. We then define i := {(i, u) | u = 0} ⊆ R. We also define +a bijection ν : R → R by +ν(i, p) = +� +(i, p − 1) +if p > 0, +(i, ri) +if p = 0. +(2.6) +7 + +Then the symplectic property (1.4) ensures that ν(µi(B)) = B [24, Lemma 3.16]. We finally +define a sequence of Y-seeds +· · · → (B, y(−1)) → (B, y(0)) → (B, y(1)) → · · · +(2.7) +in Q>0(y) by y(0) := y and (B, y(u + 1)) = ν(µi(B, y(u))). The sequence (2.7) gives a solution +of the Y-system: +Lemma 2.3. [24, Theorem 3.13] (yi,0(u))i∈I,u∈Z satisfies the Y-system associated with A±(z). +This solution is universal in the following sense. +Lemma 2.4. [24, Theorem 3.19] Suppose that a family (Yi(u))i∈I,u∈Z satisfies the Y-system +associated with A±(z). Define a semifield homomorphism f : Q>0(y) → P by +f(yi,p) := Yi(p) +� +j∈I +p +� +q=0 +Yj(p − q)−[nij;q]+� +1 + Yj(p − q) +�nij;q. +(2.8) +Then f(yi,0(u)) = Yi(u) for any i, u. +Corollary 2.5. A±(z) ∈ Y is of finite type if and only if there are different integers u, v such +that y(u) = y(v) in (2.7). +2.3 +Periodicity and reddening sequences +Similarly to (2.7), we define a sequence of Y-seeds +· · · → (B, y(−1)) → (B, y(0)) → (B, y(1)) → · · · +(2.9) +by the same formulas but now in Trop(y) rather than Q(y). +Definition 2.6. We say that the Y-system associated with A±(z) ∈ Y is positive (resp. negative) +reddening if there is a positive integer u such that all the exponents in yi(u) (resp. yi(−u)) in +(2.9) are nonpositive for any i. We denote by h+ (resp. h−) the least such positive integer u. +Equivalently, the Y-system is positive (resp. negative) reddening if and only if all the en- +tries in the C-matrix associated with the sequence of mutations (B, y(0)) → (B, y(u)) (resp. +(B, y(0)) → (B, y(−u))) are nonpositive for some u > 0. +Proposition 2.7. Suppose that the Y-system associated with A±(z) is positive and negative +reddening. Then A±(z) is of finite type. +Proof. We verify the equivalent condition in Corollary 2.5. By [2, Proposition 2.10], there are +bijections σ, σ′ : R → R such that yi(h+) = y−1 +σ(i) and yi(−h−) = y−1 +σ′(i) for any i (in other +words, the C-matrices associated with them are the minus of permutation matrices). Now the +claim follows from the separation formula for y-variables [10, Proposition 3.13] and the result +on C-matrices shown by Cao, Huang, and Li [5, Theorem 2.5]. See also [28, Theorem 5.2] for +the corresponding statement dealing with permutations that is actually suitable here. +Theorem 2.8. The Y-system associated with each A±(z) in Table 1 is positive and negative +reddening. +Proof. The quiver B associated with A±(z) is given in Table 3. We can verify the assertion by +concrete calculation on the quiver. The numbers h± are given in Table 1. +8 + +Quiver B +A±(z) +(1, 0) +(2, 1) +(1, 1) +(2, 0) +(1) +(1, 0) +(2, 1) +(2, 3) +(2, 5) +(1, 1) +(2, 0) +(2, 2) +(2, 4) +(2) +(1, 0) +(2, 1) +(2, 3) +(2, 5) +(2, 7) +(2, 9) +(1, 1) +(2, 0) +(2, 2) +(2, 4) +(2, 6) +(2, 8) +(3) +(1, 0) +(2, 1) +(2, 0) +(1, 1) +(4) +(2, 0) +(1, 1) +(1, 0) +(2, 2) +(2, 1) +(5) +(1, 0) +(2, 1) +(2, 0) +(1, 1) +(6) +Table 3: Quivers associated with A±(z) in Table 1. Each quiver is preserved by the mutation at +(∗, 0) followed by the permutation (i, p) �→ (i, p − 1) (the second argument is considered modulo +ri), which yields Y-system. For (1)–(3), this operation interchanges the connected components +(see Section 3.1). +Theorem 1.5 (1) now follows from Proposition 2.7 and Theorem 2.8. +Remark 2.9. A connected component of each quiver in Table 3 has the following cluster type: +(1) A2 +(2) A4 +(3) E6 +(4) D4 +(5) A5 +(6) A4 +These are of finite type in the sense of [8], which also implies Theorem 1.5 (1). We remark, +however, that this observation is somewhat misleading since the quiver associated with a periodic +Y-system of general rank is typically of infinite type. +It might be better to think that the +appearance of only finite type quivers happens “by chance” due to the smallness of 2, the rank +of Y-systems considered in this paper. +Theorem 2.8 also gives quantum dilogarithm identifies associated with Donaldson-Thomas +invariants. For any reddening sequence i starting from a quiver B, we can define a quantity +E(i) by using the quantum dilogarithm. We refer to [20, Remark 6.6] as the definition. This +quantity coincides with Kontsevich-Soibelman’s refined Donaldson-Thomas invariant associated +with B [20, 25]. In particular, E(i) does not depend on i, which gives the quantum dilogarithm +identifies. In our case, we have: +9 + +Corollary 2.10. For each A±(z) in Table 1, we have +E(µh+) = E(µ−h−), +where µ := ν ◦ µi is the sequence of mutations (together with the permutation) (B, y(0)) → +(B, y(1)) in (2.7). +For example, the pair (1) in Table 1 yields the famous pentagon identity of the quantum +dilogarithm. +3 +Classification +3.1 +Change of slices +We need to introduce an appropriate equivalence relation on the set Y, which identifies essentially +the same Y-systems. Before we get into the definition, we will see a typical example. Consider +the following Y-system: +Y1(u)Y1(u − 2) = (1 + Y2(u − 1)−1)−1 +Y2(u)Y2(u − 2) = (1 + Y1(u − 1)−1)−1 +(3.1) +which corresponds to A±(z) ∈ Y given by (1) in Table 1. This system of equations are defined +on the set [1, 2] × Z, but actually can be defined on each component of the following disjoint +union: +[1, 2] × Z = +1� +k=0 +{(i, u) | i − u ≡ k mod 2}. +We informally call the algebraic relation defined on each subset the slice of the whole Y-system. +If (Yi(u)) is a solution of the Y-system for i − u ≡ 0 mod 2, then (Yi(u + 1)) is a solution of the +Y-system for i − u ≡ 1 mod 2. Thus it is enough to consider only one slice when considering +solutions. Now we consider another Y-system: +Y ′ +1(u)Y ′ +1(u − 3) = (1 + Y ′ +2(u − 2)−1)−1 +Y ′ +2(u)Y ′ +2(u − 3) = (1 + Y ′ +1(u − 1)−1)−1. +(3.2) +which corresponds to A′ +±(z) ∈ Y given by +A′ ++(z) := +� +1 + z3 +−z2 +−z +1 + z3 +� +, +A′ +−(z) := +� +1 + z3 +0 +0 +1 + z3 +� +. +The Y-system (3.2) is decomposed into three slices: +[1, 2] × Z = +2� +k=0 +{(i, u) | i − u ≡ k mod 3}. +We see that for any solution of (3.1) for i − u ≡ 0 mod 2, +Y ′ +1(u) := Y1 +�2 +3u − 1 +3 +� +, +Y ′ +2(u) := Y2 +�2 +3u +� +is a solution of (3.2) for i − u ≡ 2 mod 3. We also obtain solutions for the other two slices by +shifting u. Conversely, any solution of (3.1) is obtained from a solution of (3.2). Therefore, it +10 + +is enough to consider one of the Y-systems (3.1) and (3.2). In particular, A±(z) is of finite type +if and only if A′ +±(z) is. +Now we work in the general setting. The idea is that each slice corresponds to each connected +component of the quiver associated with the matrix B defined by (2.5). Let A±(z) ∈ Y, and +assume that it is indecomposable. By [24, Proposition 3.24], we have a decomposition of the +matrix B and its index set R: +B = +t−1 +� +u=0 +B(u), +R = +t−1 +� +u=0 +R(u) +such that each B(u) is indecomposable and we have a cyclic sequence of mutations +B(0) +ν|R(0)◦µi(0) +−−−−−−−→ B(1) −→ · · · −→ B(t − 1) +ν|R(t−1)◦µi(t−1) +−−−−−−−−−−→ B(0) +(3.3) +where i(u) := i ∩ R(u). We say that two pairs A±(z) and A′ +±(z) are related by change of slices +if they yield the same cyclic sequence (3.3) up to a change of indices and the commutativity of +mutations (2.3). (This commutativity is already implicitly used to justify the notation µi(u) as +stated below (2.3).) +Example 3.1. The pairs A±(z) and A′ +±(z) associated with (3.1) and (3.2), respectively, are +related by change of slices. Indeed, we see that the sequence (3.3) for (3.1) is +(1, 0) +(2, 1) +ν◦µ(0,0) +−−−−−→ (1, 1) +(2, 0) +ν◦µ(1,0) +−−−−−→ (1, 0) +(2, 1) , +whereas the sequence (3.3) for (3.2) is +(1, 0) +(2, 1) +ν′◦µ(0,0) +−−−−−→ (1, 2) +(2, 0) +ν′◦µ(1,0) +−−−−−→ (1, 1) +(2, 2) +ν′ +−→ (1, 0) +(2, 1) . +These are the same sequence up to a change of indices. +3.2 +Proof of the classification +In this section, we will prove Theorem 1.5 (2). We first recall the following result. +Lemma 3.2 ([24, Theorem 5.5]). Let A±(z) ∈ Y. Assume that A±(z) is of finite type. Then +there is a vector v ∈ RI such that v > 0, vA+(1) > 0, and vA−(1) > 0. +In particular, +tr A±(1) > 0 and det A±(1) > 0. +By Lemma 3.2, A+(1) and A−(1) are equal to one of the following matrices: +� 2 +−1 +−1 +2 +� +, +� 2 +−1 +−2 +2 +� +, +� 2 +−1 +−3 +2 +� +, +� 2 +−1 +−1 +1 +� +, +� 2 +0 +−n +2 +� +, +� 2 +0 +−n +1 +� +, +� 1 +0 +−n +1 +� +up to a permutation of the indices. We give several lemmas about impossible pairs. Before +giving lemmas, we note that +n+ +ij;p = 0 +or +n− +ij;p = 0 +(3.4) +for any i, j, p. +Lemma 3.3. It is impossible that A±(z) ∈ Y has the following forms: +(1) A+(1) = +�2 +−a +∗ +∗ +� +, A−(1) = +�2 +−b +∗ +∗ +� +for odd a, b. +11 + +(2) A+(1) = +�2 +−a +∗ +∗ +� +, A−(1) = +�1 +−b +∗ +∗ +� +for odd a, b. +(3) A+(1) = +�1 +−1 +∗ +∗ +� +, A−(1) = +�1 +−1 +∗ +∗ +� +. +(4) A+(1) = +�1 +0 +∗ +∗ +� +, A−(1) = +�1 +∗ +∗ +∗ +� +. +Proof. For (2), we can set +A+(z) = +�1 + zr +−f(z) +∗ +∗ +� +, +A−(z) = +�1 + zr − za +−g(z) +∗ +∗ +� +. +By the symplectic property (1.4), we have +za + za−r + f(z)g(z−1) = z−a + zr−a + g(z)f(z−1). +(3.5) +Since 0 < a and a − r < 0 by (1.2), the sum of the coefficients of the terms in f(z)g(z−1) +with positive exponents is equal to that with negative exponents. Since f(1)g(1) (=ab) is odd, +f(z)g(z−1) should contain the constant term z0, which contradicts (3.4). The proof for (1) is +similar. +For (3), we can set +A+(z) = +� +1 + zr − za +−zb +∗ +∗ +� +, +A−(z) = +� +1 + zr − zc +−zd +∗ +∗ +� +with 0 < a, b, c, d < r. Without loss of generality, we can assume a < c. By (1.4), we have +z−c + zr−c + za + za−r + zc−a + zd−b = zc + zc−r + z−a + zr−a + za−c + zb−d +Since c − a > 0, we see that c − a is equal to c, r − a, or b − d. However, the first two cases are +impossible by (1.2). Thus c − a = b − d, which implies that +z−c + zr−c + za + za−r = zc + zc−r + z−a + zr−a. +Since a > 0, we see that a is equal to c or r − a. However, a = c is impossible by (3.4). Thus +a = r − a, which implies that +z−c + zr−c = zc + zc−r. +Since c > 0, we see that c = r − c. However, this implies that a = r/2 = c, which is impossible +by (3.4). +For (4), we can set +A+(z) = +�1 + zr − za +0 +∗ +∗ +� +, +A−(z) = +� +1 + zr − zb +∗ +∗ +∗ +� +. +By (1.4), we have +za + za−r + z−b + zr−b + zb−a = z−a + zr−a + zb + zb−r + za−b. +Comparing the number of the terms with positive and negative exponents, we should have a = b. +This is impossible by (3.4). +12 + +Lemma 3.4. It is impossible that indecomposable A±(z) ∈ Y has the form +A+(1) = +�∗ +0 +∗ +∗ +� +, +A−(1) = +�∗ +0 +∗ +∗ +� +. +Proof. We can set +A±(z) = +� +1 + zr1 − f±(z) +0 +−g±(z) +1 + zr2 − h±(z) +� +. +Since g+(z) ̸= 0 or g−(z) ̸= 0, we can pick the least integer c among the exponents in g+(z) and +g−(z). Without loss of generality, we can assume g+(1) contains the term zc. By (1.4), we have +f+(z)g−(z−1) + (1 + zr1)g+(z−1) = f−(z)g+(z−1) + (1 + zr1)g−(z−1). +(3.6) +The left-hand side in (3.6) contains the term zr1−c, but any exponent in the right-hand side is +strictly smaller that r1 − c by (1.2) and (3.4), which is a contradiction. +We now search for possible pairs A±(1) case by case using the symplectic property (1.4) at +z = 1 together with Lemma 3.3 and 3.4: +• Case: A+(1) = +� 2 +−1 +−1 +2 +� +. The possibilities for A−(1) are: +�2 +0 +0 +2 +� +, +�1 +0 +0 +1 +� +. +• Case: A+(1) = +� 2 +−1 +−1 +2 +� +. The possibilities for A−(1) are: +�2 +0 +0 +2 +� +, +�1 +0 +0 +1 +� +. +• Case: A+(1) = +� 2 +−1 +−2 +2 +� +. The possibilities for A−(1) are: +� 2 +0 +−1 +2 +� +, +�1 +0 +0 +2 +� +. +• Case: A+(1) = +� 2 +−1 +−3 +2 +� +. The possibilities for A−(1) are: +� 2 +0 +−2 +2 +� +. +• Case: A+(1) = +� 2 +−1 +−1 +1 +� +. The possibilities for A−(1) are: +�2 +0 +0 +2 +� +, +�1 +0 +0 +1 +� +. +13 + +• Case: A+(1) = +� 2 +0 +−n +2 +� +. The possibilities for A±(1) are: +��2 +0 +0 +2 +� +, +� 2 +−1 +−1 +2 +�� +, +�� 2 +0 +−1 +2 +� +, +� 2 +−1 +−2 +2 +�� +, +��2 +0 +0 +2 +� +, +� 1 +−1 +−1 +2 +�� +. +• Case: A+(1) = +� 2 +0 +−n +1 +� +. The possibilities for A±(1) are: +��2 +0 +0 +1 +� +, +� 2 +−2 +−1 +2 +�� +. +• Case: A+(1) = +� 1 +0 +−n +1 +� +. The possibilities for A±(1) are: +��2 +0 +0 +1 +� +, +� 2 +−2 +−1 +2 +�� +. +In summary, the remaining possible pairs, up to a permutation of the indices and an change +of sign, are given in the following table: +A+(1) +A−(1) +� 2 +−1 +−1 +2 +� +�2 +0 +0 +2 +� +� 2 +−1 +−2 +2 +� +� 2 +0 +−1 +2 +� +� 2 +−1 +−3 +2 +� +� 2 +0 +−2 +2 +� +A+(1) +A−(1) +� 2 +−1 +−1 +2 +� +�2 +0 +0 +2 +� +� 2 +−1 +−2 +2 +� +� 2 +0 +−1 +2 +� +� 2 +−1 +−3 +2 +� +� 2 +0 +−2 +2 +� +(3.7) +We now start searching for possible A±(z). +Lemma 3.5. Let n ≥ 1. Suppose that +A+(1) = +� 2 +−1 +−n +2 +� +, +A−(1) = +� +2 +0 +−(n − 1) +2 +� +. +Then +A+(z) = +� +[2]r +−z−a +−zr−a[n]2r +[2](2n−1)r +� +, +A−(z) = +� +[2]r +0 +−z2r−a[n − 1]2r +[2](2n−1)r +� +for some r, a, where [n]r is the z-integer defined by +[n]r := 1 − zrn +1 − zr . +(3.8) +Proof. We can set +A+(z) = +� +[2]r1 +−za +− �n +i=1 zbi +[2]r2 +� +, +A−(z) = +� +[2]r1 +0 +− �n−1 +i=1 zci +[2]r2 +� +. +14 + +Without loss of generality, we can assume that +b1 ≤ b2 ≤ · · · ≤ bn, +c1 ≤ c2 ≤ · · · ≤ cn−1. +By the symplectic property (1.4), we have +n−1 +� +i=1 +(z−ci + zr1−ci) + za + za−r2 = +n +� +i=1 +(z−bi + zr1−bi). +Comparing the degree by using the conditions (1.2) and (3.4), we obtain the system of linear +equations +a = r1 − b1, +a − r2 = −bn, +r1 = ci − bi = bi+1 − ci +(i = 1, . . . , n − 1), +which implies that +r2 = (2n − 1)r1, +bi = (2i − 1)r1 − a, +ci = 2ir1 − a. +Lemma 3.6. Suppose that +A+(1) = +� 2 +−1 +−1 +2 +� +, +A−(1) = +�1 +0 +0 +1 +� +. +Then +A+(z) = +� +1 + z2r +−za +−z2r−a +1 + z2r +� +, +A−(z) = +� +1 + z2r − zr +0 +0 +1 + z2r − zr +� +for some r, a. +Proof. We can set +A+(z) = +� +1 + zr1 +−za +−zb +1 + zr2 +� +, +A−(z) = +� +1 + zr1 − zc +0 +0 +1 + zr2 − zd +� +. +By (1.4), we have r1 = r2 = a + b = 2c = 2d. +Lemma 3.7. Suppose that +A+(1) = +� 2 +−1 +−2 +2 +� +, +A−(1) = +�1 +0 +0 +2 +� +. +Then +A+(z) = +� +1 + z2r +−za +−z2r−a − z3r−a +1 + z3r +� +, +A−(z) = +� +1 + z2r − zr +0 +0 +1 + z2r +� +for some r, a. +Proof. We can set +A+(z) = +� +1 + zr1 +−za +−zb1 − zb2 +1 + zr2 +� +, +A−(z) = +�1 + zr1 − zc +0 +0 +1 + zr2 +� +. +Without loss of generality, we can assume b1 ≤ b2. By (1.4), we have r1 = 2c, r2 = 3c, b1 = 2c−a, +and b2 = 3c − a. +15 + +Lemma 3.8. Suppose that +A+(1) = +� 2 +−1 +−1 +1 +� +, +A−(1) = +�2 +0 +0 +2 +� +. +Then +A+(z) = +� +1 + z2r +−za +−z2r−a +1 + z2r +� +, +A−(z) = +� +1 + z2r +0 +0 +1 + z2r +� +for some r, a. +Proof. We can set +A+(z) = +� +1 + zr1 +−za +−zb +1 + zr2 +� +, +A−(z) = +�1 + zr1 +0 +0 +1 + zr2 +� +. +By (1.4), we have r1 = r2 = a + b = 2c. +Proof of Theorem 1.5 (2). The remaining possibilities for finite type A±(z) ∈ Y, up to a per- +mutation of indices and change of sign, are the six families of the pairs given in Lemma 3.5–3.8, +which contain the parameters r, a. We can verify that these six families belong to Y, and they +can be reduced to the pairs in Table 1 by change of slices. +References +[1] George E. Andrews. An analytic generalization of the Rogers-Ramanujan identities for odd moduli. Proc. +Nat. Acad. Sci. U.S.A., 71:4082–4085, 1974. +[2] Thomas Br¨ustle, Gr´egoire Dupont, and Matthieu P´erotin. On maximal green sequences. Int. Math. Res. +Not. IMRN, (16):4547–4586, 2014. +[3] Frank Calegari, Stavros Garoufalidis, and Don Zagier. Bloch groups, algebraic K-theory, units, and Nahm’s +conjecture. arXiv preprint arXiv:1712.04887, 2017. +[4] Corina Calinescu, Antun Milas, and Michael Penn. Vertex algebraic structure of principal subspaces of basic +A(2) +2n -modules. J. Pure Appl. Algebra, 220(5):1752–1784, 2016. +[5] Peigen Cao, Min Huang, and Fang Li. A conjecture on C-matrices of cluster algebras. Nagoya Math. J., +238:37–46, 2020. +[6] Ivan Cherednik and Boris Feigin. Rogers-Ramanujan type identities and Nil-DAHA. Adv. Math., 248:1050– +1088, 2013. +[7] Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15(2):497–529, +2002. +[8] Sergey Fomin and Andrei Zelevinsky. Cluster algebras. II. Finite type classification. Invent. Math., 154(1):63– +121, 2003. +[9] Sergey Fomin and Andrei Zelevinsky. Y -systems and generalized associahedra. Ann. of Math. 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Publ. Res. Inst. Math. Sci., 49(1):1–42, +2013. +[16] Rei Inoue, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, and Tomoki Nakanishi. Periodicities of T-systems +and Y-systems, dilogarithm identities, and cluster algebras II: types Cr, F4, and G2. Publ. Res. Inst. Math. +Sci., 49(1):43–85, 2013. +[17] Victor G. Kac and Dale H. Peterson. Infinite-dimensional Lie algebras, theta functions and modular forms. +Adv. in Math., 53(2):125–264, 1984. +[18] Akishi Kato and Yuji Terashima. +Quiver mutation loops and partition q-series. +Comm. Math. Phys., +336(2):811–830, 2015. +[19] Bernhard Keller. On cluster theory and quantum dilogarithm identities. In Representations of algebras and +related topics, EMS Ser. Congr. Rep., pages 85–116. Eur. Math. Soc., Z¨urich, 2011. +[20] Bernhard Keller. Cluster algebras and derived categories. In Derived categories in algebraic geometry, EMS +Ser. Congr. Rep., pages 123–183. Eur. Math. Soc., Z¨urich, 2012. +[21] Bernhard Keller. The periodicity conjecture for pairs of Dynkin diagrams. Ann. of Math. (2), 177(1):111–170, +2013. +[22] A. Kuniba and T. Nakanishi. Spectra in conformal field theories from the Rogers dilogarithm. Modern Phys. +Lett. A, 7(37):3487–3494, 1992. +[23] Chul-hee Lee. Nahm’s conjecture and Y -systems. Commun. Number Theory Phys., 7(1):1–14, 2013. +[24] Yuma Mizuno. Difference equations arising from cluster algebras. J. Algebraic Combin., 54(1):295–351, 2021. +[25] Kentaro Nagao. +Quantum dilogarithm identities. +In Infinite analysis 2010—Developments in quantum +integrable systems, RIMS Kˆokyˆuroku Bessatsu, B28, pages 165–170. Res. Inst. Math. Sci. (RIMS), Kyoto, +2011. +[26] Werner Nahm. Conformal field theory and torsion elements of the Bloch group. In Frontiers in number +theory, physics, and geometry. II, pages 67–132. Springer, Berlin, 2007. +[27] Tomoki Nakanishi. Periodicities in cluster algebras and dilogarithm identities. In Representations of algebras +and related topics, EMS Ser. Congr. Rep., pages 407–443. Eur. Math. Soc., Z¨urich, 2011. +[28] Tomoki Nakanishi. Synchronicity phenomenon in cluster patterns. J. Lond. Math. Soc., II. Ser., 103(3):1120– +1152, 2021. +[29] Tomoki Nakanishi and Salvatore Stella. +Wonder of sine-Gordon Y -systems. +Trans. Amer. Math. Soc., +368(10):6835–6886, 2016. +[30] F. Ravanini, A. Valleriani, and R. Tateo. Dynkin TBAs. Internat. J. Modern Phys. A, 8(10):1707–1727, +1993. +[31] John R. Stembridge. Admissible W -graphs and commuting Cartan matrices. Adv. in Appl. Math., 44(3):203– +224, 2010. +[32] R. Tateo. New functional dilogarithm identities and sine-Gordon Y -systems. Phys. Lett. B, 355(1-2):157–164, +1995. +[33] Masha Vlasenko and Sander Zwegers. Nahm’s conjecture: asymptotic computations and counterexamples. +Commun. Number Theory Phys., 5(3):617–642, 2011. +[34] Liuquan Wang. +Identities on Zagier’s rank two examples for Nahm’s conjecture. +arXiv preprint +arXiv:2210.10748, 2022. +[35] Don Zagier. The dilogarithm function. In Frontiers in number theory, physics, and geometry. II, pages 3–65. +Springer, Berlin, 2007. +[36] Al. B. Zamolodchikov. On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering +theories. Phys. Lett. B, 253(3-4):391–394, 1991. +Department of Mathematics and Informatics, Faculty of Science, Chiba University, Chiba 263- +8522, Japan. +E-mail address, Y. Mizuno: ymizuno@math.s.chiba-u.ac.jp +17 + diff --git a/1tFQT4oBgHgl3EQfEzWQ/content/tmp_files/load_file.txt b/1tFQT4oBgHgl3EQfEzWQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6076ad9012aef4ba9474a2fd985905c8bc0b7915 --- /dev/null +++ b/1tFQT4oBgHgl3EQfEzWQ/content/tmp_files/load_file.txt @@ -0,0 +1,878 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf,len=877 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='13239v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='QA] 30 Jan 2023 Periodic Y-systems and Nahm sums: the rank 2 case Yuma Mizuno Abstract We classify periodic Y-systems of rank 2 satisfying the symplectic property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We find that there are six such Y-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In all cases, the periodicity follows from the existence of two reddening sequences associated with the time evolution of the Y-systems in positive and negative directions, which gives rise to quantum dilogarithm identities associated with Donaldson-Thomas invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We also consider q-series called the Nahm sums associated with these Y-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We see that they are included in Zagier’s list of rank 2 Nahm sums that are likely to be modular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' It was recently shown by Wang that they are indeed modular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1 Background The Y-system is a system of algebraic relations satisfied by coefficients of a cluster algebra, which has the following form: Yi(u)Yi(u − ri) = � j∈I ri−1 � p=1 Yj(u − p)[nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p]+� 1 + Yj(u − p) �−nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) where I is a finite index set, Yi(u) for i ∈ I, u ∈ Z are commuting variables, ri ∈ Z≥1, and nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We also use the notation [n]+ := max(0, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Such equations are first discovered by Zamolodchikov in the study of thermodynamic Bethe ansats [36], prior to the discovery of cluster algebras by Fomin and Zelevinsky [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The most striking feature of Zamolodchikov’s Y- systems, as well as their generalizations [22, 30] defined shortly after the Zamolodchikov’s work, is that they are periodic, which was fully proved by applying the theory of cluster algebras [9, 10, 15, 16, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' A systematic treatment of the Y-systems in the general setting of cluster algebras, including the Y-systems arising from the thermodynamic Bethe ansatz as spacial cases, was given by Nakanishi [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' This approach was further developed in [24], and it was shown that the algebraic relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) arises from a cluster algebra if and only if the data ri, nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p have a certain symplectic property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' This allows the “axiomatic” study of Y-systems without explicitly referring to cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In this general setting, however, the Y-system is typically not periodic, and so the study of periodic Y-systems as a generalization of Zamolodchikov’s Y-systems would be further developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In particular, the classification problem for periodic Y-systems is a challenging open problem (see the last comments in [27, Section 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' There are several classification results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Fomin and Zelevinsky [10] showed that the classification when ri = 2, nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p ≤ 0, and nii;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p = 0 for any i, j, p coincides with the Cartan-Killing classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Galashin and Pylyavskyy [12] generalized this result to show that the classification when ri = 2 and nii;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p = 0 for any i, p coincides with the classification of ADE bigraphs of Stembridge [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' On the other hand, the situation is more complicated when ri > 2 for some i, and so far there has been no comprehensive classification results except when |I| = 1 1 where it is not difficult to give a complete classification thanks to the work by Fordy and Marsh [11] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' see [24, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In this paper, we make a first attempt to give a classification result involving the case ri > 2 for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Precisely, we classify the periodic Y-systems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) with |I| = 2 satisfying the symplectic property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We would like to emphasize that we consider general ri, nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p in the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The result is given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We also discuss the relation to Nahm’s conjecture on q-series [26, 35] in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2 Main result Let I be a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We denote by Y0 the set of pairs (r, n) where r = (ri)i∈I and n = (nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p)i,j∈I,p∈N are families of integers satisfying ri ≥ 1 for any i and n± ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p = 0 unless 0 < p < ri (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2) for any i, j, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let (r, n) ∈ Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let P be a semifield, and (Yi(u))i∈I,u∈Z be a family of elements in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We say that (Yi(u)) satisfies the Y-system associated with the pair (r, n) if the relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) holds for any i, u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) itself is called the Y-system associated with (r, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We also say that (Yi(u)) is a solution of the Y-system if it satisfies the Y-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' It is useful to think a pair (r, n) ∈ Y0 as a triple of matrices with polynomial entries by the map (r, n) �→ (N0(z), N+(z), N−(z)) : Y0 → (MatI×I N[z])3 defined by N0(z) := diag(1 + zri)i∈I, N±(z) := � � p∈N n± ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='pzp � i,j∈I (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3) where we set n± ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p := [±nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We also define the map (r, n) �→ A±(z) : Y0 → (MatI×I Z[z])2 by A±(z) := N0(z) − N±(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Since this map is injective by the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2), we will identify Y0 with the image of this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For example, we will use the term “the Y-system associated with A±(z) ∈ Y0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We say that A±(z) ∈ Y0 satisfies the symplectic property if A+(z)A−(z−1)T = A−(z)A+(z−1)T, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4) where T is the transpose of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We denote by Y the subset of Y0 consisting of pairs satisfying the symplectic property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The pair A±(z) ∈ Y0 satisfies the simplectic property if and only if the Y-system associated with A±(z) ∈ Y0 is realized as the exchange relations of coefficients in a cluster algebra [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We review this fact in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We say that a solution of a Y-system is periodic if there is a positive integer Ω > 0 such that Yi(u + Ω) = Yi(u) for any i, u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We say that a pair A±(z) ∈ Y is of finite type if any solution (in any semifield) of the Y-system associated with this pair is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In this case, we also say that Y-system itself is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The purpose of this paper is to classify periodic Y-systems of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Before stating the result, we give a few remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We say that A±(z) ∈ YI is decomposable if it is a direct sum of some A′ ±(z) ∈ YI′ and A′′ ±(z) ∈ YI′′ with nonempty I′ and I′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We say that A±(z) ∈ YI is indecomposable if it is not decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' It is enough to consider indecomposable pairs in the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We also note that A±(z) is of finite type if and only if A±(z)op := A∓(z) is of finite type by the correspondence between solutions Yi(u) �→ Yi(u)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The main results are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='summarized as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1 + z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1 + z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='Table 1: Finite type classification for Y-systems of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The numbers h± are the length of reddening sequences in positive and negative directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Suppose that I = {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (1) Any pair A±(z) ∈ Y in Table 1 is of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (2) Any indecomposable pair A±(z) ∈ Y of finite type is reduced to exactly one pair in Table 1 by permuting the indices, changing sign, and changing slices (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1), if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The claim (1) can be proved by concrete calculation in a suitable universal algebra since A±(z) in Table 1 is concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We, however, give another proof involving cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We give a quiver and a sequence of mutations for each A±(z) in Table 1 that yields the Y-system as the exchange relation of coefficients in the cluster algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' See Table 3 for quivers and mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We can verify that some iteration of this sequence of mutations, as well as its inverse, is a reddening sequence (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Thanks to the deep results in the theory of cluster algebras, this property is enough to imply the periodicity (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The number h± in Table 1 are the length of reddening sequences in positive and negative directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' This verification of the periodicity is interesting not only because it is computationally more efficient, but also because it leads to nontrivial dilogarithm identities associated with Donaldson-Thomas invariants (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The claim (2) is proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2 by the following steps: Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We recall the result in [24] that asserts that A±(1) satisfies a certain positivity, which in particular implies that tr A±(1) and det A±(1) are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' This allows us to significantly reduce the candidates for finite type A±(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For a fixed A+(1) in the candidates obtained in Step 1, we search for A−(1) satisfying the symplectic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4) at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' During the search in Step 2, we discard the pair A±(1) that cannot be endowed with the parameter z (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' At this point, we have six candidates up to a permutation of the indices and a change of sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For each A±(1) in the six candidates, we try to endow with the parameter z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' It turns out that this is possible for all the six candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We give all possible A±(z) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 3 Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We finally check that each remaining candidate reduces to one of A±(z) in Table 1 by change of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Most of the Y-systems obtained from Table 1 are already known in the litera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (1)op and (6)op are Zamolodchikov’s Y-system of type A2 [36] and T2 (“tadpole”) [30], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (2)op is the reduced sine-Gordon Y-system associated with the continued fraction 3/4 = [1, 3] = 1/(1+ 1/3), and (5)op with z replaced by z2 is the reduced sine-Gordon Y-system associated with 3/5 = [1, 1, 2] = 1/(1 + 1/(1 + 1/2)) [32] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (4) is the “half” of the Y-system associated with the pair (A2, A2) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (3) appears to be new: Y1(u)Y1(u − 2) = 1 1 + Y2(u − 1)−1 Y2(u)Y2(u − 10) = � 1 + Y1(u − 3) �� 1 + Y1(u − 7) � � 1 + Y1(u − 1)−1�� 1 + Y1(u − 5)−1�� 1 + Y1(u − 9)−1� although it is implicitly given in the author’s previous work [24, Table 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The pair A±(z) ∈ Y is called the T-datum in [24] since it describes the T-systems, which is a companion to the Y-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We do not use this term since we only consider the Y-systems in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Moreover, the definition of the T-datum in [24] allows to have a non- diagonal N0 and have a nontrivial symmetrizer D, which is more general than the definition in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' See also Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4 for the Y-systems involving nontrivial symmetrizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' There is another expression of the Y-system using a pair of matrices A±(z) directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let A±(z) ∈ Y0, and define aij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p ∈ Z by A±(z) = �� p∈N a± ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='pzp � i,j∈I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let (P ± i (u))i∈I,u∈Z be a family of elements in a multiplicative abelian group P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We say that (P ± i (u)) satisfies the multiplicative Y-system associated with A±(z) if � j∈I � p∈N P + j (u − p)a+ ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p = � j∈I � p∈N P − j (u − p)a− ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5) for any i, u (schematically, “A+(z) · log P + = A−(z) · log P −” under the action z : u �→ u − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The solution (P ± i (u)) is called normalized if P is endowed with a semifield structure, and P + i (u) + P − i (u) = 1 for any i, u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We have a one-to-one correspondence between solutions of the Y-system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) and normalized solutions of the multiplicative Y-system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The correspondence is given by Yi(u) �→ P + i (u) P − i (u), P + i (u) �→ Yi(u) 1 + Yi(u), P − i (u) �→ 1 1 + Yi(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In the setting of cluster algebras, this correspondence is nothing but the normalization of the coefficients described by Fomin and Zelevinsky [7, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3 Relation to Nahm sums Consider the q-series defined by G(q) = ∞ � n=0 qn2 (q)n , H(q) = ∞ � n=0 qn2+n (q)n , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='6) 4 where (q)n := (1 − q)(1 − q2) · · · (1 − qn) is the q-Pochhammer symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The famous Rogers- Ramanujan identities express these q-series as the following infinite products: G(q) = � n≡±1 mod 5 1 1 − qn, H(q) = � n≡±2 mod 5 1 1 − qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' These expressions in particular implies that q−1/60G(q) and q11/60H(q) are modular functions on some finite index subgroup of SL(2, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In fact, it is a rare case that an infinite sum of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='6) is modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' It is known that the q-series ∞ � n=0 q 1 2 an2+bn+c (q)n (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7) with a, b, c ∈ Q is modular only if a = 1/2, 1, or 2 [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Nahm [26] considered higher rank generalization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7), which we call the Nahm sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let I be a finite set, and suppose that A ∈ QI×I is a symmetric positive definite matrix, B ∈ QI is a vector, and C ∈ Q is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The Nahm sum is the q-series defined by fA,B,C(q) := � n∈NI q 1 2 nTAn+nTB+C � a(q)ni .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' When |I| ≥ 2, it is not well understood when fA,B,C(q) is modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Nahm gave a conjecture providing a criterion on the modularity of fA,B,C(q) in terms of torsion elements in the Bloch group [26, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' See [3, 33] for the development of this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Nahm used Zamolodchikov’s periodicity to provide an evidence of the conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In fact, there is a natural way to give a candidate of modular Nahm sums from finite type A±(z) ∈ Y in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Precisely, the matrix K := A+(1)−1A−(1) is always symmetric and positive definite for finite type A±(z) ∈ Y, and it is conjectured that it gives a modular Nahm sum fK,0,C(q) for some C [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (This construction is essentially the same as that in [18], except that they did not prove that K is symmetric and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' A special case can also be found in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=') We note that the symplectic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4) at z = 1 plays an important role here since it implies that K is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' On the other hand, the positive definiteness is related to the periodicity of the Y-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Based on our classification, we find that: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Suppose that I = {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The Nahm sum fK,0,C(q) is modular for any finite type A±(z) ∈ Y, where C is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In fact, every K from finite type A±(z) is included in the Zagier’s list [35, Table 2] for rank 2 candidates of modular Nahm sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' There are Rogers-Ramanujan type identities that enable us to write each Nahm sum in the list in terms of theta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The proof of the desired identities was partially given in [1, 4, 6, 33, 35], and was recently completed by Wang [34] except for one candidate that does not appears in our construction from Y-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' See Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We can define the refinement f (s) A±(z)(q) of the Nahm sum fK,0,0(q), which is parametrized by s ∈ H for an abelian group H of order det A+(1) such that it reduces to the original one by taking summation [24, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='12]: fK,0,0(q) = � s∈H f (s) A±(z)(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' It is conjectured that each f (s) A±(z)(q) is already modular after multiplying qC for some C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We note that the symplectic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4) at z = 1 again plays an important role in the definition of the refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We will discuss this refinement for rank 2 case in more detail elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We remark that similar refinement also appears in the context of 3-dimensional quantum topology [13, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 5 A±(z) K −24C RR A±(z) K −24C RR (1) � 4/3 2/3 2/3 4/3 � 4 5 [6] (1)op � 1 −1/2 −1/2 1 � 6 5 [33] (2) �3/2 1 1 2 � 5 7 [34] (2)op � 1 −1/2 −1/2 3/4 � 9 7 [34] (3), (6) �2 2 2 4 � 4 7 [1] (3)op, (6)op � 1 −1/2 −1/2 1/2 � 10 7 [34] (4) � 2/3 1/3 1/3 2/3 � 1 [35] (4)op � 2 −1 −1 2 � 1 [35] (5) �1 1 1 2 � 3 4 [34] (5)op � 2 −1 −1 1 � 5 4 [4] Table 2: The list of the matrix K = A+(1)−1A−(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The Nahm sum fK,0,C(q) is modular, which can be proved by using Rogers-Ramanujan type identities (RR for short) given in the references in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4 Remarks on higher rank and skew-symmetrizable case We have seen that the following properties hold for rank 2 case: (P1) We have reddening sequences in both positive and negative directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (P2) The map A±(z) �→ A+(1)−1A−(1) gives modular Nahm sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We expect that the properties (P1) and (P2) also hold for any finite type A±(z) ∈ Y of general rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The followings are some known examples: For the Y-system associated with the untwisted quantum affine algebras Uq(X(1) r ) with level ℓ restriction [22], (P1) holds with h+ = tℓ and h− = t · (dual Coxeter number of Xr) where t = 1, 2, or 3 is the multiplicity in the Dynkin diagram of Xr [15, 16], and (P2) holds under the assumption [14, Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3] by the result of Kac and Peterson [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For the Y-system associated with a pair of finite type simply laced Dynkin type (Xr, X′ r′) [30], (P1) holds with h+ = (Coxeter number of Xr) and h− = (Coxeter number of X′ r′) [19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For the (reduced) sine-Gordon Y-system associated with the continued fraction p/q = [nF, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' , n1] = 1/(nF + 1/(· · · + 1/n1)) [29, 32], (P1) appears to hold with h+ = 2p and h− = 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For the Y-system associated with an admissible ADE bigraph (Γ, ∆) [12], (P1) appears to hold with h+ = (Coxeter number of Γ) and h− = (Coxeter number of ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Moreover, we can consider Y-systems associated with skew-symmetrizable cluster algebras rather than skew-symmetric ones discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In this case, the symplectic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4) becomes A+(z)DA−(z−1)T = A−(z)DA+(z−1)T, where D is a diagonal matrix called symmetrizer [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We also expect that the properties (P1) and (P2) also hold for skew-symmetrizable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' See [24, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='12] for the definition of the Nahm sum in skew-symmetrizable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' This work is supported by JSPS KAKENHI Grant Number JP21J00050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 6 2 Y-systems and cluster algebras 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1 Preliminaries on cluster algebras In this paper, a semifield is a multiplicative abelian group equipped with an addition that is commutative, associative, and distributive with respect to the multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let I be a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The set of all nonzero rational functions in the variables y = (yi)i∈I with natural number coefficients is a semifield with respect to the usual addition and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' This semifield is called the universal semifield, and denoted by Q>0(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We have a canonical bijection Homsemifield(Q>0(y), P) ∼= Homset(I, P) for any set I and semifield P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let I be a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The tropical semifield Trop(y) is the multiplicative free abelian group generated by the variables y = (yi)i∈I equipped with the addition defined by � i yai i + � i ybi i = � i ymin(ai,bi) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let I be a finite set and P be a semifield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' A Y-seed is a pair (B, y) where B = (Bij)i,j∈I is a skew-symmetric integer matrix and y = (yi)i∈I is a tuple of elements in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We sometimes represent B as the quiver whose signed adjacency matrix is B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For a Y-seed (B, y) and k ∈ I, the mutation in direction k transforms (B, y) into the new Y-seed µk(B, y) = (B′, y′) given by B′ ij := � −Bij if i = k or j = k, Bij + [−Bik]+Bkj + Bik[Bkj]+ otherwise, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) y′ i := � yk if i = k, yiy[Bki]+ k (1 + yk)−Bki otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2) A mutation is involutive, that is, µk(B, y) = (B′, y′) implies (B, y) = µk(B′, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We have the commutativity µiµj = µjµi if Bij = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3) which allows us to write µi for a set i ⊆ I such that Bij = 0 for any i, j ∈ i to mean the successive mutations along arbitrarily chosen order on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For a Y-seed (B, y) and a bijection ν : I → I, we define a new Y-seed ν(B, y) = (B′, y′) by B′ ν(i)ν(j) := Bij and y′ ν(i) := yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2 Solving Y-systems by cluster algebras Let A±(z) ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We will construct a solution of the Y-system associated with A±(z) based on [24, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We first define a subset R ⊆ I × Z by R := {(i, u) ∈ I × Z | 0 ≤ u < ri}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4) and define a skew-symmetric R × R integer matrix B by B(i,p)(j,q) = −nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p−q + nji;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='q−p + � k∈I min(p,q) � v=0 � n+ ik;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p−vn− jk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='q−v − n− ik;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p−vn+ jk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='q−v � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5) where we understand nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p = 0 if p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We then define i := {(i, u) | u = 0} ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We also define a bijection ν : R → R by ν(i, p) = � (i, p − 1) if p > 0, (i, ri) if p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='6) 7 Then the symplectic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4) ensures that ν(µi(B)) = B [24, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We finally define a sequence of Y-seeds · · → (B, y(−1)) → (B, y(0)) → (B, y(1)) → · · · (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7) in Q>0(y) by y(0) := y and (B, y(u + 1)) = ν(µi(B, y(u))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7) gives a solution of the Y-system: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' [24, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='13] (yi,0(u))i∈I,u∈Z satisfies the Y-system associated with A±(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' This solution is universal in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' [24, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='19] Suppose that a family (Yi(u))i∈I,u∈Z satisfies the Y-system associated with A±(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Define a semifield homomorphism f : Q>0(y) → P by f(yi,p) := Yi(p) � j∈I p � q=0 Yj(p − q)−[nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='q]+� 1 + Yj(p − q) �nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='8) Then f(yi,0(u)) = Yi(u) for any i, u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' A±(z) ∈ Y is of finite type if and only if there are different integers u, v such that y(u) = y(v) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3 Periodicity and reddening sequences Similarly to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7), we define a sequence of Y-seeds · · → (B, y(−1)) → (B, y(0)) → (B, y(1)) → · · · (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='9) by the same formulas but now in Trop(y) rather than Q(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We say that the Y-system associated with A±(z) ∈ Y is positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' negative) reddening if there is a positive integer u such that all the exponents in yi(u) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' yi(−u)) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='9) are nonpositive for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We denote by h+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' h−) the least such positive integer u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Equivalently, the Y-system is positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' negative) reddening if and only if all the en- tries in the C-matrix associated with the sequence of mutations (B, y(0)) → (B, y(u)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (B, y(0)) → (B, y(−u))) are nonpositive for some u > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Suppose that the Y-system associated with A±(z) is positive and negative reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Then A±(z) is of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We verify the equivalent condition in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By [2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='10], there are bijections σ, σ′ : R → R such that yi(h+) = y−1 σ(i) and yi(−h−) = y−1 σ′(i) for any i (in other words, the C-matrices associated with them are the minus of permutation matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Now the claim follows from the separation formula for y-variables [10, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='13] and the result on C-matrices shown by Cao, Huang, and Li [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' See also [28, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2] for the corresponding statement dealing with permutations that is actually suitable here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The Y-system associated with each A±(z) in Table 1 is positive and negative reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The quiver B associated with A±(z) is given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We can verify the assertion by concrete calculation on the quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The numbers h± are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 8 Quiver B A±(z) (1, 0) (2, 1) (1, 1) (2, 0) (1) (1, 0) (2, 1) (2, 3) (2, 5) (1, 1) (2, 0) (2, 2) (2, 4) (2) (1, 0) (2, 1) (2, 3) (2, 5) (2, 7) (2, 9) (1, 1) (2, 0) (2, 2) (2, 4) (2, 6) (2, 8) (3) (1, 0) (2, 1) (2, 0) (1, 1) (4) (2, 0) (1, 1) (1, 0) (2, 2) (2, 1) (5) (1, 0) (2, 1) (2, 0) (1, 1) (6) Table 3: Quivers associated with A±(z) in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Each quiver is preserved by the mutation at (∗, 0) followed by the permutation (i, p) �→ (i, p − 1) (the second argument is considered modulo ri), which yields Y-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For (1)–(3), this operation interchanges the connected components (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5 (1) now follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' A connected component of each quiver in Table 3 has the following cluster type: (1) A2 (2) A4 (3) E6 (4) D4 (5) A5 (6) A4 These are of finite type in the sense of [8], which also implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We remark, however, that this observation is somewhat misleading since the quiver associated with a periodic Y-system of general rank is typically of infinite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' It might be better to think that the appearance of only finite type quivers happens “by chance” due to the smallness of 2, the rank of Y-systems considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='8 also gives quantum dilogarithm identifies associated with Donaldson-Thomas invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For any reddening sequence i starting from a quiver B, we can define a quantity E(i) by using the quantum dilogarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We refer to [20, Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='6] as the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' This quantity coincides with Kontsevich-Soibelman’s refined Donaldson-Thomas invariant associated with B [20, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In particular, E(i) does not depend on i, which gives the quantum dilogarithm identifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In our case, we have: 9 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For each A±(z) in Table 1, we have E(µh+) = E(µ−h−), where µ := ν ◦ µi is the sequence of mutations (together with the permutation) (B, y(0)) → (B, y(1)) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For example, the pair (1) in Table 1 yields the famous pentagon identity of the quantum dilogarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 3 Classification 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1 Change of slices We need to introduce an appropriate equivalence relation on the set Y, which identifies essentially the same Y-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Before we get into the definition, we will see a typical example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Consider the following Y-system: Y1(u)Y1(u − 2) = (1 + Y2(u − 1)−1)−1 Y2(u)Y2(u − 2) = (1 + Y1(u − 1)−1)−1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) which corresponds to A±(z) ∈ Y given by (1) in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' This system of equations are defined on the set [1, 2] × Z, but actually can be defined on each component of the following disjoint union: [1, 2] × Z = 1� k=0 {(i, u) | i − u ≡ k mod 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We informally call the algebraic relation defined on each subset the slice of the whole Y-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' If (Yi(u)) is a solution of the Y-system for i − u ≡ 0 mod 2, then (Yi(u + 1)) is a solution of the Y-system for i − u ≡ 1 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Thus it is enough to consider only one slice when considering solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Now we consider another Y-system: Y ′ 1(u)Y ′ 1(u − 3) = (1 + Y ′ 2(u − 2)−1)−1 Y ′ 2(u)Y ′ 2(u − 3) = (1 + Y ′ 1(u − 1)−1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2) which corresponds to A′ ±(z) ∈ Y given by A′ +(z) := � 1 + z3 −z2 −z 1 + z3 � , A′ −(z) := � 1 + z3 0 0 1 + z3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The Y-system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2) is decomposed into three slices: [1, 2] × Z = 2� k=0 {(i, u) | i − u ≡ k mod 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We see that for any solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) for i − u ≡ 0 mod 2, Y ′ 1(u) := Y1 �2 3u − 1 3 � , Y ′ 2(u) := Y2 �2 3u � is a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2) for i − u ≡ 2 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We also obtain solutions for the other two slices by shifting u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Conversely, any solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) is obtained from a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Therefore, it 10 is enough to consider one of the Y-systems (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In particular, A±(z) is of finite type if and only if A′ ±(z) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Now we work in the general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The idea is that each slice corresponds to each connected component of the quiver associated with the matrix B defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let A±(z) ∈ Y, and assume that it is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By [24, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='24], we have a decomposition of the matrix B and its index set R: B = t−1 � u=0 B(u), R = t−1 � u=0 R(u) such that each B(u) is indecomposable and we have a cyclic sequence of mutations B(0) ν|R(0)◦µi(0) −−−−−−−→ B(1) −→ · · · −→ B(t − 1) ν|R(t−1)◦µi(t−1) −−−−−−−−−−→ B(0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3) where i(u) := i ∩ R(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We say that two pairs A±(z) and A′ ±(z) are related by change of slices if they yield the same cyclic sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3) up to a change of indices and the commutativity of mutations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (This commutativity is already implicitly used to justify the notation µi(u) as stated below (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=') Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The pairs A±(z) and A′ ±(z) associated with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2), respectively, are related by change of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Indeed, we see that the sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3) for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='1) is (1, 0) (2, 1) ν◦µ(0,0) −−−−−→ (1, 1) (2, 0) ν◦µ(1,0) −−−−−→ (1, 0) (2, 1) , whereas the sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3) for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2) is (1, 0) (2, 1) ν′◦µ(0,0) −−−−−→ (1, 2) (2, 0) ν′◦µ(1,0) −−−−−→ (1, 1) (2, 2) ν′ −→ (1, 0) (2, 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' These are the same sequence up to a change of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2 Proof of the classification In this section, we will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We first recall the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2 ([24, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let A±(z) ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Assume that A±(z) is of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Then there is a vector v ∈ RI such that v > 0, vA+(1) > 0, and vA−(1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In particular, tr A±(1) > 0 and det A±(1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2, A+(1) and A−(1) are equal to one of the following matrices: � 2 −1 −1 2 � , � 2 −1 −2 2 � , � 2 −1 −3 2 � , � 2 −1 −1 1 � , � 2 0 −n 2 � , � 2 0 −n 1 � , � 1 0 −n 1 � up to a permutation of the indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We give several lemmas about impossible pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Before giving lemmas, we note that n+ ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p = 0 or n− ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='p = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4) for any i, j, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' It is impossible that A±(z) ∈ Y has the following forms: (1) A+(1) = �2 −a ∗ ∗ � , A−(1) = �2 −b ∗ ∗ � for odd a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 11 (2) A+(1) = �2 −a ∗ ∗ � , A−(1) = �1 −b ∗ ∗ � for odd a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (3) A+(1) = �1 −1 ∗ ∗ � , A−(1) = �1 −1 ∗ ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (4) A+(1) = �1 0 ∗ ∗ � , A−(1) = �1 ∗ ∗ ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For (2), we can set A+(z) = �1 + zr −f(z) ∗ ∗ � , A−(z) = �1 + zr − za −g(z) ∗ ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By the symplectic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4), we have za + za−r + f(z)g(z−1) = z−a + zr−a + g(z)f(z−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5) Since 0 < a and a − r < 0 by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2), the sum of the coefficients of the terms in f(z)g(z−1) with positive exponents is equal to that with negative exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Since f(1)g(1) (=ab) is odd, f(z)g(z−1) should contain the constant term z0, which contradicts (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The proof for (1) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For (3), we can set A+(z) = � 1 + zr − za −zb ∗ ∗ � , A−(z) = � 1 + zr − zc −zd ∗ ∗ � with 0 < a, b, c, d < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Without loss of generality, we can assume a < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4), we have z−c + zr−c + za + za−r + zc−a + zd−b = zc + zc−r + z−a + zr−a + za−c + zb−d Since c − a > 0, we see that c − a is equal to c, r − a, or b − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' However, the first two cases are impossible by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Thus c − a = b − d, which implies that z−c + zr−c + za + za−r = zc + zc−r + z−a + zr−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Since a > 0, we see that a is equal to c or r − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' However, a = c is impossible by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Thus a = r − a, which implies that z−c + zr−c = zc + zc−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Since c > 0, we see that c = r − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' However, this implies that a = r/2 = c, which is impossible by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' For (4), we can set A+(z) = �1 + zr − za 0 ∗ ∗ � , A−(z) = � 1 + zr − zb ∗ ∗ ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4), we have za + za−r + z−b + zr−b + zb−a = z−a + zr−a + zb + zb−r + za−b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Comparing the number of the terms with positive and negative exponents, we should have a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' This is impossible by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 12 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' It is impossible that indecomposable A±(z) ∈ Y has the form A+(1) = �∗ 0 ∗ ∗ � , A−(1) = �∗ 0 ∗ ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We can set A±(z) = � 1 + zr1 − f±(z) 0 −g±(z) 1 + zr2 − h±(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Since g+(z) ̸= 0 or g−(z) ̸= 0, we can pick the least integer c among the exponents in g+(z) and g−(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Without loss of generality, we can assume g+(1) contains the term zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4), we have f+(z)g−(z−1) + (1 + zr1)g+(z−1) = f−(z)g+(z−1) + (1 + zr1)g−(z−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='6) The left-hand side in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='6) contains the term zr1−c, but any exponent in the right-hand side is strictly smaller that r1 − c by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4), which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We now search for possible pairs A±(1) case by case using the symplectic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4) at z = 1 together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4: Case: A+(1) = � 2 −1 −1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The possibilities for A−(1) are: �2 0 0 2 � , �1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Case: A+(1) = � 2 −1 −1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The possibilities for A−(1) are: �2 0 0 2 � , �1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Case: A+(1) = � 2 −1 −2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The possibilities for A−(1) are: � 2 0 −1 2 � , �1 0 0 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Case: A+(1) = � 2 −1 −3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The possibilities for A−(1) are: � 2 0 −2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Case: A+(1) = � 2 −1 −1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The possibilities for A−(1) are: �2 0 0 2 � , �1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 13 Case: A+(1) = � 2 0 −n 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The possibilities for A±(1) are: ��2 0 0 2 � , � 2 −1 −1 2 �� , �� 2 0 −1 2 � , � 2 −1 −2 2 �� , ��2 0 0 2 � , � 1 −1 −1 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Case: A+(1) = � 2 0 −n 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The possibilities for A±(1) are: ��2 0 0 1 � , � 2 −2 −1 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Case: A+(1) = � 1 0 −n 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The possibilities for A±(1) are: ��2 0 0 1 � , � 2 −2 −1 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' In summary, the remaining possible pairs, up to a permutation of the indices and an change of sign, are given in the following table: A+(1) A−(1) � 2 −1 −1 2 � �2 0 0 2 � � 2 −1 −2 2 � � 2 0 −1 2 � � 2 −1 −3 2 � � 2 0 −2 2 � A+(1) A−(1) � 2 −1 −1 2 � �2 0 0 2 � � 2 −1 −2 2 � � 2 0 −1 2 � � 2 −1 −3 2 � � 2 0 −2 2 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7) We now start searching for possible A±(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Suppose that A+(1) = � 2 −1 −n 2 � , A−(1) = � 2 0 −(n − 1) 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Then A+(z) = � [2]r −z−a −zr−a[n]2r [2](2n−1)r � , A−(z) = � [2]r 0 −z2r−a[n − 1]2r [2](2n−1)r � for some r, a, where [n]r is the z-integer defined by [n]r := 1 − zrn 1 − zr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We can set A+(z) = � [2]r1 −za − �n i=1 zbi [2]r2 � , A−(z) = � [2]r1 0 − �n−1 i=1 zci [2]r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 14 Without loss of generality, we can assume that b1 ≤ b2 ≤ · · · ≤ bn, c1 ≤ c2 ≤ · · · ≤ cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By the symplectic property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4), we have n−1 � i=1 (z−ci + zr1−ci) + za + za−r2 = n � i=1 (z−bi + zr1−bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Comparing the degree by using the conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4), we obtain the system of linear equations a = r1 − b1, a − r2 = −bn, r1 = ci − bi = bi+1 − ci (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' , n − 1), which implies that r2 = (2n − 1)r1, bi = (2i − 1)r1 − a, ci = 2ir1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Suppose that A+(1) = � 2 −1 −1 2 � , A−(1) = �1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Then A+(z) = � 1 + z2r −za −z2r−a 1 + z2r � , A−(z) = � 1 + z2r − zr 0 0 1 + z2r − zr � for some r, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We can set A+(z) = � 1 + zr1 −za −zb 1 + zr2 � , A−(z) = � 1 + zr1 − zc 0 0 1 + zr2 − zd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4), we have r1 = r2 = a + b = 2c = 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Suppose that A+(1) = � 2 −1 −2 2 � , A−(1) = �1 0 0 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Then A+(z) = � 1 + z2r −za −z2r−a − z3r−a 1 + z3r � , A−(z) = � 1 + z2r − zr 0 0 1 + z2r � for some r, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We can set A+(z) = � 1 + zr1 −za −zb1 − zb2 1 + zr2 � , A−(z) = �1 + zr1 − zc 0 0 1 + zr2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Without loss of generality, we can assume b1 ≤ b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4), we have r1 = 2c, r2 = 3c, b1 = 2c−a, and b2 = 3c − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' 15 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Suppose that A+(1) = � 2 −1 −1 1 � , A−(1) = �2 0 0 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Then A+(z) = � 1 + z2r −za −z2r−a 1 + z2r � , A−(z) = � 1 + z2r 0 0 1 + z2r � for some r, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We can set A+(z) = � 1 + zr1 −za −zb 1 + zr2 � , A−(z) = �1 + zr1 0 0 1 + zr2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='4), we have r1 = r2 = a + b = 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' The remaining possibilities for finite type A±(z) ∈ Y, up to a per- mutation of indices and change of sign, are the six families of the pairs given in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='5–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='8, which contain the parameters r, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' We can verify that these six families belong to Y, and they can be reduced to the pairs in Table 1 by change of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' References [1] George E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Algebra, 220(5):1752–1784, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' [5] Peigen Cao, Min Huang, and Fang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' A conjecture on C-matrices of cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Nagoya Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=', 238:37–46, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' [6] Ivan Cherednik and Boris Feigin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Rogers-Ramanujan type identities and Nil-DAHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=', 248:1050– 1088, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' [7] Sergey Fomin and Andrei Zelevinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=', 15(2):497–529, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' [8] Sergey Fomin and Andrei Zelevinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Finite type classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content=' Invent.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='chiba-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} +page_content='jp 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQfEzWQ/content/2301.13239v1.pdf'} diff --git a/2dE0T4oBgHgl3EQfdwCH/content/tmp_files/2301.02381v1.pdf.txt b/2dE0T4oBgHgl3EQfdwCH/content/tmp_files/2301.02381v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f8571139e82035e1885ab1ab3dbbd531ad140dc6 --- /dev/null +++ b/2dE0T4oBgHgl3EQfdwCH/content/tmp_files/2301.02381v1.pdf.txt @@ -0,0 +1,2864 @@ +arXiv:2301.02381v1 [math.NT] 6 Jan 2023 +Existence of primitive pairs with two +prescribed traces over finite fields +Aakash Choudhary∗and R. K. Sharma † +Department of Mathematics, +Indian Institute of Technology Delhi-110016, India +Abstract +Given F = Fpt, a field with pt elements, where p is a prime power, +t ≥ 7, n are positive integers and f = f1/f2 is a rational func- +tion, where f1, f2 are relatively prime, irreducible polynomials with +deg(f1) + deg(f2) = n in F[x]. We construct a sufficient condition on +(p, t) which guarantees primitive pairing (ǫ, f(ǫ)) exists in F such that +TrFpt/Fp(ǫ) = a and TrFpt/Fp(f(ǫ)) = b for any prescribed a, b ∈ Fp. +Further, we demonstrate for any positive integer n, such a pair defi- +nitely exists for large t. The scenario when n = 2 is handled separately +and we verified that such a pair exists for all (p, t) except from possible +71 values of p. A result for the case n = 3 is given as well. +Keywords: Character, Finite fields, Primitive elements. +2020 Mathematics Subject Classification: 12E20, 11T23 +1 +Introduction +Let Fp represent a field of finite order p, where p = qr for some prime q and +r, a positive integer. The multiplicative group of Fp is cyclic, it is denoted +by F∗ +p and a generator of F∗ +p is referred to as a primitive element in Fp. +∗email: achoudhary1396@gmail.com +†email: rksharmaiitd@gmail.com +1 + +The field Fp has φ(p − 1) primitive elements, where φ is the Euler’s totient +function. Let Fpt denote an extension of Fp of degree t for some positive +integer t. A necessary and sufficient condition for an element ǫ ∈ F∗ +pt to be +primitive is that it is a root of an irreducible polynomial of degree t over Fp +and such an irreducible polynomial is referred to as primitive polynomial. +For ǫ ∈ Fpt, the trace of ǫ over Fp denoted by TrFpt/Fp(ǫ), is defined as +TrFpt/Fp(ǫ) = ǫ + ǫp + ǫp2 + · · · + ǫpt−1. +In Cryptographic schemes such as Elgamel encryption scheme and the +Diffie-Hellman key exchange, primitive elements serve as the fundamental +building blocks. Numerous applications of primitive elements can be found +in Coding theory and Cryptography [10], making the study of primitive el- +ements and primitive polynomials an active research field. Please refer to +[9] for more information about the existence of primitive elements in finite +fields. For any rational function f(x) ∈ Fp(x) and ǫ ∈ Fp we call the pair +(ǫ, f(ǫ)), a primitive pair if both ǫ and f(ǫ) are primitive elements in Fp. +In general, if ǫ is primitive, f(ǫ) need not be primitive. For instance, take +x2 + 3x + 2 ∈ F7[x], then 3, 5 are primitive elements in F7 but none of f(3) +and f(5) are. +In 1985, Cohen [3] introduced the term ”primitive pair” and he verified +the existence of primitive pairs (ǫ, f(ǫ)) in Fp for linear polynomials f(x) = +x + k ∈ Fp[x]. Since then many researchers have conducted studies in this +area [12, 13, 7, 14]. +Most recently, Cohen, Sharma and Sharma [4] have +supplied a condition that ensures the occurrence of primitive pair (ǫ, f(ǫ)) in +Fp for non-exceptional rational function f, i.e., f is not of the form cxjgk(x), +where j ∈ Z, k > 1 that divides p − 1 and c ∈ F∗ +p, for any g(x) ∈ Fp(x). +Jungnickel, Vanstone [8] identified a sufficient condition for the occurrence +of primitive elements ǫ ∈ Fpt with a prescribed trace of ǫ. Later Cohen [5] +extended the result with some exceptions. +Chou and Cohen [2], in 2014, +addressed the issue of the existence of primitive element ǫ ∈ Fpt such that +TrFpt/Fp(ǫ) = TrFpt/Fp(ǫ−1) = 0. Cao and Wang [1], for t ≥ 29, established +a condition for the existence of primitive pair (ǫ, f(ǫ)) with f(x) = x2+1 +x +∈ +Fpt(x) such that for prescribed a, b ∈ F∗ +p, TrFpt/Fp(ǫ) = a and TrFpt/Fp(ǫ−1) = +b. In 2018, Gupta, Sharma and Cohen [7], for the same rational function +and prescribed a ∈ Fp, presented a condition that ensures the existence +of primitive pair (ǫ, f(ǫ)) in Fpt with TrFpt/Fp(ǫ) = a for t ≥ 5. Then in +2019, Gupta and Sharma [14] extended the result to the rational function +2 + +ΓM(x) = +a11x2+a12x+a13 +a22x+a23 +, where M = +� +a11 +a12 +a13 +0 +a22 +a23 +� +∈ M2×3(Fpt) is any +matrix of rank 2, and if ΓM(x) = λx or λx2 for some λ ∈ Fpt, then λ = 1. +In 2021, Sharma and Sharma [11] examined the rational function f = f1/f2 +in Fpt(x), where f1 and f2 are relatively prime, irreducible polynomials and +proved that for prescribed a, b ∈ Fp, the existence of primitive pair (ǫ, f(ǫ)) +in Fpt such that TrFpt/Fp(ǫ) = a and TrFpt/Fp(ǫ−1) = b for t ≥ 7. +Prior to this article, for primitive pairs, traces were considered for ǫ and +ǫ−1. In this article, we will consider the trace onto the element ǫ and its +image under f, i.e, f(ǫ). Some terminology and conventions are introduced +for explanation. We say that a non-zero polynomial f over Fp[x] has degree +k ≥ 0, if f(x) = akxk + ak−1xk−1 + · · ·+ a1x + a0, where ak ̸= 0 and we write +the degree of f as deg(f) = k. Next, we suppose that, for a rational function +f(x) = f1(x) +f2(x) ∈ Fp(x), f1 and f2 are relatively prime, irreducible polynomials +and define the degree-sum as degsum(f) = deg(f1) + deg(f2). We will now +define various sets that will play a crucial role in this article. +1. We define Rp,t(n1, n2) to represent the set of all rational function f(x) = +f1(x) +f2(x) ∈ Fpt(x) such that f1 and f2 are relatively prime, irreducible +polynomials over Fpt with deg(f1) = n1 and deg(f2) = n2. +2. Denote An1,n2 as the set consisting of pairs (p, t) ∈ N × N such that +for any f ∈ Rp,t(n1, n2) and prescribed a, b ∈ Fp, Fpt contains an ele- +ment ǫ such that (ǫ, f(ǫ)) is a primitive pair with TrFpt/Fp(ǫ) = a and +TrFpt/Fp(f(ǫ)) = b. +3. Define, Rp,t(n) = � +n1+n2=n Rp,t(n1, n2) and An = � +n1+n2=n An1,n2. +First, in this paper, for n ∈ N, we consider f(x) ∈ Rp,t(n) and a, b ∈ Fp, +and then verify that there exists an element ǫ ∈ Fpt such that (ǫ, f(ǫ)) is +a primitive pair in Fpt with TrFpt/Fp(ǫ) = a and TrFpt/Fp(f(ǫ)) = b, i.e., we +provide a sufficient condition on pt such that (p, t) ∈ An. Furthermore, using +a sieve variation of this sufficient condition, we prove the following result: +Theorem 1.1. Let t, q, r, p ∈ N be such that q is a prime number, t ≥ 7 +and p = qr. Suppose p and t assumes none of the following values: +1. 2 ≤ p ≤ 16 or p = 19, 23, 25, 27, 31, 37, 43, 49, 61, 67, 79 and t = 7; +3 + +2. 2 ≤ p ≤ 31 or p = 32, 37, 41, 43, 47, 83 and t = 8; +3. 2 ≤ p ≤ 8 or p = 11, 16 and t = 9; +4. p = 2, 3, 4, 5, 7 and t = 10, 12; +5. p = 2, 3, 4 and t = 11; +6. p = 2 and t = 14, 15, 16, 18, 20, 24. +Then (p, t) ∈ A2. +Note:- The exceptions in above theorem need not be true exceptions, they +are possible exceptions. +SageMath [16] is used to perform all nontrivial calculations required +throughout this article. +2 +Preliminaries +In this section, we present some basic concepts, notations, and results that +will be used in forthcoming sections of this article. Throughout the article, +t is a positive integer, p is an arbitrary prime power and Fp is a finite field +of order p. +2.1 +Definitions +1. A character of a finite abelian group G is a homomorphism χ from the +set G into Z1, where Z1 is the set of all elements of complex field C with +absolute value 1. The trivial character of G denoted by χ0, is defined +as χ0(g) = 1 for all g ∈ G. In addition, the set of all characters of G, +denoted by ˆG, forms a group under multiplication, which is isomorphic +to G. The order of a character χ is the least positive integer d such +that χd = χ0. For a finite field Fpt, a character of the additive group +Fpt is called an additive character and that of the multiplicative group +F∗ +pt is called a multiplicative character. +2. For u, a divisor of pt−1, an element ζ ∈ F∗ +pt is called u-free, if whenever +ζ = ξs, where ξ ∈ Fpt and s|u implies s = 1. We see that an element +ζ ∈ F∗ +pt is (pt − 1)-free if and only if it is a primitive element of Fpt. +4 + +For more information on characters, primitive elements and finite fields, we +refer the reader to [9]. +The following conclusion holds as a particular case of [15, Lemma 10]: +Lemma 2.1. Let u be a divisor of pt − 1, ζ ∈ F∗ +pt, then we have: +� +s|u +µ(s) +φ(s) +� +χs +χs(ζ) = +� +u +φ(u) +if ζ is u − free, +0 +otherwise +where µ(.) is the Mobius function and φ(.) is the Euler function, χs runs +through all the φ(s) multiplicative characters over F∗ +pt with order s. +Therefore for u, a divisor of pt − 1 +ρu : ǫ �→ θ(u) +� +s|u +µ(s) +φ(s) +� +χs +χs(ǫ) +(1) +gives a characteristic function for the subset of u-free elements of F∗ +pt, where +θ(u) = φ(u)/u. +Also for a ∈ Fp, +τa : ǫ �→ 1 +p +� +ψ∈ ˆ +Fp +ψ(TrFpt/Fp(ǫ) − a) +(2) +is a characteristic function for the subset of Fpt whose elements satisfy +TrFpt/Fp(ǫ) = a. From [9, Theorem 5.7], any additive character ψ of Fp can +be derived by ψ(a) = ψ0(ua), where ψ0 is the canonical additive character of +Fp and u is an element of Fp corresponding to ψ. Thus +τa = 1 +p +� +ψ∈ ˆ +Fp +ψ0(TrFpt/Fp(uǫ) − ua) += 1 +p +� +u∈Fp +ˆψ0(uǫ)ψ0(−ua) +(3) +where ˆψ0 is the additive character of Fpt defined by ˆψ0(ǫ) = ψ0(TrFpt/Fp(ǫ)). +In next theorem, we will make major use of the results given below by Wang +and Fu [6] in 2014. +5 + +Lemma 2.2. [6, Theorem 5.5] Let F(x) ∈ Fpd(x) be a rational function. +Write F(x) = �k +j=1 fj(x)rj, where fj(x) ∈ Fpd[x] are irreducible polynomials +and rj are non zero integers. +Let χ be a multiplicative character of Fpd. +Suppose that the rational function �d−1 +i=1 f(xpi) is not of the form h(x)ord(χ) ∈ +Fpd(x), where ord(χ) is the order of χ, Then we have +���� +� +ǫ∈Fp,f(ǫ)̸=0,∞ +χ(F(ǫ)) +���� ≤ +� +d +k +� +j=1 +deg(fj) − 1 +� +p +1 +2. +Lemma 2.3. [6, Theorem 5.6] Let f(x), g(x) ∈ Fpt(x) be rational functions. +Write f(x) = �k +j=1 fj(x)rj, where fj(x) ∈ Fpt[x] are irreducible polynomials +and rj are non-zero integers. Let D1 = �k +j=1 deg(fj), D2 = max{deg(g), 0}, +D3 is the degree of denominator of g(x) and D4 is the sum of degrees of those +irreducible polynomials dividing denominator of g but distinct from fj(x)( j= +1,2,...,k). Let χ be a multiplicative character of Fpt, and let ψ be a nontrivial +additive character of Fpt. Suppose g(x) is not of the form v(x)pt − v(x) in +Fpt(x). Then we have +���� +� +ǫ∈Fpt,f(ǫ)̸=0,∞,g(ǫ)̸=∞ +χ(f(ǫ))ψ(g(ǫ)) +���� ≤ (D1 + D2 + D3 + D4 − 1)p +t +2. +Evidently, both the sufficient condition (Theorem 3.1) and its sieving +variation (Theorem 3.4) are entirely dependent on pt and the degrees of the +numerator and denominator polynomials of the rational function. It is easy +to see that the Trace part of the main result in [11] is a special case of our +finding for f(x) = 1 +x. +For every κ ∈ N, we will use ω(κ) to represent the number of distinct +prime divisors of κ, and W(κ) to represent the number of square free divisors +of κ. Clearly, W(κ) = 2ω(κ). +3 +Sufficient Condition +Let k1, k2, p, t ∈ N be such that p is a prime power and k1, k2 are positive +integers which divide pt − 1. Let a, b ∈ Fp, f(x) ∈ Rp,t(n). Let Af,a,b(k1, k2) +represents the set consisting of all those elements ǫ ∈ Fpt such that ǫ is k1-free, +f(ǫ) is k2-free, TrFpt/Fp(ǫ) = a, and TrFpt/Ft(f(ǫ)) = b. +We now verify the sufficient condition as follows: +6 + +Theorem 3.1. Suppose t, n, p ∈ N and p is a prime power. Suppose that +p +t +2−2 > (2n + 1)W(pt − 1)2. +Then (p, t) ∈ An. +Proof. In order to prove this result it suffices to demonstrate that Af,a,b(k1, k2) > +0 for every f(x) ∈ Rp,t(n) and for every prescribed a, b ∈ Fp . Suppose that +f(x) ∈ Rp,t(n) be a rational function and that a, b ∈ Fp. Let P represent the +collection of zeroes and poles of f(x) ∈ Fpt and P +′ = P ∪ {0}. Let k1, k2 be +divisors of pt − 1. Then by definition, Af,a,b(k1, k2) will be given by +Af,a,b(k1, k2) = +� +ǫ∈Fpt−P ′ +ρk1(ǫ)ρk2(f(ǫ))τa(ǫ)τb(f(ǫ)). +Using the characteristic functions (1) and (3) defined in the previous section, +we obtain +Af,a,b(k1, k2) = θ(k1)θ(k2) +p2 +� +s1|k1,s2|k2 +µ(s1)µ(s2) +φ(s1)φ(s2) +� +s1,s2 +χf,a,b(s1, s2) +where θ(ki) = φ(ki) +ki +; i = 1, 2 and +χf,a,b(s1, s2) = +� +u,v∈Fp +ψ0(−au − bv) +� +ǫ∈Fpt−P ′ +χs1(ǫ)χs2(ǫ0) ˆψ0(uǫ + vǫ0) +where ǫ0 = f(ǫ). It follows from [9, Example 5.1] that, for any divisors s1, s2 +of pt − 1, there exist integers m1, m2 with 0 < m1, m2 < pt − 1 such that +χs1(x) = χpt−1(xm1) and χs2(x) = χpt−1(xm2). Thus +χf,a,b(s1, s2) = +� +u,v∈Fp +ψ0(−au − bv) +� +ǫ∈Fpt−P ′ +χpt−1(ǫm1f(ǫ)m2) ˆψ0(uǫ + vǫ0) +(4) += +� +u,v∈Fp +ψ0(−au − bv) +� +ǫ∈Fpt−P ′ +χpt−1(F1(ǫ)) ˆψ0(F2(ǫ)), +(5) +where F1(x) = xm1f(x)m2 ∈ Fpt(x) and F2(x) = ux + vf(x) ∈ Fpt(x). +7 + +First we consider the situation when F2(x) = l(x)pt −l(x) for some l(x) ∈ +Fpt(x), where l(x) = l1(x) +l2(x) with (l1, l2) = 1. We have, ux+vf1(x) +f2(x) = l1(x)pt +l2(x)pt − +l1(x) +l2(x), that is, +f2(x)(l1(x)pt − l1(x)l2(x)pt−1) = l2(x)pt(uxf2(x) + vf1(x)). +Since (l1(x)pt − l1(x)l2(x)pt−1, l2(x)pt) = 1, it implies that, l2(x)pt divides +f2(x), which can only happen if l2(x) is constant. That is, we have +c−(pt)f2(x)(l1(x)pt − l1(x)cpt−1) = uxf2(x) + vf1(x) +where c = l2. Now, the above equation only applies if v = 0. Substituting +it to the equation above yields, c−(pt)(l1(x)pt − l1(x)cpt−1) = ux, which can +happen only if l1 is constant and u = 0. Moreover, if F1(x) ̸= r(x)pt−1 for +any r(x) ∈ Fpt(x), then it follows form Lemma 2.2 that +|χf,a,b(s1, s2)| ≤ np +t +2 +2. +(6) +And, when F1(x) = r(x)pt−1 for some r(x) ∈ Fpt(x), where r(x) = r1(x) +r2(x) is +such that (r1, r2) = 1. Following [11], it happens only if m1 = m2 = 0, a +contradiction. +If F2(x) ̸= d(x)pt − d(x) for any d(x) ∈ Fpt(x) then, +Case 1 : When n1 ≤ n2. Then in accordance with Lemma 2.3 we have D2 += 1, and +|χf,a,b(s1, s2)| ≤ (2n + 1)p +t +2 +2. +(7) +Case 2 : When n1 > n2. We have D2 = n1 − n2 and +|χf,a,b(s1, s2)| ≤ 2np +t +2+2. +(8) +Thus, if (χs1, χs2, u, v) ̸= (χ1, χ1, 0, 0) then based on the discussion above, +and using (6), (7) and (8), we get, |χf,a,b(s1, s2)| ≤ (2n + 1)p +t +2+2. From this +and the definition of Af,a,b(k1, k2), we get +Af,a,b(k1, k2) ≥ θ(k1)θ(k2) +p2 +((pt − |P +′|) − (2n + 1)p +t +2+2(W(k1)W(k2) − 1)) +(9) +≥ θ(k1)θ(k2) +p2 +((pt − (n + 1)) − (2n + 1)p +t +2+2(W(k1)W(k2) − 1)) +(10) +8 + +Therefore, if p +t +2−2 > (2n + 1)W(k1)W(k2), then Af,a,b(k1, k2) > 0 for every +f(x) ∈ Rp,t(n) and prescribed a, b ∈ Fp. Considering k1 = k2 = pt − 1, result +follows. +Now, we provide the bounds for the absolute values for Af,a,b(mk, k) − +θ(m)Af,a,b(k, k) and Af,a,b(k, mk) − θ(m)Af,a,b(k, k). Proofs are omitted as +they follow from the idea of [7]. +Lemma 3.2. Let k be a positive integer that divides pt −1 and m is a prime +dividing pt − 1 but not k. Then +|Af,a,b(mk, k) − θ(m)Af,a,b(k, k)| ≤ θ(k)2θ(m) +p2 +(2n + 1)W(k)2p +t +2 +2 +and +|Af,a,b(k, mk) − θ(m)Af,a,b(k, k)| ≤ θ(k)2θ(m) +p2 +(2n + 1)W(k)2p +t +2+2. +Lemma 3.3. Let k be a positive integer that divides pt−1 and {q1, q2, . . . , qm} +be the collection of all primes dividing pt − 1 but not k. Then +Af,a,b(pt−1, pt−1) ≥ +m +� +i=1 +Af,a,b(k, qik)+ +m +� +i=1 +Af,a,b(qik, k)−(2m−1)Af,a,b,(k, k). +Sieve variation of sufficient condition (Theorem 3.1) is given below, proof +of which is not given as it follows from Lemmas 3.2, 3.3 and ideas in [7]. +Theorem 3.4. Let t, n, p, k ∈ N be such that k divides pt − 1, where p is +a prime power. Assume {q1, q2, . . . , qm} is the collection of all those primes +that divide pt −1 but not k. Suppose δ = 1 −2 �m +i=1 +1 +qi +and ∆ = 2m − 1 +δ ++ 2. +If δ > 0 and +p +t +2 −2 > (2n + 1)∆W(k)2 +then (p, t) ∈ An. +Lemma 3.5. Suppose that κ ∈ N is such that ω(κ) ≥ 1547, then W(κ) ≤ +κ1/12. +9 + +Proof. Let V = {2, 3, 5, . . ., 12983} is the set of first 1547 primes. We see +that the product of all elements of V exceeds K = 6.57 × 105588. Let κ = +κ1κ2, where κ1 and κ2 are co-prime integers such that all prime divisors +of κ1 come from the least 1547 prime divisors of κ and remaining prime +divisors are divisors of κ2. Hence, κ1/12 +1 +> K1/12 > 5.42 × 10465, whereas +W(κ1) < 4.93 × 10465. The conclusion follows, since ρ1/12 > 2 for all primes +ρ > 12983. +We shall need Theorem 3.4 and Lemma 3.5 for calculation work in the +next section. +4 +Proof of Theorem 1.1 +Proof will be carried out for the situation t ≥ 7, since according to [2] +there is no primitive element ǫ, for t ≤ 4, such that TrFpt/Fp(ǫ) = 0 and +TrFpt/Fp(ǫ−1) = 0. The cases t = 5 and 6 necessitate substantial computation +and appear to demand a different technique. As a result, we postpone further +examination of these situations. +We assume initially that, ω(pt − 1) ≥ 1547. +Using Theorem 3.1 and +Lemma 3.5, if p +t +2 −2 > 5p +t +6, that is, if pt > 5 +3t +t−6 then (p, t) ∈ A2. +But +t ≥ 7 gives +3t +t−6 ≤ 21. Hence, if pt > 521 then (p, t) ∈ A2, and this holds +true for ω(pt − 1) ≥ 1547. Therefore, we may suppose ω(pt − 1) ≤ 1546. +We shall use sieve variation in order to carry forward computational work. +Let 62 ≤ ω(pt − 1) ≤ 1546. +To use Theorem 3.4 assume k to be the +product of least 62 primes that divide pt − 1, that is, W(k) = 262, then +m ≤ 1485 and δ assumes its least positive value when {q1, q2, . . . , q1485} = +{307, 311, 313, . . ., 12979}. This yields δ > 0.004174 and ∆ < 710770.7395. +Hence 5∆W(k)2 < 7.558211 × 1043. Let Z = 7.558211 × 1043. By sieve +variation, (p, t) ∈ A2 if q +t +2 −2 > Z i.e., if pt > Z +2t +t−4. Since t ≥ 7, it gives +2t +t−4 ≤ 14 +3 . Therefore, (p, t) ∈ A2 under the condition that pt > 5.834 × 10204. +Hence, ω(pt − 1) ≥ 95 implies (p, t) ∈ A2. In a similar manner (p, t) ∈ A3, +A4 if ω(pt − 1) ≥ 95, and (p, t) ∈ A5 if ω(pt − 1) ≥ 96. +10 + +Table 1. +Sr.No. +a ≤ ω(pt − 1) ≤ b +W(k) +δ > +∆ < +5∆W(k)2 < +1 +a = 13, b = 94 +213 +0.04481712 +3594.3767988 +1,206,072,718,756 +2 +a = 7, b = 34 +27 +0.04609692 +1151.7513186 +94,351,469 +3 +a = 6, b = 25 +26 +0.08241088 +450.9698124 +9,235,862 +4 +a = 6, b = 23 +26 +0.12550135 +264.9453729 +5,426,082 +5 +a = 6, b = 22 +26 +0.14959773 +209.2223842 +4,284,875 +6 +a = 5, b = 19 +25 +0.07663431 +354.3225878 +1,814,132 +7 +a = 5, b = 17 +25 +0.13927194 +167.1445296 +855,780 +8 +a = 5, b = 16 +25 +0.17317025 +123.2679422 +631,132 +9 +a = 5, b = 15 +25 +0.21090610 +92.0874844 +471,488 +Using the values in the Table 1 above and repeating the process of sieve +variation, we determine that (p, t) ∈ A2 if pt > (4284875) +14 +3 or pt > 8.8929 × +1030 for t ≥ 7 and since t ≥ 8 implies +2t +t−4 ≤ 4, so (p, t) ∈ A2 if pt > 3.371×1026 +for t ≥ 8. +Therefore, for t ≥ 8 it is sufficient that ω(pt − 1) ≥ 20, We deduce, +utilising sieve variation repeatedly for values in the second section of the +preceding table that, (p, t) ∈ A2 if pt > 1.084 × 1025. +Similarly, ω(pt − 1) ≥ 18 is sufficient for inclusion of (p, t) in A2, and based +on the table above (p, t) ∈ A2 if pt > 2.2725 × 1021 for t ≥ 9, and (p, t) ∈ A2 +if pt > 8.158 × 1018 for t ≥ 10. +Hence (p, t) ∈ A2 unless t = 7 and p < 26382, t = 8 and p < 1347, t = 9 +and p < 237, t = 10 and p < 78, t = 11 and p < 53, t = 12 and p < 38, +t = 13 and p < 29, t = 14 and p < 23, t = 15 and p < 19, t = 16 and p < 16, +t = 17 and p < 13, t = 18 and p < 12, t = 19 and p < 10, t = 20 and p < 9, +t = 21, 22 and p < 8, t = 23, 24 and p < 7, t = 25, 26, 27 and p = 2, 3, 4, 5. +28 ≤ t ≤ 31 and p = 2, 3, 4. 32 ≤ t ≤ 39 and for p = 2, 3. 40 ≤ t ≤ 62 and +p = 2. +From the preceding discussion for every (p, t), we validated Theorem 3.1 +and compiled a list of 570 potential exceptions (listed in the Appendix). +11 + +Then, for these potential exceptions, we discover that sieve variation is true +for the large majority of prime powers, with the exception of those mentioned +in Theorem 1.1. +(see Appendix). +Theorem 1.1 derives from this. +Using +similar reasoning, it is possible to find a subset of An for any n ∈ N. In +particular, for A3, we have the following result. +Theorem 4.1. Suppose t q, r, p ∈ N be such that q is a prime number, t +≥ 7 and p = qr. Let p and t assumes none of the following values: +1. 2 ≤ p ≤ 31 or p = 37, 41, 43, 49, 61, 67, 71, 79, 103, 121 and t = 7; +2. 2 ≤ p ≤ 47 or p = 53, 83 and t = 8; +3. 2 ≤ p ≤ 7 or p = 9, 11, 16 and t = 9; +4. 2 ≤ p ≤ 8 and t = 10; +5. p = 2, 3, 4 and t = 11; +6. 2 ≤ p ≤ 7 and t = 12; +7. p = 2 and t = 14, 15, 16, 18, 20, 24. +Then (p, t) ∈ A3. +References +[1] Cao, X., Wang, P. (2014). Primitive elements with prescribed trace. +AAECC 25(5):339–345. +[2] Chou, W. S., Cohen, S. D. (2001). Primitive elements with zero traces. +Finite Fields Appl. 7(1):125–141. +[3] Cohen, S. D. (1985). Consecutive primitive roots in a finite field. Proc. +Amer. Math. Soc. 93(2):189–197. +[4] Cohen, S .D., Sharma, H., Sharma, R. K. (2021). Primitive values of ra- +tional functions at primitive elements of a finite field. J. Number Theory +219:237–246. +[5] Cohen, S. D., Presern, M. (2005). Primitive finite field elements with +prescribed trace. Southeast Asian Bull. Math. 29(2):283–300. +12 + +[6] Fu, L., Wan, D. (2014). A class of incomplete character sums. Q. J. +Math. 65(4):1195–1211. +[7] Gupta, A., Sharma, R. K., Cohen, S. D. (2018). Primitive element pairs +with one prescribed trace over a finite field. Finite Fields Appl. 54:1–14. +[8] Jungnickel, D., Vanstone, S. A. (1989). On primitive polynomials over +finite fields. J. Algebra 124(2):337–353. +[9] Lidl, R., Niederreiter, H. (1997). Finite Field, Vol. 20. Cambridge (UK): +Cambridge University Press. +[10] Paar, C., Pelzl, J. (2010). Public-Key Cryptosystems Based on the Dis- +crete Logarithm Problem, pp. 205–238. Berlin, Heidelberg: Springer. +[11] Sharma, H., Sharma, R.K. (2021). Existence of primitive pairs with +prescribed traces over finite fields. Commun. Algebra 49(4):1773-1780. +[12] Sharma, R.K., Awasthi, A., Gupta, A. (2018). Existence of pair of prim- +itive elements over finite fields of characteristic 2. J. Number Theory +193:386–394. +[13] Sharma, R. K., Gupta, A. (2017). Existence of some special primitive +normal elements over finite fields. Finite Fields Appl. 46:280–303. +[14] Sharma, R. K., Gupta, A. (2019). Pair of primitive elements with pre- +scribed traces over finite fields. Commun. Algebra 47(3):1278–1286. +[15] Shuqin, F., Wenbao, H. (2004). Character sums over galois rings and +primitive polynomials over finite fields. Finite Fields Appl. 10(1):36–52. +[16] The Sage Developers, SageMath, the Sage mathematics software system +(version 9.0), https:// www.sagemath.org, 2020. +13 + +Appendix +List of 570 values of (p, t) for which the condition of +Theorem 3.1 of the this article fails: +For t=7: +p= 2, 4, 8, 16, 32, 64, 256, 512, 1024, 4096, 3, 9, 27, 81, 243, 729, 6561, 5, +25, 125, 625, 3125, 15625, 7, 49, 343, 2401, 11, 121, 1331, 14641, 13, 169, +2197, 17, 19, 361, 23, 529, 29, 31, 37, 41, 1681, 43, 1849, 47, 53, 59, 3481, 61, +67, 4489, 71, 79, 6241, 83, 6889, 97, 9409, 101, 103, 10609, 107, 109, 11881, +113, 127, 131, 17161, 139, 19321, 151, 22801, 157, 181, 191, 197, 199, 211, +223, 227, 229, 233, 239, 241, 263, 269, 271, 277, 281, 283, 293, 311, 331, 359, +367, 389, 397, 401, 409, 431, 439, 463, 491, 499, 509, 547, 571, 593, 601, 607, +613, 619, 631, 643, 659, 661, 683, 691, 709, 727, 733, 739, 743, 877, 919, 953, +967, 1021, 1051, 1063, 1093, 1123, 1151, 1171, 1181, 1231, 1283, 1301, 1303, +1321, 1381, 1399, 1453, 1481, 1483, 1499, 1523, 1531, 1597, 1607, 1693, 1723, +1741, 1759, 1789, 1801, 1823, 1877, 1879, 1951, 2003, 2141, 2161, 2281, 2311, +2381, 2591, 2713, 2731, 2791, 2887, 2971, 3041, 3083, 3191, 3221, 3229, 3271, +3301, 3307, 3313, 3499, 3547, 3571, 3739, 3851, 3911, 4013, 4159, 4219, 4241, +4243, 4261, 4327, 4421, 4423, 4567, 4621, 4663, 4691, 4751, 4957, 5419, 5923, +5981, 6067, 6211, 6491, 6577, 7159, 7759, 8009, 8053, 8191, 8807, 9103, 9403, +9421, 9463, 9719, 9767, 9871, 9901, 9967, 10427, 10949, 10957, 10979, 11311, +11593, 11621, 12959, 14323, 15313, 15511, 16381, 17431, 17491, 19483, 19687, +19891, 20011, 20441, 21391, 22543, 23143, 23671, 24181, 24683, 25171, 25411. +For t=8: +p = 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 3, 9, 27, 81, 243, 729, 5, 25, 125, +7, 49, 343, 11, 121, 1331, 13, 169, 17, 19, 361, 23, 529, 29, 841, 31, 961, 37, +41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, +131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 211, 223, +227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, +313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, +419, 421, 433, 439, 443, 457, 461, 463, 467, 491, 499, 509, 521, 523, 541, 547, +557, 563, 569, 571, 587, 593, 599, 601, 617, 619, 631, 647, 653, 659, 661, 683, +691, 701, 709, 727, 733, 739, 743, 757, 773, 787, 797, 809, 811, 823, 827, 829, +839, 853, 857, 859, 863, 881, 887, 911, 919, 929, 937, 941, 947, 953, 967, 971, +977, 983, 991, 1009, 1013, 1021, 1033, 1039, 1061, 1063, 1069, 1087, 1091, +14 + +1093, 1097, 1103, 1109, 1117, 1123, 1201, 1213, 1217, 1223, 1231, 1277, 1279, +1283, 1289, 1291, 1301, 1303, 1319, 1321. +For t=9: +p = 2, 4, 8, 16, 32, 3, 9, 27, 81, 5, 25, 125, 7, 49, 11, 121, 13, 169, 17, 19, 23, +29, 31, 37, 43, 47, 53, 61, 79, 83, 137, 139, 149, 157, 211. +For t=10: +p = 2, 4, 8, 16, 32, 64, 3, 9, 27, 5, 25, 7, 49, 11, 13, 17, 19, 23, 29, 31, 37, 41, +53, 59, 61, 67. +For t=11: +p = 2, 4, 16, 3, 9, 7, 13. +For t=12: +p = 2, 4, 8, 16, 32, 3, 9, 27, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37. +For t=14: +p = 2, 4, 3, 5, 7. +For t=15: +p = 2, 4, 16, 3, 9, 5. +For t=16: +p = 2, 4, 8, 3, 5. +For t=18: +p = 2, 3, 4, 5. +For t=20: +p = 2, 4, 8. +15 + +For t=24: +p = 2, 3, 4. +For t=22, 28, 30, 36: +p = 2. +16 + +List consisting values of k, m corresponding to (p, t) for +which the condition of theorem 3.1 fails but sieve vari- +ation is satisfied for some choice of k in this article. +for t=7: +Sr.No. +p +k +m +1 +32 +1 +4 +2 +64 +3 +5 +3 +256 +3 +7 +4 +512 +1 +6 +5 +1024 +3 +8 +6 +4096 +3 +11 +7 +81 +2 +5 +8 +243 +2 +4 +9 +729 +2 +7 +10 +6561 +2 +8 +11 +125 +2 +4 +12 +625 +6 +5 +13 +3125 +2 +7 +14 +15625 +6 +10 +15 +343 +6 +4 +16 +2401 +6 +6 +17 +121 +6 +4 +18 +1331 +2 +8 +19 +14641 +6 +8 +20 +169 +6 +4 +21 +2197 +6 +6 +22 +17 +2 +1 +23 +361 +6 +5 +24 +529 +6 +6 +25 +29 +2 +2 +26 +41 +2 +3 +27 +1681 +6 +6 +28 +1849 +6 +5 +29 +47 +2 +3 +30 +53 +2 +3 +31 +59 +2 +5 +32 +3481 +6 +8 +33 +4489 +6 +8 +Sr.No. +p +k +m +34 +71 +2 +4 +35 +6241 +6 +8 +36 +83 +2 +3 +37 +6889 +6 +7 +38 +97 +6 +3 +39 +9409 +6 +7 +40 +101 +2 +3 +41 +103 +6 +4 +42 +10609 +6 +7 +43 +107 +2 +4 +44 +109 +6 +3 +45 +11881 +6 +7 +46 +113 +2 +3 +47 +127 +6 +4 +48 +131 +2 +5 +49 +17161 +6 +9 +50 +139 +6 +4 +51 +19321 +6 +9 +52 +151 +6 +3 +53 +22801 +6 +9 +54 +157 +6 +3 +55 +181 +6 +4 +56 +191 +2 +6 +57 +197 +2 +4 +58 +199 +6 +5 +59 +211 +6 +4 +60 +223 +6 +5 +61 +227 +2 +4 +62 +229 +6 +3 +63 +233 +2 +4 +64 +239 +2 +5 +65 +241 +6 +3 +66 +263 +2 +4 +17 + +Sr.No. +q +l +s +67 +269 +2 +6 +68 +271 +6 +3 +69 +277 +6 +4 +70 +281 +2 +5 +71 +283 +6 +3 +72 +293 +2 +4 +73 +311 +2 +5 +74 +331 +6 +4 +75 +359 +2 +6 +76 +367 +6 +4 +77 +389 +2 +6 +78 +397 +6 +5 +79 +401 +2 +5 +80 +409 +6 +4 +81 +431 +2 +7 +82 +439 +6 +4 +83 +463 +6 +4 +84 +491 +2 +5 +85 +499 +6 +4 +86 +509 +2 +5 +87 +547 +6 +4 +88 +571 +6 +4 +89 +593 +2 +5 +90 +601 +6 +4 +91 +607 +6 +4 +92 +613 +6 +5 +93 +619 +6 +4 +94 +631 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+p +k +m +1 +2 +15 +6 +24 + diff --git a/2dE0T4oBgHgl3EQfdwCH/content/tmp_files/load_file.txt b/2dE0T4oBgHgl3EQfdwCH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..222f3843073ee961d1ca6f1c5e6d03981d7cb03f --- /dev/null +++ b/2dE0T4oBgHgl3EQfdwCH/content/tmp_files/load_file.txt @@ -0,0 +1,2525 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf,len=2524 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='02381v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='NT] 6 Jan 2023 Existence of primitive pairs with two prescribed traces over finite fields Aakash Choudhary∗and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Sharma † Department of Mathematics, Indian Institute of Technology Delhi-110016, India Abstract Given F = Fpt, a field with pt elements, where p is a prime power, t ≥ 7, n are positive integers and f = f1/f2 is a rational func- tion, where f1, f2 are relatively prime, irreducible polynomials with deg(f1) + deg(f2) = n in F[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We construct a sufficient condition on (p, t) which guarantees primitive pairing (ǫ, f(ǫ)) exists in F such that TrFpt/Fp(ǫ) = a and TrFpt/Fp(f(ǫ)) = b for any prescribed a, b ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Further, we demonstrate for any positive integer n, such a pair defi- nitely exists for large t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' The scenario when n = 2 is handled separately and we verified that such a pair exists for all (p, t) except from possible 71 values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' A result for the case n = 3 is given as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Keywords: Character, Finite fields, Primitive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2020 Mathematics Subject Classification: 12E20, 11T23 1 Introduction Let Fp represent a field of finite order p, where p = qr for some prime q and r, a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' The multiplicative group of Fp is cyclic, it is denoted by F∗ p and a generator of F∗ p is referred to as a primitive element in Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' ∗email: achoudhary1396@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='com †email: rksharmaiitd@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='com 1 The field Fp has φ(p − 1) primitive elements, where φ is the Euler’s totient function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let Fpt denote an extension of Fp of degree t for some positive integer t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' A necessary and sufficient condition for an element ǫ ∈ F∗ pt to be primitive is that it is a root of an irreducible polynomial of degree t over Fp and such an irreducible polynomial is referred to as primitive polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For ǫ ∈ Fpt, the trace of ǫ over Fp denoted by TrFpt/Fp(ǫ), is defined as TrFpt/Fp(ǫ) = ǫ + ǫp + ǫp2 + · · · + ǫpt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In Cryptographic schemes such as Elgamel encryption scheme and the Diffie-Hellman key exchange, primitive elements serve as the fundamental building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Numerous applications of primitive elements can be found in Coding theory and Cryptography [10], making the study of primitive el- ements and primitive polynomials an active research field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Please refer to [9] for more information about the existence of primitive elements in finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For any rational function f(x) ∈ Fp(x) and ǫ ∈ Fp we call the pair (ǫ, f(ǫ)), a primitive pair if both ǫ and f(ǫ) are primitive elements in Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In general, if ǫ is primitive, f(ǫ) need not be primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For instance, take x2 + 3x + 2 ∈ F7[x], then 3, 5 are primitive elements in F7 but none of f(3) and f(5) are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In 1985, Cohen [3] introduced the term ”primitive pair” and he verified the existence of primitive pairs (ǫ, f(ǫ)) in Fp for linear polynomials f(x) = x + k ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Since then many researchers have conducted studies in this area [12, 13, 7, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Most recently, Cohen, Sharma and Sharma [4] have supplied a condition that ensures the occurrence of primitive pair (ǫ, f(ǫ)) in Fp for non-exceptional rational function f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', f is not of the form cxjgk(x), where j ∈ Z, k > 1 that divides p − 1 and c ∈ F∗ p, for any g(x) ∈ Fp(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Jungnickel, Vanstone [8] identified a sufficient condition for the occurrence of primitive elements ǫ ∈ Fpt with a prescribed trace of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Later Cohen [5] extended the result with some exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Chou and Cohen [2], in 2014, addressed the issue of the existence of primitive element ǫ ∈ Fpt such that TrFpt/Fp(ǫ) = TrFpt/Fp(ǫ−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Cao and Wang [1], for t ≥ 29, established a condition for the existence of primitive pair (ǫ, f(ǫ)) with f(x) = x2+1 x ∈ Fpt(x) such that for prescribed a, b ∈ F∗ p, TrFpt/Fp(ǫ) = a and TrFpt/Fp(ǫ−1) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In 2018, Gupta, Sharma and Cohen [7], for the same rational function and prescribed a ∈ Fp, presented a condition that ensures the existence of primitive pair (ǫ, f(ǫ)) in Fpt with TrFpt/Fp(ǫ) = a for t ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Then in 2019, Gupta and Sharma [14] extended the result to the rational function 2 ΓM(x) = a11x2+a12x+a13 a22x+a23 , where M = � a11 a12 a13 0 a22 a23 � ∈ M2×3(Fpt) is any matrix of rank 2, and if ΓM(x) = λx or λx2 for some λ ∈ Fpt, then λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In 2021, Sharma and Sharma [11] examined the rational function f = f1/f2 in Fpt(x), where f1 and f2 are relatively prime, irreducible polynomials and proved that for prescribed a, b ∈ Fp, the existence of primitive pair (ǫ, f(ǫ)) in Fpt such that TrFpt/Fp(ǫ) = a and TrFpt/Fp(ǫ−1) = b for t ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Prior to this article, for primitive pairs, traces were considered for ǫ and ǫ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In this article, we will consider the trace onto the element ǫ and its image under f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='e, f(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Some terminology and conventions are introduced for explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We say that a non-zero polynomial f over Fp[x] has degree k ≥ 0, if f(x) = akxk + ak−1xk−1 + · · ·+ a1x + a0, where ak ̸= 0 and we write the degree of f as deg(f) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Next, we suppose that, for a rational function f(x) = f1(x) f2(x) ∈ Fp(x), f1 and f2 are relatively prime, irreducible polynomials and define the degree-sum as degsum(f) = deg(f1) + deg(f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We will now define various sets that will play a crucial role in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We define Rp,t(n1, n2) to represent the set of all rational function f(x) = f1(x) f2(x) ∈ Fpt(x) such that f1 and f2 are relatively prime, irreducible polynomials over Fpt with deg(f1) = n1 and deg(f2) = n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Denote An1,n2 as the set consisting of pairs (p, t) ∈ N × N such that for any f ∈ Rp,t(n1, n2) and prescribed a, b ∈ Fp, Fpt contains an ele- ment ǫ such that (ǫ, f(ǫ)) is a primitive pair with TrFpt/Fp(ǫ) = a and TrFpt/Fp(f(ǫ)) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Define, Rp,t(n) = � n1+n2=n Rp,t(n1, n2) and An = � n1+n2=n An1,n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' First, in this paper, for n ∈ N, we consider f(x) ∈ Rp,t(n) and a, b ∈ Fp, and then verify that there exists an element ǫ ∈ Fpt such that (ǫ, f(ǫ)) is a primitive pair in Fpt with TrFpt/Fp(ǫ) = a and TrFpt/Fp(f(ǫ)) = b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', we provide a sufficient condition on pt such that (p, t) ∈ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Furthermore, using a sieve variation of this sufficient condition, we prove the following result: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let t, q, r, p ∈ N be such that q is a prime number, t ≥ 7 and p = qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Suppose p and t assumes none of the following values: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2 ≤ p ≤ 16 or p = 19, 23, 25, 27, 31, 37, 43, 49, 61, 67, 79 and t = 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2 ≤ p ≤ 31 or p = 32, 37, 41, 43, 47, 83 and t = 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2 ≤ p ≤ 8 or p = 11, 16 and t = 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' p = 2, 3, 4, 5, 7 and t = 10, 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' p = 2, 3, 4 and t = 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' p = 2 and t = 14, 15, 16, 18, 20, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Then (p, t) ∈ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Note:- The exceptions in above theorem need not be true exceptions, they are possible exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' SageMath [16] is used to perform all nontrivial calculations required throughout this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2 Preliminaries In this section, we present some basic concepts, notations, and results that will be used in forthcoming sections of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Throughout the article, t is a positive integer, p is an arbitrary prime power and Fp is a finite field of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1 Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' A character of a finite abelian group G is a homomorphism χ from the set G into Z1, where Z1 is the set of all elements of complex field C with absolute value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' The trivial character of G denoted by χ0, is defined as χ0(g) = 1 for all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In addition, the set of all characters of G, denoted by ˆG, forms a group under multiplication, which is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' The order of a character χ is the least positive integer d such that χd = χ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For a finite field Fpt, a character of the additive group Fpt is called an additive character and that of the multiplicative group F∗ pt is called a multiplicative character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For u, a divisor of pt−1, an element ζ ∈ F∗ pt is called u-free, if whenever ζ = ξs, where ξ ∈ Fpt and s|u implies s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We see that an element ζ ∈ F∗ pt is (pt − 1)-free if and only if it is a primitive element of Fpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 4 For more information on characters, primitive elements and finite fields, we refer the reader to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' The following conclusion holds as a particular case of [15, Lemma 10]: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let u be a divisor of pt − 1, ζ ∈ F∗ pt, then we have: � s|u µ(s) φ(s) � χs χs(ζ) = � u φ(u) if ζ is u − free, 0 otherwise where µ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=') is the Mobius function and φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=') is the Euler function, χs runs through all the φ(s) multiplicative characters over F∗ pt with order s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Therefore for u, a divisor of pt − 1 ρu : ǫ �→ θ(u) � s|u µ(s) φ(s) � χs χs(ǫ) (1) gives a characteristic function for the subset of u-free elements of F∗ pt, where θ(u) = φ(u)/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Also for a ∈ Fp, τa : ǫ �→ 1 p � ψ∈ ˆ Fp ψ(TrFpt/Fp(ǫ) − a) (2) is a characteristic function for the subset of Fpt whose elements satisfy TrFpt/Fp(ǫ) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' From [9, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='7], any additive character ψ of Fp can be derived by ψ(a) = ψ0(ua), where ψ0 is the canonical additive character of Fp and u is an element of Fp corresponding to ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Thus τa = 1 p � ψ∈ ˆ Fp ψ0(TrFpt/Fp(uǫ) − ua) = 1 p � u∈Fp ˆψ0(uǫ)ψ0(−ua) (3) where ˆψ0 is the additive character of Fpt defined by ˆψ0(ǫ) = ψ0(TrFpt/Fp(ǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In next theorem, we will make major use of the results given below by Wang and Fu [6] in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='5] Let F(x) ∈ Fpd(x) be a rational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Write F(x) = �k j=1 fj(x)rj, where fj(x) ∈ Fpd[x] are irreducible polynomials and rj are non zero integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let χ be a multiplicative character of Fpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Suppose that the rational function �d−1 i=1 f(xpi) is not of the form h(x)ord(χ) ∈ Fpd(x), where ord(χ) is the order of χ, Then we have ���� � ǫ∈Fp,f(ǫ)̸=0,∞ χ(F(ǫ)) ���� ≤ � d k � j=1 deg(fj) − 1 � p 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='6] Let f(x), g(x) ∈ Fpt(x) be rational functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Write f(x) = �k j=1 fj(x)rj, where fj(x) ∈ Fpt[x] are irreducible polynomials and rj are non-zero integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let D1 = �k j=1 deg(fj), D2 = max{deg(g), 0}, D3 is the degree of denominator of g(x) and D4 is the sum of degrees of those irreducible polynomials dividing denominator of g but distinct from fj(x)( j= 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=',k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let χ be a multiplicative character of Fpt, and let ψ be a nontrivial additive character of Fpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Suppose g(x) is not of the form v(x)pt − v(x) in Fpt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Then we have ���� � ǫ∈Fpt,f(ǫ)̸=0,∞,g(ǫ)̸=∞ χ(f(ǫ))ψ(g(ǫ)) ���� ≤ (D1 + D2 + D3 + D4 − 1)p t 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Evidently, both the sufficient condition (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1) and its sieving variation (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='4) are entirely dependent on pt and the degrees of the numerator and denominator polynomials of the rational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' It is easy to see that the Trace part of the main result in [11] is a special case of our finding for f(x) = 1 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For every κ ∈ N, we will use ω(κ) to represent the number of distinct prime divisors of κ, and W(κ) to represent the number of square free divisors of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Clearly, W(κ) = 2ω(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 3 Sufficient Condition Let k1, k2, p, t ∈ N be such that p is a prime power and k1, k2 are positive integers which divide pt − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let a, b ∈ Fp, f(x) ∈ Rp,t(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let Af,a,b(k1, k2) represents the set consisting of all those elements ǫ ∈ Fpt such that ǫ is k1-free, f(ǫ) is k2-free, TrFpt/Fp(ǫ) = a, and TrFpt/Ft(f(ǫ)) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We now verify the sufficient condition as follows: 6 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Suppose t, n, p ∈ N and p is a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Suppose that p t 2−2 > (2n + 1)W(pt − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Then (p, t) ∈ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In order to prove this result it suffices to demonstrate that Af,a,b(k1, k2) > 0 for every f(x) ∈ Rp,t(n) and for every prescribed a, b ∈ Fp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Suppose that f(x) ∈ Rp,t(n) be a rational function and that a, b ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let P represent the collection of zeroes and poles of f(x) ∈ Fpt and P ′ = P ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let k1, k2 be divisors of pt − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Then by definition, Af,a,b(k1, k2) will be given by Af,a,b(k1, k2) = � ǫ∈Fpt−P ′ ρk1(ǫ)ρk2(f(ǫ))τa(ǫ)τb(f(ǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Using the characteristic functions (1) and (3) defined in the previous section, we obtain Af,a,b(k1, k2) = θ(k1)θ(k2) p2 � s1|k1,s2|k2 µ(s1)µ(s2) φ(s1)φ(s2) � s1,s2 χf,a,b(s1, s2) where θ(ki) = φ(ki) ki ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' i = 1, 2 and χf,a,b(s1, s2) = � u,v∈Fp ψ0(−au − bv) � ǫ∈Fpt−P ′ χs1(ǫ)χs2(ǫ0) ˆψ0(uǫ + vǫ0) where ǫ0 = f(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' It follows from [9, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1] that, for any divisors s1, s2 of pt − 1, there exist integers m1, m2 with 0 < m1, m2 < pt − 1 such that χs1(x) = χpt−1(xm1) and χs2(x) = χpt−1(xm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Thus χf,a,b(s1, s2) = � u,v∈Fp ψ0(−au − bv) � ǫ∈Fpt−P ′ χpt−1(ǫm1f(ǫ)m2) ˆψ0(uǫ + vǫ0) (4) = � u,v∈Fp ψ0(−au − bv) � ǫ∈Fpt−P ′ χpt−1(F1(ǫ)) ˆψ0(F2(ǫ)), (5) where F1(x) = xm1f(x)m2 ∈ Fpt(x) and F2(x) = ux + vf(x) ∈ Fpt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 7 First we consider the situation when F2(x) = l(x)pt −l(x) for some l(x) ∈ Fpt(x), where l(x) = l1(x) l2(x) with (l1, l2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We have, ux+vf1(x) f2(x) = l1(x)pt l2(x)pt − l1(x) l2(x), that is, f2(x)(l1(x)pt − l1(x)l2(x)pt−1) = l2(x)pt(uxf2(x) + vf1(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Since (l1(x)pt − l1(x)l2(x)pt−1, l2(x)pt) = 1, it implies that, l2(x)pt divides f2(x), which can only happen if l2(x) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' That is, we have c−(pt)f2(x)(l1(x)pt − l1(x)cpt−1) = uxf2(x) + vf1(x) where c = l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Now, the above equation only applies if v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Substituting it to the equation above yields, c−(pt)(l1(x)pt − l1(x)cpt−1) = ux, which can happen only if l1 is constant and u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Moreover, if F1(x) ̸= r(x)pt−1 for any r(x) ∈ Fpt(x), then it follows form Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='2 that |χf,a,b(s1, s2)| ≤ np t 2 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (6) And, when F1(x) = r(x)pt−1 for some r(x) ∈ Fpt(x), where r(x) = r1(x) r2(x) is such that (r1, r2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Following [11], it happens only if m1 = m2 = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' If F2(x) ̸= d(x)pt − d(x) for any d(x) ∈ Fpt(x) then, Case 1 : When n1 ≤ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Then in accordance with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='3 we have D2 = 1, and |χf,a,b(s1, s2)| ≤ (2n + 1)p t 2 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (7) Case 2 : When n1 > n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We have D2 = n1 − n2 and |χf,a,b(s1, s2)| ≤ 2np t 2+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (8) Thus, if (χs1, χs2, u, v) ̸= (χ1, χ1, 0, 0) then based on the discussion above, and using (6), (7) and (8), we get, |χf,a,b(s1, s2)| ≤ (2n + 1)p t 2+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' From this and the definition of Af,a,b(k1, k2), we get Af,a,b(k1, k2) ≥ θ(k1)θ(k2) p2 ((pt − |P ′|) − (2n + 1)p t 2+2(W(k1)W(k2) − 1)) (9) ≥ θ(k1)θ(k2) p2 ((pt − (n + 1)) − (2n + 1)p t 2+2(W(k1)W(k2) − 1)) (10) 8 Therefore, if p t 2−2 > (2n + 1)W(k1)W(k2), then Af,a,b(k1, k2) > 0 for every f(x) ∈ Rp,t(n) and prescribed a, b ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Considering k1 = k2 = pt − 1, result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Now, we provide the bounds for the absolute values for Af,a,b(mk, k) − θ(m)Af,a,b(k, k) and Af,a,b(k, mk) − θ(m)Af,a,b(k, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Proofs are omitted as they follow from the idea of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let k be a positive integer that divides pt −1 and m is a prime dividing pt − 1 but not k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Then |Af,a,b(mk, k) − θ(m)Af,a,b(k, k)| ≤ θ(k)2θ(m) p2 (2n + 1)W(k)2p t 2 +2 and |Af,a,b(k, mk) − θ(m)Af,a,b(k, k)| ≤ θ(k)2θ(m) p2 (2n + 1)W(k)2p t 2+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let k be a positive integer that divides pt−1 and {q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' , qm} be the collection of all primes dividing pt − 1 but not k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Then Af,a,b(pt−1, pt−1) ≥ m � i=1 Af,a,b(k, qik)+ m � i=1 Af,a,b(qik, k)−(2m−1)Af,a,b,(k, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Sieve variation of sufficient condition (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1) is given below, proof of which is not given as it follows from Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='3 and ideas in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let t, n, p, k ∈ N be such that k divides pt − 1, where p is a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Assume {q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' , qm} is the collection of all those primes that divide pt −1 but not k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Suppose δ = 1 −2 �m i=1 1 qi and ∆ = 2m − 1 δ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' If δ > 0 and p t 2 −2 > (2n + 1)∆W(k)2 then (p, t) ∈ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Suppose that κ ∈ N is such that ω(κ) ≥ 1547, then W(κ) ≤ κ1/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let V = {2, 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', 12983} is the set of first 1547 primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We see that the product of all elements of V exceeds K = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='57 × 105588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let κ = κ1κ2, where κ1 and κ2 are co-prime integers such that all prime divisors of κ1 come from the least 1547 prime divisors of κ and remaining prime divisors are divisors of κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Hence, κ1/12 1 > K1/12 > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='42 × 10465, whereas W(κ1) < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='93 × 10465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' The conclusion follows, since ρ1/12 > 2 for all primes ρ > 12983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We shall need Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='4 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='5 for calculation work in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 4 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1 Proof will be carried out for the situation t ≥ 7, since according to [2] there is no primitive element ǫ, for t ≤ 4, such that TrFpt/Fp(ǫ) = 0 and TrFpt/Fp(ǫ−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' The cases t = 5 and 6 necessitate substantial computation and appear to demand a different technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' As a result, we postpone further examination of these situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We assume initially that, ω(pt − 1) ≥ 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='5, if p t 2 −2 > 5p t 6, that is, if pt > 5 3t t−6 then (p, t) ∈ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' But t ≥ 7 gives 3t t−6 ≤ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Hence, if pt > 521 then (p, t) ∈ A2, and this holds true for ω(pt − 1) ≥ 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Therefore, we may suppose ω(pt − 1) ≤ 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' We shall use sieve variation in order to carry forward computational work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let 62 ≤ ω(pt − 1) ≤ 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' To use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='4 assume k to be the product of least 62 primes that divide pt − 1, that is, W(k) = 262, then m ≤ 1485 and δ assumes its least positive value when {q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' , q1485} = {307, 311, 313, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', 12979}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' This yields δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='004174 and ∆ < 710770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='7395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Hence 5∆W(k)2 < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='558211 × 1043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let Z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='558211 × 1043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' By sieve variation, (p, t) ∈ A2 if q t 2 −2 > Z i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', if pt > Z 2t t−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Since t ≥ 7, it gives 2t t−4 ≤ 14 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Therefore, (p, t) ∈ A2 under the condition that pt > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='834 × 10204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Hence, ω(pt − 1) ≥ 95 implies (p, t) ∈ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In a similar manner (p, t) ∈ A3, A4 if ω(pt − 1) ≥ 95, and (p, t) ∈ A5 if ω(pt − 1) ≥ 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 10 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' a ≤ ω(pt − 1) ≤ b W(k) δ > ∆ < 5∆W(k)2 < 1 a = 13, b = 94 213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='04481712 3594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='3767988 1,206,072,718,756 2 a = 7, b = 34 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='04609692 1151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='7513186 94,351,469 3 a = 6, b = 25 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='08241088 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='9698124 9,235,862 4 a = 6, b = 23 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='12550135 264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='9453729 5,426,082 5 a = 6, b = 22 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='14959773 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='2223842 4,284,875 6 a = 5, b = 19 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='07663431 354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='3225878 1,814,132 7 a = 5, b = 17 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='13927194 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1445296 855,780 8 a = 5, b = 16 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='17317025 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='2679422 631,132 9 a = 5, b = 15 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='21090610 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='0874844 471,488 Using the values in the Table 1 above and repeating the process of sieve variation, we determine that (p, t) ∈ A2 if pt > (4284875) 14 3 or pt > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='8929 × 1030 for t ≥ 7 and since t ≥ 8 implies 2t t−4 ≤ 4, so (p, t) ∈ A2 if pt > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='371×1026 for t ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Therefore, for t ≥ 8 it is sufficient that ω(pt − 1) ≥ 20, We deduce, utilising sieve variation repeatedly for values in the second section of the preceding table that, (p, t) ∈ A2 if pt > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='084 × 1025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Similarly, ω(pt − 1) ≥ 18 is sufficient for inclusion of (p, t) in A2, and based on the table above (p, t) ∈ A2 if pt > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='2725 × 1021 for t ≥ 9, and (p, t) ∈ A2 if pt > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='158 × 1018 for t ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Hence (p, t) ∈ A2 unless t = 7 and p < 26382, t = 8 and p < 1347, t = 9 and p < 237, t = 10 and p < 78, t = 11 and p < 53, t = 12 and p < 38, t = 13 and p < 29, t = 14 and p < 23, t = 15 and p < 19, t = 16 and p < 16, t = 17 and p < 13, t = 18 and p < 12, t = 19 and p < 10, t = 20 and p < 9, t = 21, 22 and p < 8, t = 23, 24 and p < 7, t = 25, 26, 27 and p = 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 28 ≤ t ≤ 31 and p = 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 32 ≤ t ≤ 39 and for p = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 40 ≤ t ≤ 62 and p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' From the preceding discussion for every (p, t), we validated Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1 and compiled a list of 570 potential exceptions (listed in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 11 Then, for these potential exceptions, we discover that sieve variation is true for the large majority of prime powers, with the exception of those mentioned in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (see Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1 derives from this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Using similar reasoning, it is possible to find a subset of An for any n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' In particular, for A3, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Suppose t q, r, p ∈ N be such that q is a prime number, t ≥ 7 and p = qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Let p and t assumes none of the following values: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2 ≤ p ≤ 31 or p = 37, 41, 43, 49, 61, 67, 71, 79, 103, 121 and t = 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2 ≤ p ≤ 47 or p = 53, 83 and t = 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2 ≤ p ≤ 7 or p = 9, 11, 16 and t = 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2 ≤ p ≤ 8 and t = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' p = 2, 3, 4 and t = 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 2 ≤ p ≤ 7 and t = 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' p = 2 and t = 14, 15, 16, 18, 20, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Then (p, t) ∈ A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' References [1] Cao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Primitive elements with prescribed trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' AAECC 25(5):339–345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [2] Chou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Cohen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Primitive elements with zero traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Finite Fields Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 7(1):125–141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [3] Cohen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Consecutive primitive roots in a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 93(2):189–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [4] Cohen, S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Sharma, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Sharma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Primitive values of ra- tional functions at primitive elements of a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Number Theory 219:237–246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [5] Cohen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Presern, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Primitive finite field elements with prescribed trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Southeast Asian Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 29(2):283–300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 12 [6] Fu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Wan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' A class of incomplete character sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 65(4):1195–1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [7] Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Sharma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Cohen, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Sharma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Existence of primitive pairs with prescribed traces over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Algebra 49(4):1773-1780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [12] Sharma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Awasthi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Existence of pair of prim- itive elements over finite fields of characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Number Theory 193:386–394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [13] Sharma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Existence of some special primitive normal elements over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Finite Fields Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 46:280–303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [14] Sharma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Pair of primitive elements with pre- scribed traces over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Algebra 47(3):1278–1286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [15] Shuqin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=', Wenbao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Character sums over galois rings and primitive polynomials over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' Finite Fields Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 10(1):36–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' [16] The Sage Developers, SageMath, the Sage mathematics software system (version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='0), https:// www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='sagemath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='org, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 13 Appendix List of 570 values of (p, t) for which the condition of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1 of the this article fails: For t=7: p= 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 16,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 1319,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 1321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For t=9: p = 2, 4, 8, 16, 32, 3, 9, 27, 81, 5, 25, 125, 7, 49, 11, 121, 13, 169, 17, 19, 23, 29, 31, 37, 43, 47, 53, 61, 79, 83, 137, 139, 149, 157, 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For t=10: p = 2, 4, 8, 16, 32, 64, 3, 9, 27, 5, 25, 7, 49, 11, 13, 17, 19, 23, 29, 31, 37, 41, 53, 59, 61, 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For t=11: p = 2, 4, 16, 3, 9, 7, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For t=12: p = 2, 4, 8, 16, 32, 3, 9, 27, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For t=14: p = 2, 4, 3, 5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For t=15: p = 2, 4, 16, 3, 9, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For t=16: p = 2, 4, 8, 3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For t=18: p = 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For t=20: p = 2, 4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 15 For t=24: p = 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' For t=22, 28, 30, 36: p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' 16 List consisting values of k, m corresponding to (p, t) for which the condition of theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='1 fails but sieve vari- ation is satisfied for some choice of k in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' for t=7: Sr.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' p k m 153 1231 6 8 154 1277 6 9 155 1279 6 8 156 1283 6 9 157 1289 6 9 158 1291 6 9 159 1303 6 8 160 1319 6 8 161 1321 6 8 162 191 6 7 163 911 30 9 164 659 30 8 165 1301 30 10 Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content=' p k m 166 281 30 6 167 410 30 6 168 421 30 8 169 937 30 7 170 1331 30 9 171 307 30 7 172 1217 30 7 173 967 30 8 174 461 30 7 175 463 30 7 176 853 30 9 177 727 30 7 178 729 30 9 for t=9: Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfdwCH/content/2301.02381v1.pdf'} +page_content='No.' metadata={'source': 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Rowe1,*, B. Yuan1, M. Buzzi1, G. Jotzu1, Y. Zhu1, M. Fechner1, M. Först1, B. Liu1,2 +D. Pontiroli3, M. Riccò3, A. Cavalleri1,4,* + +1 Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany +2 Paul Scherrer Institute, Villigen, Switzerland +3 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Italy +4 Department of Physics, Clarendon Laboratory, University of Oxford, United Kingdom +* e-mail: edward.rowe@mpsd.mpg.de, andrea.cavalleri@mpsd.mpg.de + +Photo-excitation at terahertz and mid-infrared frequencies has emerged as a new +way to manipulate functionalities in quantum materials, in some cases creating +non-equilibrium phases that have no equilibrium analogue. In K3C60, a metastable +zero-resistance phase was documented with optical properties and pressure de- +pendences compatible with non-equilibrium high temperature superconductivity. +Here, we report the discovery of a dominant energy scale for this phenomenon, +along with the demonstration of a giant increase in photo-susceptibility near +10 THz excitation frequency. At these drive frequencies a metastable supercon- +ducting-like phase is observed up to room temperature for fluences as low as +~400 µJ/cm2. These findings shed light on the microscopic mechanism underlying +photo-induced superconductivity. They also trace a path towards steady state op- +eration, currently limited by the availability of a suitable high-repetition rate opti- +cal source at these frequencies. + + + + +2 +The search for new non-equilibrium functional phases in quantum materials, such as op- +tically induced ferroelectricity1,2, magnetism3-5, charge density wave order6,7, non-trivial +topology8,9 and superconductivity10-18, has become a central research theme in condensed +matter physics. In the case of K3C60 (Fig. 1a), mid infrared optical pulses have been exten- +sively documented to yield an unconventional non-equilibrium phase which exhibits met- +astable zero-resistance14, an extraordinarily high mobility and a superconducting-like gap +in the optical conductivity12,14 that reduce with applied pressure13, and nonlinear I-V char- +acteristics19. All these observations are indicative of non-equilibrium high temperature +superconductivity, observed at base temperatures far exceeding the highest equilibrium +superconducting critical temperature of any alkali-doped fulleride (Fig. 1b). +Typical experimental results reported to date are displayed in Fig. 2c. K3C60 powders were +held at a base temperature T = 100 K ≫ T! = 20 K and irradiated with 1 ps-long pulses +with 170 meV photon energy (l ~ 7.3 µm, ) ~ 41 THz) at a fluence of 18 mJ/cm². This +strong excitation regime yielded a long-lived transient state with dramatic changes in +both the real and imaginary parts of the optical conductivity, measured using phase + +Figure 1. Crystal structure and phase diagram K3C60. (a) Crystal structure of the organic molec- +ular solid K3C60. C60 molecules are situated at the vertices of a face-centered-cubic lattice. Potassium +atoms (red) occupy the interstitial voids. (b) Pressure-temperature phase diagram of the fcc-A3C60 +alkali-doped fulleride family of compounds. Physical pressure tunes the spacing between the C60 +molecules. The grey line indicates the boundary between the insulating and metallic/superconduct- +ing compounds. The blue shaded area indicates where superconductivity is observed at equilibrium. +The star indicates the K3C60 compound investigated in this work, which superconducts at tempera- +tures ! ≤ !! = 20K. + +Insulator +Superconductor +3 +sensitive terahertz time-domain spectroscopy. The transient optical properties displayed +in Fig. 2c are reminiscent of those of the equilibrium superconducting state measured in +the same material at T ≪ T! = 20 K (cf. Fig. 2b), and are suggestive of transient high +temperature superconductivity. These signatures consist of a saturated reflectivity, a gap +in the real part of the optical conductivity +"(-), and an imaginary conductivity +#(-) +which diverges towards low frequencies as ~1/-. The divergent +#(-) implies (through +Kramers-Kronig relations) the presence of a peak in +" centred at zero frequency, with a +width limited by the lifetime of the state which also determines the carrier mobility. +These data were obtained by accounting for the inhomogeneous excitation of the probed +volume using a multilayer model. Here we show the results of this reconstruction under +the assumption of a linear (open symbols) and sublinear (filled symbols)20 dependence of +the photo-induced changes in the terahertz refractive index on the mid-infrared pump +fluence, as detailed in supplementary section S6. Allowing for a finite temperature super- +conductor, in which a varying density of uncondensed quasi-particles also contributes to +the terahertz response, the superconducting-like nature of the transient state is inde- +pendent of the specific choice of assumption. Only quantitative differences, associated +with the relative densities of induced superfluid and heated quasi-particles, which can be +extracted by fitting with a two-fluid model, emerge. +Note that the enhancement of conductivity observed in these experiments is not con- +nected to an increase in the carrier density, but is solely caused by a transfer of spectral +weight from the real part (resistive) to the imaginary part (inductive) of the conductivity, +and hence reflects a colossal increase in the carrier mobility at constant density. +Three spectrally-integrated figures of merit are extracted from the snapshots of R(-, 2), ++"(-, 2) and +#(-, 2), and plotted as a function of pump-probe time delay 2 in Fig. 2e, +showing the time evolution of the system. + + +4 + +Figure 2: Photo-induced metastable superconductivity in K3C60 generated with intense 170 +meV excitation pulses. (a) Schematic of the experimental set-up. Pump pulses with 170 meV pho- +ton energy were generated in an optical parametric amplifier (OPA) and subsequent difference fre- +quency generation (DFG) of the signal and idler beams. These pulses were stretched to a duration +of ~1 ps by linear propagation in a highly dispersive CaF2 rod. The photoinduced changes in the far- +infrared optical properties of K3C60 were detected with phase-sensitive transient THz time-domain +spectroscopy. (b) Reflectivity (sample–diamond interface), real and imaginary part of the optical +conductivity of K3C60 measured upon cooling across the equilibrium superconducting transition. +The blue shading indicates the change of spectral weight in these quantities across the thermally +driven superconducting transition. (c) Same quantities measured at equilibrium (red lines) and +10 ps after excitation (filled and open symbols). The data in filled (open) symbols are obtained ac- +counting for the inhomogeneous excitation of the probed volume under the assumption of a square +root (linear) fluence dependence of the photo-induced changes in the complex refractive index of +the material (Supplementary Section S6). The blue shading indicates the change of spectral weight +in these quantities after photo-excitation. The blue solid lines are fits to the transient optical data +with a Drude-Lorentz model (Supplementary Section S7). These data were acquired at a base tem- +perature T = 100 K with an excitation fluence of 18 mJ cm-2. (d) Same quantities as in (c) but meas- +ured at a base temperature T = 295 K with an excitation fluence of 18 mJ cm-2. (e) Time dependence +of the average reflectivity, average real part of the optical conductivity &"((), and light-induced “su- +perfluid density” extracted from a two-fluid model fit and expressed as a fraction of the total charge +carrier density. All quantities are evaluated in the region of the photo-induced gap (5–10 meV). +Filled and open symbols indicate the results of two different reconstructions as in (c). The red dotted +lines indicate the value of the corresponding quantity at equilibrium. These data were acquired at a +base temperature T = 100 K with a fluence of 18 mJ cm−2 and a pump-pulse duration of ~1 ps. + +a +3-Stage OPA +WLG +DFG +Ti:SaOscillator +Ti:SaAmplifierx2 +THz gen. +b +d +e +1.0 +Reflectivity* +Reflectivity +1.0 +0.5 +0.5 +25K > Tc +Equilibrium +Equilibrium +18 mJ cm-2 +5K< Tc +Photoexcited +Photoexcited +0.0 +0.0 +900 +Equil. +T = 100 K +T = 295 K +T= i0 ps += iops +cm +0% +Gapping +cm +600 +150 +300 +* +6 +6 +100% +0 +900 +1.0 +cm +600 +0.5 +300 +(S +02 +102040 +102040 +04 +102040 +C +0 5 10 +50 +100 +Energy (meV) Energy (meV) Energy (meV) +Time (ps) +5 +The first two quantities are the frequency-averaged values of the reflectivity and of +"(-) +below the energy gap, for which a zero-temperature superconductor with infinite lifetime +would give values of 1 and 0 respectively. The third figure of merit is the fractional super- +fluid density which is proportional +to the divergence of +#(-). This is determined by fitting the photoexcited optical proper- +ties with a two-fluid model where one fluid represents the remaining normal carriers with +a finite scattering rate and the other has zero scattering rate, giving a superconducting- +like contribution. Details of this fitting procedure are given in supplementary section S7. +For low excitation fluences the system becomes superconducting-like after photoexcita- +tion, and relaxes on a time scale of a few picoseconds. As already seen in the spectrally +resolved measurements, for high excitation fluences the system enters a metastable re- +gime in which the superconducting-like optical properties persist for much longer times, +up to several nanoseconds. +We note that the temperature dependence reported in Ref. 12 shows transient supercon- +ducting-like optical properties up to a temperature of 150-200 K. For higher tempera- +tures the gapping and extracted superfluid density are severely reduced. Examples of such +spectra measured at room temperature are shown in Fig. 2d. Nevertheless, the pressure +scaling reported in Ref. 13 suggests that traces of non-equilibrium superconductivity may +survive up to higher temperatures, raising the prospect that with more effective driving a +full manifestation of the metastable superconducting-like state may be possible at 300 K. +To date, these experiments have been limited to excitation photon energies between 80 +and 165 meV (20-40 THz), such that a more comprehensive search for a dominant excita- +tion frequency scale has remained out of reach. Many potentially important resonances at +lower frequencies (ℎ4 < 80 meV) have remained unexplored, primarily due to the lack of +a suitable high-intensity pump source that operates in this range. In the present work, we +explore excitation at energies between 24 and 80 meV (6-20 THz). + + +6 + +Figure 3: Photo-induced metastable superconductivity in K3C60 generated with 41 meV exci- +tation pulses. (a) Schematic of the experimental set-up. Pump pulses with 41 meV (10 THz) photon +energy are generated in a twin optical parametric amplifier (OPA) and subsequent chirped-pulse +difference frequency generation (DFG) of the two stretched signal beams. The photoinduced +changes in the far-infrared optical properties of K3C60 are detected with phase-sensitive transient +THz time-domain spectroscopy. (b) Reflectivity (sample–diamond interface), real and imaginary +part of the optical conductivity of K3C60 measured at equilibrium (red lines) and 50 ps after excita- +tion (filled and open symbols). The data in filled (open) symbols are obtained accounting for the +inhomogeneous excitation of the probed volume under the assumption of a square root (linear) flu- +ence dependence of the photo-induced changes in the complex refractive index of the material. The +blue shading indicates the change of spectral weight in these quantities after photo-excitation. These +data were acquired at a base temperature T = 100 K with pump pulses tuned to 41 meV (10 THz) +center frequency and excitation fluence of 0.4 mJ cm-2. (c) Same quantities as in (b) but measured +10 ps after photoexcitation at a base temperature T = 295 K. (d) Same quantities as in (c) but meas- +ured 50 ps after photoexcitation. (e) Time dependence of the average reflectivity, average real part +of the optical conductivity &"((), and light-induced “superfluid density” extracted from a two-fluid +model fit and expressed as a fraction of the total charge carrier density. All quantities are evaluated +in the region of the photo-induced gap (5–10 meV). Filled and open symbols indicate the results of +two different reconstruction as in (b). The inset in the top panel highlights the early time delays +region where light amplification (* > 1) is observed (red shading). The red dotted lines indicate the +value of the corresponding quantity at equilibrium. These data were acquired at a base temperature +T = 100 K with pump pulses tuned to 45 meV (11 THz) photon energy and excitation fluence of +0.5 mJ cm-2 + +a +2x3-stageOPAs +Pulsestretchers +WLG +DFG +THz gen. +Ti:SaAmplifier +b +e +Reflectivity +Reflectivity* +.0 +0.5 +0.5 +Equilibrium +Equilibrium +Equilibrium +Photoexcited +Photoexcited +Photoexcited +0.0 +0.0 +900 +T=100K +T= 295K +T= 295K +0% +cm +T = 50 ps +T = 10 ps +T = 50 ps +600 +cm +Gapping +150 +0.5 mJ cm-2 +300 +* +100% +0 +6 +900 +1.0 +600 +ntot +0.5 +nsf +300 +6 +0 +4 +102040 +4102040 +41020 +0 5 10 +50 +100 +Energy (meV) +Energy (meV) +E +Energy (meV) +Time (ps) +7 +This energy range hosts a number of excitations, both vibrational (phonons) and elec- +tronic in nature, including a broad polaronic peak seen in +" centered at approximately +60 meV (15 THz). The possible relevance of this excitation has been highlighted in Ref. 21, +although this prediction could not be tested to date. +To achieve wide tuneability, we made use of a terahertz source based on chirped pulse +difference frequency generation, mixing the near-infrared signal beams of two phase- +locked optical parametric amplifiers22. This source, illustrated schematically in Fig. 3a and +described in detail in supplementary section S4, was used to generate narrow-bandwidth +pulses with photon energies spanning the range from 24 to 145 meV (6-35 THz). All meas- +urements reported here were carried out with an excitation bandwidth of ~4 meV +(1 THz) and ~600 fs pulse duration. The same probing protocol as that reported in Fig. 2 +was utilized here to detect changes in the complex optical properties for probe energies +spanning 4-18 meV (1-4.5 THz). +Figures 3b-d show reflectivity and complex conductivity spectra measured after photoex- +citation with pulses tuned to 41 meV photon energy (l ~ 30 µm, ) ~ 10 THz) at base tem- +peratures of 100 K and room temperature, respectively. Figure 3e displays the time-evo- +lution of the optical properties. The response is very similar to the case reported in Fig. 2 +for 170 meV (41 THz) excitation, manifested on metastable timescales but persisting here +up to room temperature – despite an almost two orders of magnitude weaker excitation +fluence. +Figure 4a shows the scaling with fluence of the below-gap averaged values of 8(-) and ++"(-), as well as the fractional superfluid density in response to photoexcitation at +170 meV (41 THz) and 41 meV (10 THz). These measurements were carried out at a +pump-probe time delay of 10 ps, and thus refer to the metastable phase. The figure shows +how all figures of merit approach their equilibrium superconducting-state values as the + + +8 +fluence increases, with the fluence required being approximately 50 times less for 41 meV +(10 THz) compared to 170 meV (41 THz) excitation. +Similar fluence dependence measurements were carried out by varying the photon en- +ergy of the pump and maintaining a constant 4 meV (1 THz) bandwidth with 600 fs pulse +duration. For all excitation photon energies between 24 meV (6 THz) and 145 meV +(35 THz) the photoinduced changes in the optical properties were qualitatively similar to +those shown in Figs. 2, 3 with only the size of the response for a given fluence differing. +From each fluence dependence we extracted a figure of merit for the photo-susceptibility, + +Figure 4: Scaling of the out-of-equilibrium features of photo-induced metastable supercon- +ductivity in K3C60. (a) Fluence dependence of the average reflectivity, average real part of the op- +tical conductivity &"((), and light-induced “superfluid density” extracted from a two-fluid model fit +and expressed as a fraction of the total charge carrier density. All quantities are evaluated in the +region of the photo-induced gap (5–10 meV). Red and blue symbols indicate measurements with +excitation pulses tuned to 41 meV (10 THz) and 170 meV (41 THz) central frequency. The red dotted +lines indicate the value of the corresponding quantity at equilibrium. These data were acquired at a +base temperature T = 100 K, at a time-delay ∆t = 10ps, and with a pump pulse duration of ~600 fs. +(b) Frequency dependence of the photo-susceptibility of K3C60 defined as the gradient of the lost +spectral weight in &" in the low-fluence limit (Supplementary Section S8) measured 10 ps and 50 ps +after photo-excitation. These measurements were carried out at a base temperature T = 100 K. +These data are obtained by accounting for the inhomogeneous excitation of the probed volume un- +der the assumption of a square root fluence dependence of the photo-induced changes in the com- +plex refractive index of the material. + +41 me +170 meV +9 +defined as the rate of growth of the +" gap with excitation fluence in the limit of low flu- +ence (see supplementary section S8). Plots of the pump-frequency-dependent photo-sus- +ceptibility are shown in Fig. 4b for both 10 ps and 50 ps pump-probe time delay. A peak +centered at 41 meV (10 THz) with approximately 16 meV FWHM bandwidth is observed +in these measurements. In the next section we will discuss three distinct energy scales +which coincide with this resonance, sequentially these relate to “on-ball” orbital excita- +tions, phonons and finally excitons. +Superconductivity in alkali-doped fullerides is believed to be mediated by a dynamical +Jahn-Teller distortion, which leads to an effective negative Hund’s coupling for the orbit- +als of a single buckyball23 and to a low spin S=1/2 state. A theoretical model based on this +assumption has been successful at providing a quantitatively correct phase diagram for +fulleride superconductors, based on ab-initio calculations24,25. Within this model, the local +ground state of the system is a six-fold degenerate low-spin state, which features intra- +orbital pairs that de-localize over two molecular orbitals. As detailed in supplementary +section S10, a first set of local excited states also features such pairs, albeit with a different +angular momentum (i.e. a different inter-orbital phase for the delocalized pair). Ab-initio +calculations predict an energy splitting of 37 meV between these two sets of states24,25. +The observed resonance may therefore be related to the creation of interorbital pairs with +local angular momentum, which may also contribute to superconductivity, as suggested +in the Suhl-Kondo mechanism26,27. However, it is not yet clear how exactly this excitation +is transformed in the presence of tunneling between neighboring C60 molecules, and why +the creation of such pairs may support metastable superconductivity at such high tem- +peratures. Furthermore, as the local parity of this excited state would be different from +that of the ground state, condensation in this configuration may give rise to a supercon- +ductor with different symmetry. This possibility, whilst tantalizing, remains speculative +and should be tested with more comprehensive ultrafast probing methods. + + +10 +Turning to phonon excitations, we also note that the 41 meV resonance frequency identi- +fied here coincides with an infrared-active T1u phonon which predominantly consists of +intramolecular motion of the C atoms. While the atomic motions of the 170 meV molecular +mode discussed previously in Ref. 12 are directed along the tangential directions of the +C603- molecule, those of the 41 meV mode are predominantly along the radial directions +(see supplementary section S9). By performing frozen phonon calculations using density +functional theory (DFT) we evaluated the different impact of these distortions on the +three t1u molecular levels at the Fermi energy, which we map out from DFT wave functions +as maximally-localized Wannier functions (supplementary section S9). In the undistorted +C603- structure these molecular levels are degenerate. Applying a distortion along a T1u +coordinate lifts this degeneracy leaving a doubly degenerate t1u orbital lowered in energy. +This electronic configuration is prone to developing a Jahn-Teller distortion that may lead +to an enhanced negative Hund’s coupling, possibly facilitating the onset of superconduc- +tivity at higher temperatures. +The strength of the induced splitting is quadratic in the phonon coordinate and is more +significant for when driving the 41 meV mode compared to the 170 meV one, suggesting +that the observed resonance may arise from a more efficient manipulation of the elec- +tronic degrees of freedom when driving the 41 meV T1u mode. +Finally, we address the electronic excitations discussed in Ref. 21, in which the existence +of a polaronic mode was predicted at the same energy scales as the resonance reported +here. However, we also note that the proposed mechanism for the formation of the non- +equilibrium superconducting-like state was one in which the quasi-particles are cooled +incoherently via coupling to the polaronic bath. As already reported in Ref. 28 and shown +here in the inset to Fig. 3e, the response of the sample in the first few picoseconds after +photoexcitation yields amplification of the terahertz probe light, which is likely to reflect +coherent dynamics of the driven degrees of freedom. Assuming that the mechanism + + +11 +proposed in Ref. 21 were to be valid, the early-time dynamics of that model would require +further investigation to understand how such coherences would arise at early times. The +amplification observed here and in Ref. 28 has so far been attributed to the existence of a +parametric resonance that couples amplitude (Higgs) modes to phase (Goldstone) modes, +an effect possible at the sample surface because of reduced screening. +We expect the significance of this discovery to be capitalized upon in future work. The +extreme efficiency improvement due to resonant enhancement, nearing two orders of +magnitude, is expected to also dramatically reduce unwanted dissipation. This, taken in +conjunction with the observed nanosecond-long lifetime suggests that excitation of the +sample with a train of pulses of only 400 µJ/cm2 delivered at 100 MHz repetition rate – as +determined by the inverse lifetime of this state - may yield continuous wave operation. +Because this effect is documented here to persist up to room temperature, continuous +wave operation would likely have important practical implications. To make this regime +experimentally accessible, single order-of-magnitude improvements in the efficiency of +the process, or in the light matter coupling strength, combined with suitable develop- +ments in high repetition rate THz sources would be required. + +Acknowledgments +The research leading to these results received funding from the European Research Council under +the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement +No. 319286 (QMAC). We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) +via the Cluster of Excellence ‘The Hamburg Centre for Ultrafast Imaging’ (EXC 1074 – project ID +194651731). We thank Michael Volkmann and Peter Licht for their technical assistance. We are +also grateful to Boris Fiedler and Birger Höhling for their support in the fabrication of the elec- +tronic devices used on the measurement setup, and to Jörg Harms for assistance with graphics. + + + + +12 +References +1 +Nova, T. F., Disa, A. 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Higgs-Mediated Optical Amplification in a Nonequilibrium Superconductor. +Physical Review X 11, 011055, (2021). + + + + +14 +Giant resonant enhancement for photo-induced +superconductivity in K3C60 +E. Rowe1,*, B. Yuan1, M. Buzzi1, G. Jotzu1, Y. Zhu1, M. Fechner1, M. Först1, B. Liu1,2 +D. Pontiroli3, M. Riccò3, A. Cavalleri1,4,* + +1 Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany +2 Paul Scherrer Institute, Villigen, Switzerland +3 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Italy +4 Department of Physics, Clarendon Laboratory, University of Oxford, United Kingdom +* e-mail: edward.rowe@mpsd.mpg.de, andrea.cavalleri@mpsd.mpg.de + + +Supplemental Material +S1. Sample growth and characterization +S2. Determination of the equilibrium optical properties +S3. High fluence mid-infrared source +S4. Frequency-tunable narrowband terahertz and mid-infrared source +S5. Measurements of the transient THz reflectivity +S6. Determination of the transient optical properties +S7. Fitting the transient optical spectra +S8. Extracting the frequency-dependent photosusceptibility +S9. Density functional theory calculations +S10. Local electronic hamiltonian calculations + + + + + + + +15 +S1. Sample growth and characterization + +The K3C60 powder pellets used in this work were prepared and characterized as reported +previously1-3. Stoichiometric amounts of ground C60 powder and potassium were placed +in a sealed pyrex vial, which was evacuated to a pressure of 10-6 mbar. Whilst keeping the +C60 powder and solid potassium separated, the vial was kept at 523 K for 72 h and then +at 623 K for 28 h such that the C60 powder was exposed to pure potassium vapor. The vial +was then opened inside an Ar glovebox (<0.1 ppm O2 and H2O), where the powder was +reground and pelletized before annealing at 623K for 5 days. X-ray diffraction +measurements were then carried out on the resulting K3C60 powder, which confirmed +that it was phase pure, with an average grain size ranging between 100 and 400 nm. The +static superconducting transition temperature was measured to be 19.8 K (in agreement +with literature values) via magnetic susceptibility measurements upon zero field cooling +and cooling in field with a field strength of 400 A/m. + + + +Figure S1.1: a. X-ray diffraction data and single f.c.c. phase Rietveld refinement for the K3C60 +powder used in this work. b. Temperature dependence of the sample magnetic susceptibility +measured by SQUID magnetometry upon cooling without (ZFC: zero field cooling) and with a +magnetic field applied (FCC: field cooled cooling). + + + +observed +(10emu/(g0e) +calculated +residual +reflections +20 +ZFC +FCC +30 +40 +50 +10 +20 +30 +40 +50 +60 +5 +10 +15 +20 +2A +tdearees +Temperature + +16 +S2. Determination of the equilibrium optical properties + +The equilibrium reflectivity was measured for photon energies between 5 meV and 500 +meV using a commercial Fourier-transform infrared spectrometer (FTIR) equipped with +a microscope at the SISSI beamline in the Elettra Synchrotron Facility (Trieste, Italy), as +reported previously1-3. The sample was pressed by a diamond window into a sealed +holder in order to obtain an optically flat interface and prevent exposure to air. This +procedure was carried out inside an Ar filled glove box (<0.1 ppm O2 and H2O) before the +sealed sample was removed and mounted on a He cooled cryostat to enable temperature +dependent measurements. The K3C60 reflectivity spectra were referenced against a gold +mirror placed at the sample position. +In order to extract the complex optical conductivity a Kramers-Kronig algorithm for +samples in contact with a transparent window4 was used. This requires data at all +frequencies, which were obtained, at low energies (<5 meV) using an extrapolation based +on a Drude-Lorentz fit, and at high energies (>500 meV) using data measured on single +crystal samples reported in Refs. 5,6. +The equilibrium properties are shown in figure S2.1 for temperatures of 100 K and 300 +K. This and further data measured at different temperatures and pressures were already +reported in Refs. 1,2 and discussed also in comparison with data obtained from single +crystals. +These data were fitted with a Drude-Lorentz model, which is given by the following +equation: +𝜎!(𝜔) + 𝑖𝜎"(𝜔) = 𝜔#" +4𝜋 +1 +𝛾$ − 𝑖𝜔 + 𝜔#,&'( +" +4𝜋 +𝜔 +𝑖.𝜔),&'( +" +− 𝜔"/ + 𝛾&'(𝜔 +Here the first term represents the Drude response of the free carriers with 𝜔# and 𝛾$ +representing the plasma frequency and scattering rate respectively, whereas the second +term captures the mid infrared absorption in the form of a Lorentz oscillator centered at +frequency 𝜔),&'( with plasma frequency 𝜔#,&'( and damping rate 𝛾&'(. The equilibrium +data reported here was used to normalize the transient optical spectra of K3C60 measured +upon photoexcitation, as discussed in detail in section S6. + + + + + +17 +Figure S2.1: Equilibrium optical properties (reflectivity, real, and imaginary part of the optical +conductivity) of K3C60 measured at a temperature of 100 K (blue) and 300 K (green). The black +dashed curve is a Drude-Lorentz fit to the optical conductivity at 100 K in the range from 3 meV +to 60 meV as described in the text. + +S3. High fluence mid-infrared source + +For the data reported in figure 2 and in figure 4(a) at 170 meV (41 THz) excitation, the +pump pulses were generated via difference frequency mixing (DFG) of the signal and idler +output of a three-stage home-built optical parametric amplifier (OPA). A commercial +Ti:Al2O3 amplifier delivering 60 fs duration pulses at 800 nm central wavelength was used +to drive the OPA, and the DFG process was performed using a 0.5 mm thick GaSe crystal, +resulting in ~100 fs long pulses. The 170 meV pulses were then propagated through a +highly dispersive 16 mm long CaF2 rod, stretching their duration to ~1 ps. The spectrum +of the pump pulses was characterized using a home built FTIR spectrometer. Their +duration was measured by cross-correlation with a synchronized, 35 fs long, 800 nm +wavelength pulse in a 50 μm thick GaSe crystal. While a certain degree of tunability is also +given by this source, its useful operation range spans between 80 and 320 meV, hence it +was only used for the high-intensity experiments at 170 meV excitation. + + + + + +900 +E +900㎡ +1.0 +300 K +100K +Reflectivity +T- +T- +600 +600 +Fit 100 K +0.5 +300 +300 +02 +0.04 +0 +10 +30 +100 +410 +30 +100 +4 +10 +30 +100 +Energy (meV) +Energy (meV) +Energy (meV) + +18 +S4. Frequency-tunable narrowband terahertz and mid-infrared source + +For the experiments that required tunability of the excitation pulses down to the THz gap, +a different source was used. This source is based on the principle of chirped-pulse +difference frequency generation (CP-DFG) in organic non-linear optical crystals, namely +DAST and DSTMS of approximately 600 μm thickness. The principle of operation of this +new source is described in detail in Ref. 7. A commercial Ti:Al2O3 amplifier is used to drive +two identical three-stage OPAs which are seeded by the same white-light, such that the +signal beams have the same phase-fluctuations. The ~100 fs signal pulses are then +chirped using a pair of transmission-grating-based stretchers as depicted in figure 3(a). +This arrangement enables continuous tuning of the pulse durations by varying the +distance between the gratings in each pair, effectively enabling continuous tuning of the +pump-pulse bandwidth. For this experiment the pump pulse bandwidth was kept +constant at 4 meV by maintaining a signal pulse duration of ~600 fs, as measured using a +home-built second harmonic-based Frequency-Resolved-Optical-Gating (FROG) device. +Frequency tuning of the generated excitation pulses was carried out both by varying the +central wavelengths of the two OPA signal beams, and by varying the time delay between +the chirped signal pulses in the DFG crystal (for fine tuning). For each measurement the +pump frequency spectrum was measured via FTIR (Fourier Transform Infrared +Spectroscopy). + +S5. Measurements of the transient THz reflectivity + +The experiments presented in Figures 2, 3, and 4 were performed on compacted K3C60 +powder pellets pressed against a diamond window to ensure an optically flat interface. +As K3C60 is water and oxygen sensitive, the pellets were sealed in an air-tight holder and +all sample handling operations were performed in an Argon filled glove box with <0.1 +ppm O2 and H2O. The sample holder was then installed at the end of a commercial Helium +cold-finger (base temperature 5K), to cool the pellets down to a temperature of 100 K. +The changes in the properties of the sample following photoexcitation were measured +using time-domain THz-spectroscopy. + + + +19 +The mid-infrared pump induced changes in the low frequency optical properties, were +retrieved using transient THz time domain spectroscopy in two different experimental +setups. The THz probe pulses were generated via optical rectification in a 0.2 mm thick +(110)-cut GaP crystal starting from 800 nm pulses with a duration of ~80 fs and 35 fs, +respectively. Whilst in one setup these 800 nm were derived from the same laser used +for pumping the source described in section S4, the 35 fs, 800 nm pulses were generated +by a second Ti:Al2O3 amplifier optically synchronized to that used to pump the high- +intensity mid-infrared source described in section S3. The THz probe pulses were then +focused onto the sample with incidence angles of 30 and 0 degrees, respectively. After +reflection from the sample, the electric field profile of the THz pulses was reconstructed +in a standard electro-optic sampling setup, using a (110)-cut 0.2 mm GaP crystal +supported on a 1 mm thick (100)-cut GaP substrate to delay internal reflections. The +setup combined with the frequency tunable narrowband source had a measurement +bandwidth that extended between 4 and 18 meV, while the other spanned between +4 meV to 29 meV. The time resolution of both setups is determined by the measurement +bandwidth and is ~250 fs and ~150 fs respectively. +To minimize the effects on the pump-probe time resolution due to the finite duration of +the THz probe pulse, the experiments were performed as described in Refs. 8, 9. The +pump-probe time delay was controlled by fixing the delay between the 800 nm gating +pulse and the mid-infrared pump pulse 𝜏. The transient THz field was then obtained by +scanning the delay 𝑡 relative to both. +In order to simultaneously retrieve both the ‘pump on’ (𝐸*+, +&- (𝑡, 𝜏)) and ‘pump off’ +(𝐸*+, +&..(𝑡)) probe fields, a differential chopping scheme was deployed. The scheme was +different for the two above mentioned setup. For the narrowband, frequency tunable +setup which operated at a repetition rate of 1 kHz, the THz probe pulse was chopped at a +frequency of 500 Hz and the mid-infrared pump pulse was chopped at ~ 357 Hz. The +electro-optic sampling signal was then fed to two lock-in amplifiers reading out 𝑉/01! at +500 Hz and 𝑉/01" at 143 Hz respectively. For the high-intensity setup, operating at 2 kHz +repetition rate, the THz probe pulse was chopped at a frequency of 1 kHz and the mid- +infrared pump was chopped at 500 Hz. In this case, the electro-optic sampling signal was +filtered by two lock-in amplifiers operating at 1 kHz and 500 Hz respectively. 𝐸*+, +&..(𝑡) and +Δ𝐸*+,(𝑡, 𝜏) were then extracted from the signals in the two lock-ins using the following +formulas: + + + +20 +𝐸*+, +&..(𝑡) = 𝑉𝐿𝐼𝐴1(𝑡, 𝜏) − 𝛼𝑉𝐿𝐼𝐴2(𝑡, 𝜏) +Δ𝐸*+,(𝑡, 𝜏) = 𝐸*+, +&- (𝑡, 𝜏) − 𝐸*+, +&..(𝑡) = 𝛼𝑉𝐿𝐼𝐴2(𝑡, 𝜏) + +where 𝛼 is a calibration constant determined experimentally on an InSb reference +sample. This is done by extracting Δ𝐸*+,(𝑡, 𝜏) as the difference of two separate +measurements of 𝐸*+, +&- (𝑡, 0) and 𝐸*+, +&..(𝑡) performed with the first lock-in amplifier and by +chopping only the THz probe pulse while leaving the mid-infrared pump pulse either +always on or always off. Equating the value of Δ𝐸*+,(𝑡, 𝜏) determined in this way to the +one with differential chopping yields the calibration constant. + +S6. Determination of the transient optical properties + +From the measured changes in the reflected probe field (see section S5), the transient +complex reflection coefficient of the sample 𝑟̃(𝜔, 𝜏) can be determined by taking the +Fourier transform along t of both 𝐸*+, +&..(𝑡) and Δ𝐸*+,(𝑡, 𝜏) and using the following +equation: + +Δ𝐸:*+,(𝜔, 𝜏) +𝐸:*+, +&..(𝜔) += 𝑟̃(𝜔, 𝜏) − 𝑟̃)(𝜔) +𝑟̃)(𝜔) + + +where 𝑟̃)(𝜔) is the equilibrium complex reflection coefficient, obtained as described in +section S2. +In the cases where the pump light penetrates in the sample several times deeper than the +probe light, one can assume that the probe pulse samples a volume in the material that +has been homogeneously excited by the pump. In this case, it is possible to directly extract +the complex-valued optical response functions by inverting the Fresnel equations. +However, in K3C60 the penetration depth of the probe electric field (~600-900 nm) +exceeds that of the pump (~500 nm at 10 THz, ~200 nm at 41 THz), such that the probe +interrogates an inhomogeneously excited volume (Figure S6.1(a)). + + + +21 +As the pump penetrates into the material, its intensity is reduced, and it will induce +progressively weaker changes in the refractive index of the sample. This situation is +modeled by “slicing” the probed thickness of the material into thin layers (figure S6.1(b)), +where we assume that the pump-induced changes in the refractive index ∆𝑛= scale +according to the pump intensity in the layer, i.e. 𝑛=(𝜔, 𝑧, 𝜏) = 𝑛=)(𝜔) + ∆𝑛=(𝜔, 𝜏, 𝐼(𝑧)). The +pump intensity 𝐼(𝑧) is assumed to follow the dependence 𝐼(𝑧) = 𝐼)𝑒2,/4!"#!, where +𝑑#56# = 𝜆#56# 4𝜋𝐼𝑚 D𝑛).𝜔#56#/E +F +. Here, the refractive index of the material at the +pump frequency, 𝑛).𝜔#56#/ is taken to be the one at equilibrium. Additionally, an +assumption is made on the functional form for the dependence of ∆𝑛= on the pump +intensity. Here, we consider two different forms given by: +(1) ∆𝑛=(𝜔, 𝜏, 𝑧) ∝ 𝐼(𝑧) +(2) ∆𝑛=(𝜔, 𝜏, 𝑧) ∝ H𝐼(𝑧) +Respectively, these equations result in the following depth-dependent functional forms +for the spatial profile of the refractive index: +(1) 𝑛=(𝑧, 𝜔, 𝜏) = 𝑛=)(𝜔) + Δ𝑛=(𝜔, 𝜏)𝑒2,/4!"#! +(2) 𝑛=(𝑧, 𝜔, 𝜏) = 𝑛=)(𝜔) + ∆𝑛=(𝜔, 𝜏)𝑒2,/"4!"#! +where Δ𝑛=(𝜔, 𝜏) represents the pump-induced change in the refractive index of the +material at the sample surface. + +Figure S6.1: a. Schematics of pump-probe penetration depth mismatch. b. Multi-layer model +with exponential decay used to calculate the pump-induced changes in the complex refractive +index 𝑛#(𝜔, 𝜏) for each pump-probe delay 𝜏. The transition from red to background (grey) +represents the decaying pump-induced changes in 𝑛#(𝜔, 𝑧). + +Sample +Sample +Probe +Pump + +22 +For each time delay 𝜏 and probe frequency 𝜔7, the complex reflection coefficient 𝑟̃(∆𝑛=) of +the multilayer stack described above is calculated using the transfer matrix method10, +keeping ∆𝑛= as a free parameter. To numerically extract the value of ∆𝑛=(𝜔, 𝜏) we minimize +the following function: + +IΔ𝐸:*+,(𝜔7) +𝐸:*+, +&..(𝜔7) +− 𝑟̃(𝜔7, Δn) − 𝑟̃)(𝜔7) +𝑟̃)(𝜔7) +I + +By then taking 𝑛=(𝜔, 𝜏) = 𝑛=)(𝜔) + Δ𝑛=(𝜔, 𝜏), one obtains the refractive index of the +material as if it had been homogeneously excited. From 𝑛=(𝜔, 𝜏) we then calculate 𝑅(𝜔, 𝜏), +𝜎!(𝜔, 𝜏) and 𝜎"(𝜔, 𝜏) as plotted in the main text. +Figures S6.2 and S6.3 display extended data sets measured at increasing pump-probe +delays with pump photon energies of 170 meV (41 THz) and 45 meV (11 THz) +respectively. Therein we report reflectivity (sample-diamond interface), real and +imaginary part of the optical conductivity after reconstruction under the assumptions of +models (1) and (2), identified with hollow and filled circles respectively. +At early delays, for both excitation mechanisms and reconstruction assumptions, the +reconstructed reflectivity is higher than one, and the real part of the optical conductivity +is negative, indicative of amplification of the incoming THz probe radiation, as discussed +previously in Ref. 11. In all cases, this non-equilibrium driven state then relaxes into a +superconducting-like state with a fully gapped 𝜎!(𝜔) and a divergence ∝ 1 𝜔 +⁄ in the +𝜎"(𝜔) spectrum. At even later delays the optical spectra are those of a finite temperature +superconductor. These optical properties can be interpreted in the context of a two fluid +model, in which a varying density of uncondensed quasi-particles also contributes to the +terahertz response. +Importantly the time-evolution of K3C60 following photo-excitation is independent of the +used reconstruction, and only the specific values of pump-probe delay up to which +amplification, fully gapped superconductor, and finite temperature superconductor +appear are affected by this choice. + + + +23 + +Figure S6.2: Comparison of linear and sub-linear reconstruction in the transient optical spectra +at 170 meV (41 THz) pump-photon energy. Reflectivity (sample-diamond interface), real, and +imaginary parts of the optical conductivity measured at equilibrium (red lines) and after +photoexcitation (blue symbols) at increasing pump-probe time delays indicated in the figure. The data +in filled (open) symbols reconstructed under the assumption of a square-root (linear) fluence +dependence of the changes in complex refractive index of the material. These data were measured at +18 mJ cm-2 excitation fluence, and at a base temperature of 100 K. + +0 ps +1 ps +2 ps +5 ps +10 ps +50 ps + +24 + +Figure S6.3: Comparison of linear and sub-linear reconstruction in the transient optical spectra +at 45 meV (11 THz) pump-photon energy. Reflectivity (sample-diamond interface), real, and +imaginary parts of the optical conductivity measured at equilibrium (red lines) and after +photoexcitation (blue symbols) at increasing pump-probe time delays indicated in the figure. The data +in filled (open) symbols reconstructed under the assumption of a square-root (linear) fluence +dependence of the changes in complex refractive index of the material. These data were measured at +0.5 mJ cm-2 excitation fluence, and at a base temperature of 100 K. + +1.5 ps +3.5 ps +5.5 ps +11.5 ps +50.5 ps + +25 +S7. Fitting of the transient optical spectra + +The transient optical conductivity spectra presented in figures 2-3 as well as for each +fluence in figure 4 were fitted with a two-fluid model according to the following equation: +𝜎=(𝜔, 𝜏) = 𝜋 +2 +Λ'(𝜏) 𝑒" +𝑚 +𝛿[𝜔 = 0] + 𝑖 Λ'(𝜏) 𝑒" +𝑚 +1 +𝜔 ++ Λ-(𝜏) 𝑒" +𝑚 +1 +𝛾$ − 𝑖𝜔 ++ R +𝐵-𝜔 +𝑖(Ω-" − 𝜔") + 𝛾-𝜔 +" +-8! + + +Here the first term captures the frequency dependent contribution from the +supercarriers with density Λ', the second term captures the Drude contribution of the +normal carriers with density Λ- and scattering rate 𝛾$. Finally, we include a sum over + +Figure S7.1: Two-fluid fit to the transient spectrum. Reflectivity, real (𝜎!) and imaginary (𝜎") +parts of the optical conductivity measured in equilibrium at 100 K (red) and 50 ps after +photoexcitation with a fluence of 0.5 mJ cm-2 at 45 meV (11 THz) photon energy. The fit to the +equilibrium data using the procedure described in this section is shown as a dashed black line and +gives zero superfluid density. The two-fluid fit to the transient data generated using the same +procedure is shown as a solid blue line and returns a superfluid fraction Λ# (Λ$ + Λ#) +⁄ += 73%. The +data in this figure was reconstructed under the assumption of a square root dependence of the +change in refractive index on excitation fluence (see supplementary section S6). + + + +26 +two Lorentz oscillators in order to capture the broad midinfrared absorption peak +centered at around 60 meV. +The transient data are fitted at each delay 𝜏 using the parameter-set that captures the +equilibrium optical conductivity spectra as a starting condition, and leaving only Λ' and +Λ- free to vary, as though the effect of the pump is to simply convert carriers from the +normal to the superconducting fluid. +Figure S7.1 shows representative fits to transient data measured at 100 K base +temperature and at 50 ps time delay, as well as to the 100 K equilibrium spectra. +Importantly, while the fit of the equilibrium data converges to a superfluid fraction +Λ' (Λ- + Λ') +⁄ + which is equal to zero, the fit to the transient data yields Λ' (Λ- +⁄ ++ Λ') = +0.73. The transient optical data was fitted at each time delay and driving frequency, +yielding the time and frequency dependence of the superfluid fractions shown in figures +2(e), 3(e), and 4(a). + +S8. Extracting the frequency-dependent photosusceptibility + +In figure 4(b) we introduce a figure of merit, referred to as the ‘photosusceptibility’, +which can be used to quantitatively compare the efficiency with which the metastable +light-induced superconducting state is generated in K3C60 for different excitation +frequencies. +For each excitation photon energy, transient optical spectra were measured at different +excitation fluences ℱ. From these fluence dependent spectra we extract the loss in +spectral weight of 𝜎!(𝜔) after photoexcitation in the 5-10 meV spectral range, calculated +as: +𝑆𝑊𝐿(ℱ) = Z +𝜎! +9:(𝜔) − 𝜎! +#;&<&(𝜔, ℱ) 𝑑𝜔 +!) meV/ℏ +B meV/ℏ + +where 𝜎! +9:(𝜔) and 𝜎! +#;&<&(𝜔, ℱ) are the 𝜎!(𝜔) spectra measured in equilibrium and upon +photoexcitation respectively. The 𝑆𝑊𝐿(ℱ) data is then fitted with the following +phenomenological function: +𝐴 \ +1 +1 + 𝐵𝑒2CDℱ +1 +− 1 +2] + + + +27 +where ℱ represents the excitation fluence and 𝐴, 𝐵 are free parameters. The +‘photosusceptibility’ plotted in figure 4(b) is equal to 𝐵, which is the gradient of this +function evaluated at zero fluence. Figure S8.1 shows the fluence-dependent data and +corresponding fit for one exemplary dataset. + +Figure S8.1: Extracting photosusceptibility from the fluence-dependent data. Lost spectral +weight in the real part of the optical conductivity between 5 and 10 meV as a function of fluence (red +circles), measured 10 ps after photoexcitation at 100 K with a pump spectrum centered at 41 meV (10 +THz). The fit is shown as a solid green line, with the gradient at zero fluence (which we define as the +photosusceptibility) shown as a dashed blue line. The data in this figure was reconstructed under the +assumption of a square root dependence of the change in refractive index on excitation fluence (see +supplementary section S6). + +S9. Density functional theory calculations + +In this section, we address how the displacement of phonon modes affects the electronic +properties of K3C60. Specifically, we consider the molecular orbitals and their response to +the change in the crystal structure. To carry out this investigation, a first-principles +approach based on density functional theory (DFT) was used. The starting point is the +unit cell of K3C60 containing sixty carbon and three potassium atoms. Before computing + + + +28 +the phonon spectrum, this unit cell is structurally relaxed, and the resulting lattice +constants and atomic coordinates are listed in table S9.1. +Next, the phonon spectrum of K3C60 is computed from the force constant matrix utilizing +a finite displacement approach12. In total, there are 186 non-translational phonon modes +covering the symmetries of point group m-3. Specifically, there are 24 Tu, 7 Eu, 23 Tg, 8 Eg, +and 8 Ag modes. Note that only the modes of Tu character are infrared active, and we list +their computed frequencies in the table S9.2. +We utilized a frozen phonon approach to estimate the impact of these distortions on the +molecular levels. Therefore, we modulated our equilibrium crystal structure with the +eigen-displacements of these modes. We then created a low energy Hamiltonian for these +structures by computing the maximally localized Wannier functions for the valence band +electrons. Note that since the three valence bands are well separated in energy from other +orbital-like bands our method does not require a disentanglement procedure. +Our calculations focused on the three degenerate t1u molecular levels at the Fermi energy, +which we mapped out from DFT wave functions as maximally-localized Wannier +functions. In the equilibrium structure, the onsite energy of these molecular levels is +degenerate; however, deforming the crystal by applying a T1u polar distortion lifts this +degeneracy. Thereby, similar to a Jahn-Teller distortion, the symmetry breaking of the +crystal structure splits the level into a double and a single degenerate orbital. For the 43.2 +meV and 173.4 meV phonon modes, this splitting manifests as a lowering in the energy +of the double degenerate orbital. A schematic visualization of this is depicted in the inset +to figure S9.1(a). Diagrams illustrating the distortion of the C60 molecule for the 43.2 meV +Lattice vectors +a +14.175 Å +Alpha +90˚ +90˚ +90˚ +b +14.175 Å +Beta +c +14.175 Å +Gamma +Atomic positions according to Space Group 202 (Fm-3) +Element +Wykoff label +X +y +c +C +H +0.00000 +0.54991 +0.24682 +C +I +0.58242 +0.10057 +0.21408 +C +I +0.66275 +0.05092 +0.18294 +K +C +0.25000 +0.25000 +0.25000 +K +A +0.00000 +0.00000 +0.00000 + +Table S9.1: Structural parameters of K3C60 from first-principles computations + + + +29 +and 173.4 meV modes (labelled ‘A’ and ‘B’ and corresponding to mode numbers 4 and 21 +in table S9.2 respectively) are shown in figure S9.1(b). +Besides this qualitative difference of the phonon-mode distortion on the molecular levels, +we also examined the strength of the induced splitting. From group-theory, the size of the +splitting scales with the square of the distortion. Figure S9.1(a) displays how the splitting +develops as a function of the fluence of the incoming THz pulse. Each phonon mode +distortion was weighted according to its eigenfrequency and mode effective charge in this + plot. For the same strength of the driving electric field, the splitting induced by phonon +A produces a more significant separation of the t1u levels compared to phonon B. Due to +the square scaling of the splitting with the electric field, this effect is further enhanced at +higher field strengths. + The computations were performed with the Vienna ab-initio simulation package +VASP.6.213-15. For the phonon calculations, we used the Phonopy software package16 and +the Wannier90 package for wannierization12. The computations further utilized +Number: +ℎ𝜈#56# (meV) +1 +2.2 +2 +14.1 +3 +42.4 +4 (A) +43.2 +5 +48.3 +6 +60.5 +7 +62.6 +8 +71.5 +9 +80.6 +10 +83.9 +11 +85.9 +12 +91.1 +13 +92.0 +14 +118.6 +15 +122.8 +16 +147.5 +17 +148.3 +18 +149.8 +19 +163.7 +20 +165.4 +21 (B) +173.4 +22 +176.9 +23 +184.8 +24 +185.3 + +Table S9.2: List of the IR active phonon modes of Tu symmetry. + + + +30 +pseudopotentials generated within the Projected Augmented Wave (PAW) method16. +Specifically, the following default potentials were used: C 2s22p2 and K 3s23p64s1. The +Generalized Gradient Approximation (GGA17) approximation for the exchange- +correlation potential was used. For the final numerical setting, a 4x4x4 Monkhorst18 +generated k-point-mesh sampling of the Brillouin zone and a plane-wave energy cutoff of +600 eV were chosen. The calculations were re-iterated self-consistently until the change +in total energy converged within 10-8 eV. + + +Figure S9.1: Effect of vibrational distortions on the t1u molecular levels from first-principle +computations. (a) shows the induced splitting of the molecular orbital of t1u symmetry at the Fermi +energy (as illustrated by the inset) as a function of drive fluence. The two curves represent the effect +of the two distinct T1u IR-phonon modes with eigenfrequencies of 43.2 (red) and 173.4 (blue) meV. +The eigen displacement of these modes are shown in (b). Note, that due to the symmetry character of +the phonon modes the t1u level split into a single and double degenerate orbital. Lastly, in (c) we show +the induced splitting as a function of frequency for a fixed fluence. Here we consider the whole +spectrum of T1u IR modes of K3C60, as listed in table S9.2. + +S10. Local electronic hamiltonian calculations + +The Hamiltonian proposed in Ref. 19 in order to model superconductivity in alkali-doped +fullerides is based on an effective negative Hund’s coupling J. It arises from a combination of +the usual Hund’s coupling with a dynamical Jahn-Teller distortion. This causes states featuring +intra-orbital pairing on a buckyball to be energetically favourable. Using ab-initio calculations, + +B + +31 +values of the intra-orbital interaction U = 0.826 eV and of J = −18.5meV were predicted for +K3C6020. The phase diagram for the A3C60 family of compounds was computed using DMFT +starting from this Hamiltonian and was found to be in quantitative agreement with experimental +data21. +The Hamiltonian can be written as: + +𝐻 = 𝐻Intra + 𝐻Inter + 𝐻Pairhop + 𝐻Spinswap + +with an intra-orbital interaction with magnitude U given by: + +𝐻Intra = 𝑈 R 𝑛7,↑𝑛7,↓ +U +7 + + +where 𝑛7,V = 𝑎7,V +W 𝑎7,V is the number operator for a spin down electron on orbital i with spin 𝜎 ∈ +{↑, ↓}. 𝑎7,V +W and 𝑎7,V are fermion creation and annihilation operators, respectively. The inter- +orbital interaction appears as: + +𝐻Inter = (𝑈 − 2𝐽) R R .1 − δ7X/𝑛7,↑𝑛X,↓ +U +X +U +7 ++ (𝑈 − 3𝐽) R R R +𝑛7,V𝑛X,V +72! +X +U +7 +V + + +with δ7X denoting the Kronecker delta. which, given that J is negative, makes these terms higher +in energy. In addition, there is a pair hopping term, which corresponds to a transfer of a pair of +electrons from one orbital to another. It is given by: + +𝐻Pairhop = 𝐽 R R .1 − δ7X/𝑎7,↑ +W 𝑎7,↓ +W 𝑎X,↓𝑎X,↑ +U +X +U +7 + + +This term was found to be crucial for the appearance of superconductivity21. Finally, there is a +“spin +swapping” +term, +where +two +opposite +spins +exchange +orbitals: + +−𝐽 R R .1 − δ7X/𝑎7,↑ +W 𝑎7,↓𝑎X,↓ +W 𝑎X,↑ +U +X +U +7 + + + + + +32 +When restricting ourselves to a Hilbert space where the three degenerate orbitals are populated +by three electrons (as is appropriate for A3C60 in the atomic limit), we can use a basis given by +the different possible arrangements in which the orbitals can be populated: + +{|↑, ↑↓ ,0⟩,|↑ ,0, ↑↓⟩,|↑↓, ↑ ,0⟩,|0, ↑, ↑↓⟩,|↑↓ ,0, ↑⟩,|0, ↑↓, ↑⟩, |↓, ↑, ↑⟩,|↑, ↓, ↑⟩,|↑, ↑, ↓⟩,|↑, ↑, ↑⟩} + +as well as a second set of states created by flipping all spins in the set above. +In this basis, the Hamiltonian takes on the form: + +𝐻m − (3𝑈 + 5𝐽)𝐼o = −𝐽 +⎝ +⎜ +⎜ +⎜ +⎜ +⎜ +⎜ +⎛ +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 ++1 +0 +0 +0 +0 +0 +0 +0 +0 ++1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 ++2 +−1 ++1 +0 +0 +0 +0 +0 +0 +0 +−1 ++2 +−1 +0 +0 +0 +0 +0 +0 +0 ++1 +−1 ++2 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 ++4⎠ +⎟ +⎟ +⎟ +⎟ +⎟ +⎟ +⎞ + + +where 𝐼o is the identity matrix, which encodes an overall energy offset. This matrix is block- +diagonal, meaning that different sectors of the Hilbert space are not coupled to each other: For +example, there is no term that destroys or creates pairs. Because of the inverted Hund’s +coupling, i.e. because J is negative, the stretched state |↑, ↑, ↑⟩ as well as its global spin-flip +partner |↓, ↓, ↓⟩ are now the most energetic local eigenstates. +The local ground state is 6-fold degenerate, with an exemplary instance given by: +|𝑔!⟩ = (|↑, ↑↓ ,0⟩ +|↑ ,0, ↑↓⟩)/√2, i.e. it is a state where one singlet pair of electrons has de- +localized over two orbitals. +The first excited manifold is 10-fold degenerate. Six of those states are of the type +|𝑒!⟩ = ((|↑, ↑↓ ,0⟩ −|↑ ,0, ↑↓⟩)/√2 i.e. identical to the ground state except for the phase of the +de-localized singlet pair (and hence corresponding to a different local angular momentum) – +as illustrated in Figure S10.1. +The energy difference between these two manifolds is given by 2J=37meV, remarkably close +to the observed resonance in the experiment. However, several questions remain in order to +determine whether an excitation of this transition is responsible for the experimental +observation: + + + +33 +Firstly, how does the light field of the laser couple to +this excitation? As the size of a buckyball is +comparable to the distance between buckyballs both +inter-site and intra-site driving terms may be +comparable in terms of the associated energy. +Understanding possible inter-site driving terms +(arising from the oscillating energy difference +between neighbouring sites, given by the electric +field multiplied with the charge and the lattice +spacing) will require a calculation featuring multiple +buckyballs. Locally, because the dynamical Jahn- +Teller distortion causes the populated orbitals to be +superpositions of several undistorted orbitals, we +may expect the electric field to lift the orbital +degeneracy, for example through an orbital offset +term of the type 𝐻offset = Δ(𝑛U,↑ + 𝑛U,↓), where Δ +encodes the amplitude of the drive and is oscillating +in time. Such a term would in fact cause an excitation from |𝑒!⟩ to |𝑔!⟩, but it would not +populate any un-paired state (which are not affected by this driving term, as all orbitals are +equally occupied). +Secondly, K3C60 has an electronic bandwidth of about 0.5eV21, meaning that the system is far +away from the atomic limit (i.e. zero inter-site tunneling). Nevertheless, because the excitation +here does not require inter-site tunneling (unlike e.g., double occupancy creation in a regular +one-band Hubbard model), it may remain sufficiently separable. +Finally, how does this excitation generate superconductivity? Indeed, the Suhl-Kondo +mechanism suggests that in a multi-band system, pairs in any local superposition can contribute +to superconductivity, but how the generation of excited-state pairs can lead to superconducting +properties starting from a normal state remains to be investigated. + + +Figure S10.1: Ground state and first +excited state of the local Hamiltonian. +The yellow lines indicate the phase of the +pair which is de-localized over two +orbitals. The energy spacing between +these two states is given by -2J + +-2J + +34 +References +1 +Mitrano, M. et al. Possible light-induced superconductivity in K3C60 at high temperature. +Nature 530, 461-464, (2016). +2 +Cantaluppi, A. et al. Pressure tuning of light-induced superconductivity in K3C60. Nature +Physics 14, 837-841, (2018). +3 +Budden, M. et al. Evidence for metastable photo-induced superconductivity in K3C60. +Nature Physics 17, 611-618, (2021). +4 +Plaskett, J. S. & Schatz, P. N. On the Robinson and Price (Kramers—Kronig) Method of +Interpreting Reflection Data Taken through a Transparent Window. The Journal of +Chemical Physics 38, 612-617, (1963). +5 +Degiorgi, L. et al. Optical properties of the alkali-metal-doped superconducting fullerenes: +K3C60 and Rb3C60. Physical Review B 49, 7012-7025, (1994). +6 +Degiorgi, L., Briceno, G., Fuhrer, M. S., Zettl, A. & Wachter, P. Optical measurements of the +superconducting gap in single-crystal K3C60 and Rb3C60. Nature 369, 541-543, (1994). +7 +Liu, B. et al. Generation of narrowband, high-intensity, carrier-envelope phase-stable +pulses tunable between 4 and 18 THz. Opt. Lett. 42, 129-131, (2017). +8 +Kindt, J. T. & Schmuttenmaer, C. A. Theory for determination of the low-frequency time- +dependent response function in liquids using time-resolved terahertz pulse spectroscopy. +The Journal of Chemical Physics 110, 8589-8596, (1999). +9 +Schmuttenmaer, C. A. Exploring Dynamics in the Far-Infrared with Terahertz +Spectroscopy. Chemical Reviews 104, 1759-1780, (2004). +10 +Born, M. & Wolf, E. Principles of Optics. 7th edn, (Cambridge University Press, 1999). +11 +Buzzi, M. et al. Higgs-Mediated Optical Amplification in a Nonequilibrium Superconductor. +Physical Review X 11, 011055, (2021). +12 +Marzari, N. & Vanderbilt, D. Maximally localized generalized Wannier functions for +composite energy bands. Physical Review B 56, 12847-12865, (1997). + + + +35 +13 +Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and +semiconductors using a plane-wave basis set. Computational Materials Science 6, 15-50, +(1996). +14 +Kresse, G. & Hafner, J. Ab initio molecular dynamics for open-shell transition metals. +Physical Review B 48, 13115-13118, (1993). +15 +Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy +calculations using a plane-wave basis set. Physical Review B 54, 11169-11186, (1996). +16 +Togo, A. & Tanaka, I. First principles phonon calculations in materials science. Scripta +Materialia 108, 1-5, (2015). +17 +Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. +Physical Review Letters 77, 3865-3868, (1996). +18 +Monkhorst, H. J. & Pack, J. D. Special points for Brillouin-zone integrations. Physical Review +B 13, 5188-5192, (1976). +19 +Capone, M., Fabrizio, M., Castellani, C. & Tosatti, E. Strongly Correlated Superconductivity. +Science 296, 2364-2366, (2002). +20 +Nomura, Y., Sakai, S., Capone, M. & Arita, R. Unified understanding of superconductivity +and Mott transition in alkali-doped fullerides from first principles. Science Advances 1, +e1500568, (2015). +21 +Nomura, Y., Sakai, S., Capone, M. & Arita, R. Exotics-wave superconductivity in alkali- +doped fullerides. Journal of Physics: Condensed Matter 28, 153001, (2016). + + diff --git a/7dFAT4oBgHgl3EQfoR1F/content/tmp_files/load_file.txt b/7dFAT4oBgHgl3EQfoR1F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b77d4888611d65319cc1e3f52a891b7aafcfdaee --- /dev/null +++ b/7dFAT4oBgHgl3EQfoR1F/content/tmp_files/load_file.txt @@ -0,0 +1,864 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf,len=863 +page_content='1 Giant resonant enhancement for photo induced superconductivity in K3C60 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Rowe1, , B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Yuan1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Buzzi1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Jotzu1, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Zhu1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Fechner1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Först1, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Liu1,2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Pontiroli3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Riccò3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Cavalleri1,4,* 1 Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany 2 Paul Scherrer Institute, Villigen, Switzerland 3 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Italy 4 Department of Physics, Clarendon Laboratory, University of Oxford, United Kingdom e mail: edward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='rowe@mpsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='de, andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='cavalleri@mpsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='de Photo-excitation at terahertz and mid-infrared frequencies has emerged as a new way to manipulate functionalities in quantum materials, in some cases creating non-equilibrium phases that have no equilibrium analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In K3C60, a metastable zero-resistance phase was documented with optical properties and pressure de- pendences compatible with non-equilibrium high temperature superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Here, we report the discovery of a dominant energy scale for this phenomenon, along with the demonstration of a giant increase in photo-susceptibility near 10 THz excitation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' At these drive frequencies a metastable supercon- ducting-like phase is observed up to room temperature for fluences as low as ~400 µJ/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These findings shed light on the microscopic mechanism underlying photo-induced superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' They also trace a path towards steady state op- eration, currently limited by the availability of a suitable high-repetition rate opti- cal source at these frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 2 The search for new non-equilibrium functional phases in quantum materials, such as op- tically induced ferroelectricity1,2, magnetism3-5, charge density wave order6,7, non-trivial topology8,9 and superconductivity10-18, has become a central research theme in condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In the case of K3C60 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 1a), mid infrared optical pulses have been exten- sively documented to yield an unconventional non-equilibrium phase which exhibits met- astable zero-resistance14, an extraordinarily high mobility and a superconducting-like gap in the optical conductivity12,14 that reduce with applied pressure13, and nonlinear I-V char- acteristics19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' All these observations are indicative of non-equilibrium high temperature superconductivity, observed at base temperatures far exceeding the highest equilibrium superconducting critical temperature of any alkali-doped fulleride (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Typical experimental results reported to date are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' K3C60 powders were held at a base temperature T = 100 K ≫ T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' = 20 K and irradiated with 1 ps-long pulses with 170 meV photon energy (l ~ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='3 µm, ) ~ 41 THz) at a fluence of 18 mJ/cm².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This strong excitation regime yielded a long-lived transient state with dramatic changes in both the real and imaginary parts of the optical conductivity, measured using phase Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Crystal structure and phase diagram K3C60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (a) Crystal structure of the organic molec- ular solid K3C60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' C60 molecules are situated at the vertices of a face-centered-cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Potassium atoms (red) occupy the interstitial voids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (b) Pressure-temperature phase diagram of the fcc-A3C60 alkali-doped fulleride family of compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Physical pressure tunes the spacing between the C60 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The grey line indicates the boundary between the insulating and metallic/superconduct- ing compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The blue shaded area indicates where superconductivity is observed at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The star indicates the K3C60 compound investigated in this work, which superconducts at tempera- tures !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ≤ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' = 20K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Insulator Superconductor 3 sensitive terahertz time-domain spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The transient optical properties displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 2c are reminiscent of those of the equilibrium superconducting state measured in the same material at T ≪ T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' = 20 K (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 2b), and are suggestive of transient high temperature superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These signatures consist of a saturated reflectivity, a gap in the real part of the optical conductivity +"(-), and an imaginary conductivity +#(-) which diverges towards low frequencies as ~1/-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The divergent +#(-) implies (through Kramers-Kronig relations) the presence of a peak in +" centred at zero frequency, with a width limited by the lifetime of the state which also determines the carrier mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These data were obtained by accounting for the inhomogeneous excitation of the probed volume using a multilayer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Here we show the results of this reconstruction under the assumption of a linear (open symbols) and sublinear (filled symbols)20 dependence of the photo-induced changes in the terahertz refractive index on the mid-infrared pump fluence, as detailed in supplementary section S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Allowing for a finite temperature super- conductor, in which a varying density of uncondensed quasi-particles also contributes to the terahertz response, the superconducting-like nature of the transient state is inde- pendent of the specific choice of assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Only quantitative differences, associated with the relative densities of induced superfluid and heated quasi-particles, which can be extracted by fitting with a two-fluid model, emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Note that the enhancement of conductivity observed in these experiments is not con- nected to an increase in the carrier density, but is solely caused by a transfer of spectral weight from the real part (resistive) to the imaginary part (inductive) of the conductivity, and hence reflects a colossal increase in the carrier mobility at constant density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Three spectrally-integrated figures of merit are extracted from the snapshots of R(-, 2), +"(-, 2) and +#(-, 2), and plotted as a function of pump-probe time delay 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 2e, showing the time evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 4 Figure 2: Photo induced metastable superconductivity in K3C60 generated with intense 170 meV excitation pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (a) Schematic of the experimental set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Pump pulses with 170 meV pho ton energy were generated in an optical parametric amplifier (OPA) and subsequent difference fre quency generation (DFG) of the signal and idler beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These pulses were stretched to a duration of ~1 ps by linear propagation in a highly dispersive CaF2 rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The photoinduced changes in the far infrared optical properties of K3C60 were detected with phase sensitive transient THz time domain spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (b) Reflectivity (sample–diamond interface), real and imaginary part of the optical conductivity of K3C60 measured upon cooling across the equilibrium superconducting transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The blue shading indicates the change of spectral weight in these quantities across the thermally driven superconducting transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (c) Same quantities measured at equilibrium (red lines) and 10 ps after excitation (filled and open symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The data in filled (open) symbols are obtained ac counting for the inhomogeneous excitation of the probed volume under the assumption of a square root (linear) fluence dependence of the photo induced changes in the complex refractive index of the material (Supplementary Section S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The blue shading indicates the change of spectral weight in these quantities after photo excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The blue solid lines are fits to the transient optical data with a Drude Lorentz model (Supplementary Section S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These data were acquired at a base tem perature T = 100 K with an excitation fluence of 18 mJ cm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (d) Same quantities as in (c) but meas ured at a base temperature T = 295 K with an excitation fluence of 18 mJ cm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (e) Time dependence of the average reflectivity, average real part of the optical conductivity &"((), and light induced “su perfluid density” extracted from a two fluid model fit and expressed as a fraction of the total charge carrier density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' All quantities are evaluated in the region of the photo induced gap (5–10 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Filled and open symbols indicate the results of two different reconstructions as in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The red dotted lines indicate the value of the corresponding quantity at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These data were acquired at a base temperature T = 100 K with a fluence of 18 mJ cm−2 and a pump pulse duration of ~1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' a 3-Stage OPA WLG DFG Ti:SaOscillator Ti:SaAmplifierx2 THz gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' b d e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 Reflectivity* Reflectivity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 25K > Tc Equilibrium Equilibrium 18 mJ cm-2 5K< Tc Photoexcited Photoexcited 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 900 Equil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' T = 100 K T = 295 K T= i0 ps = iops cm 0% Gapping cm 600 150 300 * 6 6 100% 0 900 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 cm 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 300 (S 02 102040 102040 04 102040 C 0 5 10 50 100 Energy (meV) Energy (meV) Energy (meV) Time (ps) 5 The first two quantities are the frequency-averaged values of the reflectivity and of +"(-) below the energy gap, for which a zero-temperature superconductor with infinite lifetime would give values of 1 and 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The third figure of merit is the fractional super- fluid density which is proportional to the divergence of +#(-).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This is determined by fitting the photoexcited optical proper- ties with a two-fluid model where one fluid represents the remaining normal carriers with a finite scattering rate and the other has zero scattering rate, giving a superconducting- like contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Details of this fitting procedure are given in supplementary section S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For low excitation fluences the system becomes superconducting-like after photoexcita- tion, and relaxes on a time scale of a few picoseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' As already seen in the spectrally resolved measurements, for high excitation fluences the system enters a metastable re- gime in which the superconducting-like optical properties persist for much longer times, up to several nanoseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' We note that the temperature dependence reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 12 shows transient supercon- ducting-like optical properties up to a temperature of 150-200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For higher tempera- tures the gapping and extracted superfluid density are severely reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Examples of such spectra measured at room temperature are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Nevertheless, the pressure scaling reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 13 suggests that traces of non-equilibrium superconductivity may survive up to higher temperatures, raising the prospect that with more effective driving a full manifestation of the metastable superconducting-like state may be possible at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' To date, these experiments have been limited to excitation photon energies between 80 and 165 meV (20-40 THz), such that a more comprehensive search for a dominant excita- tion frequency scale has remained out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Many potentially important resonances at lower frequencies (ℎ4 < 80 meV) have remained unexplored, primarily due to the lack of a suitable high-intensity pump source that operates in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In the present work, we explore excitation at energies between 24 and 80 meV (6-20 THz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 6 Figure 3: Photo-induced metastable superconductivity in K3C60 generated with 41 meV exci- tation pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (a) Schematic of the experimental set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Pump pulses with 41 meV (10 THz) photon energy are generated in a twin optical parametric amplifier (OPA) and subsequent chirped-pulse difference frequency generation (DFG) of the two stretched signal beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The photoinduced changes in the far-infrared optical properties of K3C60 are detected with phase-sensitive transient THz time-domain spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (b) Reflectivity (sample–diamond interface), real and imaginary part of the optical conductivity of K3C60 measured at equilibrium (red lines) and 50 ps after excita- tion (filled and open symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The data in filled (open) symbols are obtained accounting for the inhomogeneous excitation of the probed volume under the assumption of a square root (linear) flu- ence dependence of the photo-induced changes in the complex refractive index of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The blue shading indicates the change of spectral weight in these quantities after photo-excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These data were acquired at a base temperature T = 100 K with pump pulses tuned to 41 meV (10 THz) center frequency and excitation fluence of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='4 mJ cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (c) Same quantities as in (b) but measured 10 ps after photoexcitation at a base temperature T = 295 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (d) Same quantities as in (c) but meas- ured 50 ps after photoexcitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (e) Time dependence of the average reflectivity, average real part of the optical conductivity &"((), and light-induced “superfluid density” extracted from a two-fluid model fit and expressed as a fraction of the total charge carrier density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' All quantities are evaluated in the region of the photo-induced gap (5–10 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Filled and open symbols indicate the results of two different reconstruction as in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The inset in the top panel highlights the early time delays region where light amplification (* > 1) is observed (red shading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The red dotted lines indicate the value of the corresponding quantity at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These data were acquired at a base temperature T = 100 K with pump pulses tuned to 45 meV (11 THz) photon energy and excitation fluence of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 mJ cm-2 a 2x3-stageOPAs Pulsestretchers WLG DFG THz gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Ti:SaAmplifier b e Reflectivity Reflectivity* .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 Equilibrium Equilibrium Equilibrium Photoexcited Photoexcited Photoexcited 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 900 T=100K T= 295K T= 295K 0% cm T = 50 ps T = 10 ps T = 50 ps 600 cm Gapping 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 mJ cm-2 300 * 100% 0 6 900 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 600 ntot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 nsf 300 6 0 4 102040 4102040 41020 0 5 10 50 100 Energy (meV) Energy (meV) E Energy (meV) Time (ps) 7 This energy range hosts a number of excitations, both vibrational (phonons) and elec- tronic in nature, including a broad polaronic peak seen in +" centered at approximately 60 meV (15 THz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The possible relevance of this excitation has been highlighted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 21, although this prediction could not be tested to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' To achieve wide tuneability, we made use of a terahertz source based on chirped pulse difference frequency generation, mixing the near-infrared signal beams of two phase- locked optical parametric amplifiers22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This source, illustrated schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 3a and described in detail in supplementary section S4, was used to generate narrow-bandwidth pulses with photon energies spanning the range from 24 to 145 meV (6-35 THz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' All meas- urements reported here were carried out with an excitation bandwidth of ~4 meV (1 THz) and ~600 fs pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The same probing protocol as that reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 2 was utilized here to detect changes in the complex optical properties for probe energies spanning 4-18 meV (1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 THz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figures 3b-d show reflectivity and complex conductivity spectra measured after photoex- citation with pulses tuned to 41 meV photon energy (l ~ 30 µm, ) ~ 10 THz) at base tem- peratures of 100 K and room temperature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figure 3e displays the time-evo- lution of the optical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The response is very similar to the case reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 2 for 170 meV (41 THz) excitation, manifested on metastable timescales but persisting here up to room temperature – despite an almost two orders of magnitude weaker excitation fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figure 4a shows the scaling with fluence of the below-gap averaged values of 8(-) and +"(-), as well as the fractional superfluid density in response to photoexcitation at 170 meV (41 THz) and 41 meV (10 THz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These measurements were carried out at a pump-probe time delay of 10 ps, and thus refer to the metastable phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The figure shows how all figures of merit approach their equilibrium superconducting-state values as the 8 fluence increases, with the fluence required being approximately 50 times less for 41 meV (10 THz) compared to 170 meV (41 THz) excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Similar fluence dependence measurements were carried out by varying the photon en- ergy of the pump and maintaining a constant 4 meV (1 THz) bandwidth with 600 fs pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For all excitation photon energies between 24 meV (6 THz) and 145 meV (35 THz) the photoinduced changes in the optical properties were qualitatively similar to those shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 2, 3 with only the size of the response for a given fluence differing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' From each fluence dependence we extracted a figure of merit for the photo-susceptibility, Figure 4: Scaling of the out-of-equilibrium features of photo-induced metastable supercon- ductivity in K3C60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (a) Fluence dependence of the average reflectivity, average real part of the op- tical conductivity &"((), and light-induced “superfluid density” extracted from a two-fluid model fit and expressed as a fraction of the total charge carrier density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' All quantities are evaluated in the region of the photo-induced gap (5–10 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Red and blue symbols indicate measurements with excitation pulses tuned to 41 meV (10 THz) and 170 meV (41 THz) central frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The red dotted lines indicate the value of the corresponding quantity at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These data were acquired at a base temperature T = 100 K, at a time-delay ∆t = 10ps, and with a pump pulse duration of ~600 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (b) Frequency dependence of the photo-susceptibility of K3C60 defined as the gradient of the lost spectral weight in &" in the low-fluence limit (Supplementary Section S8) measured 10 ps and 50 ps after photo-excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These measurements were carried out at a base temperature T = 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These data are obtained by accounting for the inhomogeneous excitation of the probed volume un- der the assumption of a square root fluence dependence of the photo-induced changes in the com- plex refractive index of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 41 me 170 meV 9 defined as the rate of growth of the +" gap with excitation fluence in the limit of low flu ence (see supplementary section S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Plots of the pump frequency dependent photo sus ceptibility are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 4b for both 10 ps and 50 ps pump probe time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' A peak centered at 41 meV (10 THz) with approximately 16 meV FWHM bandwidth is observed in these measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In the next section we will discuss three distinct energy scales which coincide with this resonance, sequentially these relate to “on ball” orbital excita tions, phonons and finally excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Superconductivity in alkali doped fullerides is believed to be mediated by a dynamical Jahn Teller distortion, which leads to an effective negative Hund’s coupling for the orbit als of a single buckyball23 and to a low spin S=1/2 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' A theoretical model based on this assumption has been successful at providing a quantitatively correct phase diagram for fulleride superconductors, based on ab initio calculations24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Within this model, the local ground state of the system is a six fold degenerate low spin state, which features intra orbital pairs that de localize over two molecular orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' As detailed in supplementary section S10, a first set of local excited states also features such pairs, albeit with a different angular momentum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' a different inter orbital phase for the delocalized pair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Ab initio calculations predict an energy splitting of 37 meV between these two sets of states24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The observed resonance may therefore be related to the creation of interorbital pairs with local angular momentum, which may also contribute to superconductivity, as suggested in the Suhl Kondo mechanism26,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' However, it is not yet clear how exactly this excitation is transformed in the presence of tunneling between neighboring C60 molecules, and why the creation of such pairs may support metastable superconductivity at such high tem peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Furthermore, as the local parity of this excited state would be different from that of the ground state, condensation in this configuration may give rise to a supercon ductor with different symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This possibility, whilst tantalizing, remains speculative and should be tested with more comprehensive ultrafast probing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 10 Turning to phonon excitations, we also note that the 41 meV resonance frequency identi- fied here coincides with an infrared-active T1u phonon which predominantly consists of intramolecular motion of the C atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' While the atomic motions of the 170 meV molecular mode discussed previously in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 12 are directed along the tangential directions of the C603- molecule, those of the 41 meV mode are predominantly along the radial directions (see supplementary section S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' By performing frozen phonon calculations using density functional theory (DFT) we evaluated the different impact of these distortions on the three t1u molecular levels at the Fermi energy, which we map out from DFT wave functions as maximally-localized Wannier functions (supplementary section S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In the undistorted C603- structure these molecular levels are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Applying a distortion along a T1u coordinate lifts this degeneracy leaving a doubly degenerate t1u orbital lowered in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This electronic configuration is prone to developing a Jahn-Teller distortion that may lead to an enhanced negative Hund’s coupling, possibly facilitating the onset of superconduc- tivity at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The strength of the induced splitting is quadratic in the phonon coordinate and is more significant for when driving the 41 meV mode compared to the 170 meV one, suggesting that the observed resonance may arise from a more efficient manipulation of the elec- tronic degrees of freedom when driving the 41 meV T1u mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Finally, we address the electronic excitations discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 21, in which the existence of a polaronic mode was predicted at the same energy scales as the resonance reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' However, we also note that the proposed mechanism for the formation of the non- equilibrium superconducting-like state was one in which the quasi-particles are cooled incoherently via coupling to the polaronic bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' As already reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 28 and shown here in the inset to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 3e, the response of the sample in the first few picoseconds after photoexcitation yields amplification of the terahertz probe light, which is likely to reflect coherent dynamics of the driven degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Assuming that the mechanism 11 proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 21 were to be valid, the early time dynamics of that model would require further investigation to understand how such coherences would arise at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The amplification observed here and in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 28 has so far been attributed to the existence of a parametric resonance that couples amplitude (Higgs) modes to phase (Goldstone) modes, an effect possible at the sample surface because of reduced screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' We expect the significance of this discovery to be capitalized upon in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The extreme efficiency improvement due to resonant enhancement, nearing two orders of magnitude, is expected to also dramatically reduce unwanted dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This, taken in conjunction with the observed nanosecond long lifetime suggests that excitation of the sample with a train of pulses of only 400 µJ/cm2 delivered at 100 MHz repetition rate – as determined by the inverse lifetime of this state may yield continuous wave operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Because this effect is documented here to persist up to room temperature, continuous wave operation would likely have important practical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' To make this regime experimentally accessible, single order of magnitude improvements in the efficiency of the process, or in the light matter coupling strength, combined with suitable develop ments in high repetition rate THz sources would be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Acknowledgments The research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007 2013)/ERC Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 319286 (QMAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) via the Cluster of Excellence ‘The Hamburg Centre for Ultrafast Imaging’ (EXC 1074 – project ID 194651731).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' We thank Michael Volkmann and Peter Licht for their technical assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' We are also grateful to Boris Fiedler and Birger Höhling for their support in the fabrication of the elec tronic devices used on the measurement setup, and to Jörg Harms for assistance with graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 12 References 1 Nova, T.' 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Nonequilibrium Superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Physical Review X 11, 011055, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 14 Giant resonant enhancement for photo induced superconductivity in K3C60 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Rowe1, , B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Yuan1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Buzzi1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Jotzu1, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Zhu1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Fechner1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Först1, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Liu1,2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Pontiroli3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Riccò3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Cavalleri1,4,* 1 Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany 2 Paul Scherrer Institute, Villigen, Switzerland 3 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Italy 4 Department of Physics, Clarendon Laboratory, University of Oxford, United Kingdom e mail: edward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='rowe@mpsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='de, andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='cavalleri@mpsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='de Supplemental Material S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Sample growth and characterization S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Determination of the equilibrium optical properties S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' High fluence mid infrared source S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Frequency tunable narrowband terahertz and mid infrared source S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Measurements of the transient THz reflectivity S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Determination of the transient optical properties S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Fitting the transient optical spectra S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Extracting the frequency dependent photosusceptibility S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Density functional theory calculations S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Local electronic hamiltonian calculations 15 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Sample growth and characterization The K3C60 powder pellets used in this work were prepared and characterized as reported previously1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Stoichiometric amounts of ground C60 powder and potassium were placed in a sealed pyrex vial, which was evacuated to a pressure of 10-6 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Whilst keeping the C60 powder and solid potassium separated, the vial was kept at 523 K for 72 h and then at 623 K for 28 h such that the C60 powder was exposed to pure potassium vapor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The vial was then opened inside an Ar glovebox (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 ppm O2 and H2O), where the powder was reground and pelletized before annealing at 623K for 5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' X-ray diffraction measurements were then carried out on the resulting K3C60 powder, which confirmed that it was phase pure, with an average grain size ranging between 100 and 400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The static superconducting transition temperature was measured to be 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='8 K (in agreement with literature values) via magnetic susceptibility measurements upon zero field cooling and cooling in field with a field strength of 400 A/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' X ray diffraction data and single f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' phase Rietveld refinement for the K3C60 powder used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Temperature dependence of the sample magnetic susceptibility measured by SQUID magnetometry upon cooling without (ZFC: zero field cooling) and with a magnetic field applied (FCC: field cooled cooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' observed (10emu/(g0e) calculated residual reflections 20 ZFC FCC 30 40 50 10 20 30 40 50 60 5 10 15 20 2A tdearees Temperature 16 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Determination of the equilibrium optical properties The equilibrium reflectivity was measured for photon energies between 5 meV and 500 meV using a commercial Fourier-transform infrared spectrometer (FTIR) equipped with a microscope at the SISSI beamline in the Elettra Synchrotron Facility (Trieste, Italy), as reported previously1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The sample was pressed by a diamond window into a sealed holder in order to obtain an optically flat interface and prevent exposure to air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This procedure was carried out inside an Ar filled glove box (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 ppm O2 and H2O) before the sealed sample was removed and mounted on a He cooled cryostat to enable temperature dependent measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The K3C60 reflectivity spectra were referenced against a gold mirror placed at the sample position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In order to extract the complex optical conductivity a Kramers-Kronig algorithm for samples in contact with a transparent window4 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This requires data at all frequencies, which were obtained, at low energies (<5 meV) using an extrapolation based on a Drude-Lorentz fit, and at high energies (>500 meV) using data measured on single crystal samples reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The equilibrium properties are shown in figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 for temperatures of 100 K and 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This and further data measured at different temperatures and pressures were already reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 1,2 and discussed also in comparison with data obtained from single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These data were fitted with a Drude-Lorentz model, which is given by the following equation: 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (𝜔) + 𝑖𝜎"(𝜔) = 𝜔#" 4𝜋 1 𝛾$ − 𝑖𝜔 + 𝜔#,&\'( " 4𝜋 𝜔 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='𝜔),&\'( " − 𝜔"/ + 𝛾&\'(𝜔 Here the first term represents the Drude response of the free carriers with 𝜔# and 𝛾$ representing the plasma frequency and scattering rate respectively, whereas the second term captures the mid infrared absorption in the form of a Lorentz oscillator centered at frequency 𝜔),&\'( with plasma frequency 𝜔#,&\'( and damping rate 𝛾&\'(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The equilibrium data reported here was used to normalize the transient optical spectra of K3C60 measured upon photoexcitation, as discussed in detail in section S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 17 Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1: Equilibrium optical properties (reflectivity, real, and imaginary part of the optical conductivity) of K3C60 measured at a temperature of 100 K (blue) and 300 K (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The black dashed curve is a Drude Lorentz fit to the optical conductivity at 100 K in the range from 3 meV to 60 meV as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' High fluence mid infrared source For the data reported in figure 2 and in figure 4(a) at 170 meV (41 THz) excitation, the pump pulses were generated via difference frequency mixing (DFG) of the signal and idler output of a three-stage home-built optical parametric amplifier (OPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' A commercial Ti:Al2O3 amplifier delivering 60 fs duration pulses at 800 nm central wavelength was used to drive the OPA, and the DFG process was performed using a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 mm thick GaSe crystal, resulting in ~100 fs long pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The 170 meV pulses were then propagated through a highly dispersive 16 mm long CaF2 rod, stretching their duration to ~1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The spectrum of the pump pulses was characterized using a home built FTIR spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Their duration was measured by cross-correlation with a synchronized, 35 fs long, 800 nm wavelength pulse in a 50 μm thick GaSe crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' While a certain degree of tunability is also given by this source, its useful operation range spans between 80 and 320 meV, hence it was only used for the high-intensity experiments at 170 meV excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 900 E 900㎡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 300 K 100K Reflectivity T T 600 600 Fit 100 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 300 300 02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='04 0 10 30 100 410 30 100 4 10 30 100 Energy (meV) Energy (meV) Energy (meV) 18 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Frequency tunable narrowband terahertz and mid infrared source For the experiments that required tunability of the excitation pulses down to the THz gap, a different source was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This source is based on the principle of chirped-pulse difference frequency generation (CP-DFG) in organic non-linear optical crystals, namely DAST and DSTMS of approximately 600 μm thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The principle of operation of this new source is described in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' A commercial Ti:Al2O3 amplifier is used to drive two identical three-stage OPAs which are seeded by the same white-light, such that the signal beams have the same phase-fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The ~100 fs signal pulses are then chirped using a pair of transmission-grating-based stretchers as depicted in figure 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This arrangement enables continuous tuning of the pulse durations by varying the distance between the gratings in each pair, effectively enabling continuous tuning of the pump-pulse bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For this experiment the pump pulse bandwidth was kept constant at 4 meV by maintaining a signal pulse duration of ~600 fs, as measured using a home-built second harmonic-based Frequency-Resolved-Optical-Gating (FROG) device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Frequency tuning of the generated excitation pulses was carried out both by varying the central wavelengths of the two OPA signal beams, and by varying the time delay between the chirped signal pulses in the DFG crystal (for fine tuning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For each measurement the pump frequency spectrum was measured via FTIR (Fourier Transform Infrared Spectroscopy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Measurements of the transient THz reflectivity The experiments presented in Figures 2, 3, and 4 were performed on compacted K3C60 powder pellets pressed against a diamond window to ensure an optically flat interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' As K3C60 is water and oxygen sensitive, the pellets were sealed in an air tight holder and all sample handling operations were performed in an Argon filled glove box with <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 ppm O2 and H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The sample holder was then installed at the end of a commercial Helium cold finger (base temperature 5K), to cool the pellets down to a temperature of 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The changes in the properties of the sample following photoexcitation were measured using time domain THz spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 19 The mid-infrared pump induced changes in the low frequency optical properties, were retrieved using transient THz time domain spectroscopy in two different experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The THz probe pulses were generated via optical rectification in a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2 mm thick (110)-cut GaP crystal starting from 800 nm pulses with a duration of ~80 fs and 35 fs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Whilst in one setup these 800 nm were derived from the same laser used for pumping the source described in section S4, the 35 fs, 800 nm pulses were generated by a second Ti:Al2O3 amplifier optically synchronized to that used to pump the high- intensity mid-infrared source described in section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The THz probe pulses were then focused onto the sample with incidence angles of 30 and 0 degrees, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' After reflection from the sample, the electric field profile of the THz pulses was reconstructed in a standard electro-optic sampling setup, using a (110)-cut 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2 mm GaP crystal supported on a 1 mm thick (100)-cut GaP substrate to delay internal reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The setup combined with the frequency tunable narrowband source had a measurement bandwidth that extended between 4 and 18 meV, while the other spanned between 4 meV to 29 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The time resolution of both setups is determined by the measurement bandwidth and is ~250 fs and ~150 fs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' To minimize the effects on the pump-probe time resolution due to the finite duration of the THz probe pulse, the experiments were performed as described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The pump-probe time delay was controlled by fixing the delay between the 800 nm gating pulse and the mid-infrared pump pulse 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The transient THz field was then obtained by scanning the delay 𝑡 relative to both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In order to simultaneously retrieve both the ‘pump on’ (𝐸*+, &- (𝑡, 𝜏)) and ‘pump off’ (𝐸*+, &.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='.(𝑡)) probe fields, a differential chopping scheme was deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The scheme was different for the two above mentioned setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For the narrowband, frequency tunable setup which operated at a repetition rate of 1 kHz, the THz probe pulse was chopped at a frequency of 500 Hz and the mid-infrared pump pulse was chopped at ~ 357 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The electro-optic sampling signal was then fed to two lock-in amplifiers reading out 𝑉/01!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' at 500 Hz and 𝑉/01" at 143 Hz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For the high-intensity setup, operating at 2 kHz repetition rate, the THz probe pulse was chopped at a frequency of 1 kHz and the mid- infrared pump was chopped at 500 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In this case, the electro-optic sampling signal was filtered by two lock-in amplifiers operating at 1 kHz and 500 Hz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 𝐸*+, &.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='.(𝑡) and Δ𝐸*+,(𝑡, 𝜏) were then extracted from the signals in the two lock-ins using the following formulas: 20 𝐸*+, &.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='.(𝑡) = 𝑉𝐿𝐼𝐴1(𝑡, 𝜏) − 𝛼𝑉𝐿𝐼𝐴2(𝑡, 𝜏) Δ𝐸*+,(𝑡, 𝜏) = 𝐸*+, &- (𝑡, 𝜏) − 𝐸*+, &.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='.(𝑡) = 𝛼𝑉𝐿𝐼𝐴2(𝑡, 𝜏) where 𝛼 is a calibration constant determined experimentally on an InSb reference sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This is done by extracting Δ𝐸*+,(𝑡, 𝜏) as the difference of two separate measurements of 𝐸*+, &- (𝑡, 0) and 𝐸*+, &.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='.(𝑡) performed with the first lock-in amplifier and by chopping only the THz probe pulse while leaving the mid-infrared pump pulse either always on or always off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Equating the value of Δ𝐸*+,(𝑡, 𝜏) determined in this way to the one with differential chopping yields the calibration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Determination of the transient optical properties From the measured changes in the reflected probe field (see section S5), the transient complex reflection coefficient of the sample 𝑟̃(𝜔, 𝜏) can be determined by taking the Fourier transform along t of both 𝐸*+, &.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='.(𝑡) and Δ𝐸*+,(𝑡, 𝜏) and using the following equation: Δ𝐸:*+,(𝜔, 𝜏) 𝐸:*+, &.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='.(𝜔) = 𝑟̃(𝜔, 𝜏) − 𝑟̃)(𝜔) 𝑟̃)(𝜔) where 𝑟̃)(𝜔) is the equilibrium complex reflection coefficient, obtained as described in section S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In the cases where the pump light penetrates in the sample several times deeper than the probe light, one can assume that the probe pulse samples a volume in the material that has been homogeneously excited by the pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In this case, it is possible to directly extract the complex-valued optical response functions by inverting the Fresnel equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' However, in K3C60 the penetration depth of the probe electric field (~600-900 nm) exceeds that of the pump (~500 nm at 10 THz, ~200 nm at 41 THz), such that the probe interrogates an inhomogeneously excited volume (Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 21 As the pump penetrates into the material, its intensity is reduced, and it will induce progressively weaker changes in the refractive index of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This situation is modeled by “slicing” the probed thickness of the material into thin layers (figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1(b)), where we assume that the pump-induced changes in the refractive index ∆𝑛= scale according to the pump intensity in the layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 𝑛=(𝜔, 𝑧, 𝜏) = 𝑛=)(𝜔) + ∆𝑛=(𝜔, 𝜏, 𝐼(𝑧)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The pump intensity 𝐼(𝑧) is assumed to follow the dependence 𝐼(𝑧) = 𝐼)𝑒2,/4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' "#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=', where 𝑑#56# = 𝜆#56# 4𝜋𝐼𝑚 D𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='𝜔#56#/E F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Here, the refractive index of the material at the pump frequency, 𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='𝜔#56#/ is taken to be the one at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Additionally, an assumption is made on the functional form for the dependence of ∆𝑛= on the pump intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Here, we consider two different forms given by: (1) ∆𝑛=(𝜔, 𝜏, 𝑧) ∝ 𝐼(𝑧) (2) ∆𝑛=(𝜔, 𝜏, 𝑧) ∝ H𝐼(𝑧) Respectively, these equations result in the following depth-dependent functional forms for the spatial profile of the refractive index: (1) 𝑛=(𝑧, 𝜔, 𝜏) = 𝑛=)(𝜔) + Δ𝑛=(𝜔, 𝜏)𝑒2,/4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='"#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (2) 𝑛=(𝑧, 𝜔, 𝜏) = 𝑛=)(𝜔) + ∆𝑛=(𝜔, 𝜏)𝑒2,/"4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='"#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' where Δ𝑛=(𝜔, 𝜏) represents the pump-induced change in the refractive index of the material at the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Schematics of pump-probe penetration depth mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Multi-layer model with exponential decay used to calculate the pump-induced changes in the complex refractive index 𝑛#(𝜔, 𝜏) for each pump-probe delay 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The transition from red to background (grey) represents the decaying pump-induced changes in 𝑛#(𝜔, 𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Sample Sample Probe Pump 22 For each time delay 𝜏 and probe frequency 𝜔7, the complex reflection coefficient 𝑟̃(∆𝑛=) of the multilayer stack described above is calculated using the transfer matrix method10, keeping ∆𝑛= as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' To numerically extract the value of ∆𝑛=(𝜔, 𝜏) we minimize the following function: IΔ𝐸:*+,(𝜔7) 𝐸:*+, &.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='.(𝜔7) − 𝑟̃(𝜔7, Δn) − 𝑟̃)(𝜔7) 𝑟̃)(𝜔7) I By then taking 𝑛=(𝜔, 𝜏) = 𝑛=)(𝜔) + Δ𝑛=(𝜔, 𝜏), one obtains the refractive index of the material as if it had been homogeneously excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' From 𝑛=(𝜔, 𝜏) we then calculate 𝑅(𝜔, 𝜏), 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (𝜔, 𝜏) and 𝜎"(𝜔, 𝜏) as plotted in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figures S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='3 display extended data sets measured at increasing pump-probe delays with pump photon energies of 170 meV (41 THz) and 45 meV (11 THz) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Therein we report reflectivity (sample-diamond interface), real and imaginary part of the optical conductivity after reconstruction under the assumptions of models (1) and (2), identified with hollow and filled circles respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' At early delays, for both excitation mechanisms and reconstruction assumptions, the reconstructed reflectivity is higher than one, and the real part of the optical conductivity is negative, indicative of amplification of the incoming THz probe radiation, as discussed previously in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In all cases, this non-equilibrium driven state then relaxes into a superconducting-like state with a fully gapped 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (𝜔) and a divergence ∝ 1 𝜔 ⁄ in the 𝜎"(𝜔) spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' At even later delays the optical spectra are those of a finite temperature superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These optical properties can be interpreted in the context of a two fluid model, in which a varying density of uncondensed quasi-particles also contributes to the terahertz response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Importantly the time-evolution of K3C60 following photo-excitation is independent of the used reconstruction, and only the specific values of pump-probe delay up to which amplification, fully gapped superconductor, and finite temperature superconductor appear are affected by this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 23 Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2: Comparison of linear and sub-linear reconstruction in the transient optical spectra at 170 meV (41 THz) pump-photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Reflectivity (sample-diamond interface), real, and imaginary parts of the optical conductivity measured at equilibrium (red lines) and after photoexcitation (blue symbols) at increasing pump-probe time delays indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The data in filled (open) symbols reconstructed under the assumption of a square-root (linear) fluence dependence of the changes in complex refractive index of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These data were measured at 18 mJ cm-2 excitation fluence, and at a base temperature of 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 0 ps 1 ps 2 ps 5 ps 10 ps 50 ps 24 Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='3: Comparison of linear and sub-linear reconstruction in the transient optical spectra at 45 meV (11 THz) pump-photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Reflectivity (sample-diamond interface), real, and imaginary parts of the optical conductivity measured at equilibrium (red lines) and after photoexcitation (blue symbols) at increasing pump-probe time delays indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The data in filled (open) symbols reconstructed under the assumption of a square-root (linear) fluence dependence of the changes in complex refractive index of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' These data were measured at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 mJ cm-2 excitation fluence, and at a base temperature of 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 ps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 ps 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 ps 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 ps 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 ps 25 S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Fitting of the transient optical spectra The transient optical conductivity spectra presented in figures 2-3 as well as for each fluence in figure 4 were fitted with a two-fluid model according to the following equation: 𝜎=(𝜔, 𝜏) = 𝜋 2 Λ\'(𝜏) 𝑒" 𝑚 𝛿[𝜔 = 0] + 𝑖 Λ\'(𝜏) 𝑒" 𝑚 1 𝜔 + Λ-(𝜏) 𝑒" 𝑚 1 𝛾$ − 𝑖𝜔 + R 𝐵-𝜔 𝑖(Ω-" − 𝜔") + 𝛾-𝜔 " -8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=" Here the first term captures the frequency dependent contribution from the supercarriers with density Λ', the second term captures the Drude contribution of the normal carriers with density Λ- and scattering rate 𝛾$." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Finally, we include a sum over Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1: Two-fluid fit to the transient spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Reflectivity, real (𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=') and imaginary (𝜎") parts of the optical conductivity measured in equilibrium at 100 K (red) and 50 ps after photoexcitation with a fluence of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 mJ cm-2 at 45 meV (11 THz) photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The fit to the equilibrium data using the procedure described in this section is shown as a dashed black line and gives zero superfluid density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The two-fluid fit to the transient data generated using the same procedure is shown as a solid blue line and returns a superfluid fraction Λ# (Λ$ + Λ#) ⁄ = 73%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The data in this figure was reconstructed under the assumption of a square root dependence of the change in refractive index on excitation fluence (see supplementary section S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 26 two Lorentz oscillators in order to capture the broad midinfrared absorption peak centered at around 60 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=" The transient data are fitted at each delay 𝜏 using the parameter-set that captures the equilibrium optical conductivity spectra as a starting condition, and leaving only Λ' and Λ- free to vary, as though the effect of the pump is to simply convert carriers from the normal to the superconducting fluid." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 shows representative fits to transient data measured at 100 K base temperature and at 50 ps time delay, as well as to the 100 K equilibrium spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=" Importantly, while the fit of the equilibrium data converges to a superfluid fraction Λ' (Λ- + Λ') ⁄ which is equal to zero, the fit to the transient data yields Λ' (Λ- ⁄ + Λ') = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The transient optical data was fitted at each time delay and driving frequency, yielding the time and frequency dependence of the superfluid fractions shown in figures 2(e), 3(e), and 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Extracting the frequency dependent photosusceptibility In figure 4(b) we introduce a figure of merit, referred to as the ‘photosusceptibility’, which can be used to quantitatively compare the efficiency with which the metastable light-induced superconducting state is generated in K3C60 for different excitation frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For each excitation photon energy, transient optical spectra were measured at different excitation fluences ℱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' From these fluence dependent spectra we extract the loss in spectral weight of 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (𝜔) after photoexcitation in the 5-10 meV spectral range, calculated as: 𝑆𝑊𝐿(ℱ) = Z 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 9:(𝜔) − 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' #;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='&<&(𝜔, ℱ) 𝑑𝜔 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=') meV/ℏ B meV/ℏ where 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 9:(𝜔) and 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' #;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='&<&(𝜔, ℱ) are the 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (𝜔) spectra measured in equilibrium and upon photoexcitation respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The 𝑆𝑊𝐿(ℱ) data is then fitted with the following phenomenological function: 𝐴 \\ 1 1 + 𝐵𝑒2CDℱ 1 − 1 2] 27 where ℱ represents the excitation fluence and 𝐴, 𝐵 are free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The ‘photosusceptibility’ plotted in figure 4(b) is equal to 𝐵, which is the gradient of this function evaluated at zero fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 shows the fluence-dependent data and corresponding fit for one exemplary dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1: Extracting photosusceptibility from the fluence-dependent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Lost spectral weight in the real part of the optical conductivity between 5 and 10 meV as a function of fluence (red circles), measured 10 ps after photoexcitation at 100 K with a pump spectrum centered at 41 meV (10 THz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The fit is shown as a solid green line, with the gradient at zero fluence (which we define as the photosusceptibility) shown as a dashed blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The data in this figure was reconstructed under the assumption of a square root dependence of the change in refractive index on excitation fluence (see supplementary section S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Density functional theory calculations In this section, we address how the displacement of phonon modes affects the electronic properties of K3C60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Specifically, we consider the molecular orbitals and their response to the change in the crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' To carry out this investigation, a first-principles approach based on density functional theory (DFT) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The starting point is the unit cell of K3C60 containing sixty carbon and three potassium atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Before computing 28 the phonon spectrum, this unit cell is structurally relaxed, and the resulting lattice constants and atomic coordinates are listed in table S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Next, the phonon spectrum of K3C60 is computed from the force constant matrix utilizing a finite displacement approach12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In total, there are 186 non-translational phonon modes covering the symmetries of point group m-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Specifically, there are 24 Tu, 7 Eu, 23 Tg, 8 Eg, and 8 Ag modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Note that only the modes of Tu character are infrared active, and we list their computed frequencies in the table S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' We utilized a frozen phonon approach to estimate the impact of these distortions on the molecular levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Therefore, we modulated our equilibrium crystal structure with the eigen-displacements of these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' We then created a low energy Hamiltonian for these structures by computing the maximally localized Wannier functions for the valence band electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Note that since the three valence bands are well separated in energy from other orbital-like bands our method does not require a disentanglement procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Our calculations focused on the three degenerate t1u molecular levels at the Fermi energy, which we mapped out from DFT wave functions as maximally-localized Wannier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In the equilibrium structure, the onsite energy of these molecular levels is degenerate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' however, deforming the crystal by applying a T1u polar distortion lifts this degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Thereby, similar to a Jahn-Teller distortion, the symmetry breaking of the crystal structure splits the level into a double and a single degenerate orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For the 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2 meV and 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='4 meV phonon modes, this splitting manifests as a lowering in the energy of the double degenerate orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' A schematic visualization of this is depicted in the inset to figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Diagrams illustrating the distortion of the C60 molecule for the 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2 meV Lattice vectors a 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='175 Å Alpha 90˚ 90˚ 90˚ b 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='175 Å Beta c 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='175 Å Gamma Atomic positions according to Space Group 202 (Fm-3) Element Wykoff label X y c C H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='54991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='24682 C I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='58242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='10057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='21408 C I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='66275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='05092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='18294 K C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='25000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='25000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='25000 K A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='00000 Table S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1: Structural parameters of K3C60 from first principles computations 29 and 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='4 meV modes (labelled ‘A’ and ‘B’ and corresponding to mode numbers 4 and 21 in table S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2 respectively) are shown in figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Besides this qualitative difference of the phonon-mode distortion on the molecular levels, we also examined the strength of the induced splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' From group-theory, the size of the splitting scales with the square of the distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1(a) displays how the splitting develops as a function of the fluence of the incoming THz pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Each phonon mode distortion was weighted according to its eigenfrequency and mode effective charge in this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For the same strength of the driving electric field, the splitting induced by phonon A produces a more significant separation of the t1u levels compared to phonon B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Due to the square scaling of the splitting with the electric field, this effect is further enhanced at higher field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The computations were performed with the Vienna ab-initio simulation package VASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='213-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For the phonon calculations, we used the Phonopy software package16 and the Wannier90 package for wannierization12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The computations further utilized Number: ℎ𝜈#56# (meV) 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2 2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='4 4 (A) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2 5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='3 6 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='6 8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='6 10 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='9 11 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='9 12 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 13 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0 14 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='6 15 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='8 16 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5 17 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='3 18 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='8 19 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='7 20 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='4 21 (B) 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='4 22 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='9 23 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='8 24 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='3 Table S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2: List of the IR active phonon modes of Tu symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 30 pseudopotentials generated within the Projected Augmented Wave (PAW) method16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Specifically, the following default potentials were used: C 2s22p2 and K 3s23p64s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The Generalized Gradient Approximation (GGA17) approximation for the exchange- correlation potential was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' For the final numerical setting, a 4x4x4 Monkhorst18 generated k-point-mesh sampling of the Brillouin zone and a plane-wave energy cutoff of 600 eV were chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The calculations were re-iterated self-consistently until the change in total energy converged within 10-8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1: Effect of vibrational distortions on the t1u molecular levels from first-principle computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' (a) shows the induced splitting of the molecular orbital of t1u symmetry at the Fermi energy (as illustrated by the inset) as a function of drive fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The two curves represent the effect of the two distinct T1u IR-phonon modes with eigenfrequencies of 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2 (red) and 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='4 (blue) meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The eigen displacement of these modes are shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Note, that due to the symmetry character of the phonon modes the t1u level split into a single and double degenerate orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Lastly, in (c) we show the induced splitting as a function of frequency for a fixed fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Here we consider the whole spectrum of T1u IR modes of K3C60, as listed in table S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Local electronic hamiltonian calculations The Hamiltonian proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 19 in order to model superconductivity in alkali-doped fullerides is based on an effective negative Hund’s coupling J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' It arises from a combination of the usual Hund’s coupling with a dynamical Jahn-Teller distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This causes states featuring intra-orbital pairing on a buckyball to be energetically favourable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Using ab-initio calculations, B 31 values of the intra-orbital interaction U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='826 eV and of J = −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5meV were predicted for K3C6020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The phase diagram for the A3C60 family of compounds was computed using DMFT starting from this Hamiltonian and was found to be in quantitative agreement with experimental data21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The Hamiltonian can be written as: 𝐻 = 𝐻Intra + 𝐻Inter + 𝐻Pairhop + 𝐻Spinswap with an intra-orbital interaction with magnitude U given by: 𝐻Intra = 𝑈 R 𝑛7,↑𝑛7,↓ U 7 where 𝑛7,V = 𝑎7,V W 𝑎7,V is the number operator for a spin down electron on orbital i with spin 𝜎 ∈ {↑, ↓}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' 𝑎7,V W and 𝑎7,V are fermion creation and annihilation operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The inter- orbital interaction appears as: 𝐻Inter = (𝑈 − 2𝐽) R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 − δ7X/𝑛7,↑𝑛X,↓ U X U 7 + (𝑈 − 3𝐽) R R R 𝑛7,V𝑛X,V 72!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' X U 7 V with δ7X denoting the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' which, given that J is negative, makes these terms higher in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In addition, there is a pair hopping term, which corresponds to a transfer of a pair of electrons from one orbital to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' It is given by: 𝐻Pairhop = 𝐽 R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 − δ7X/𝑎7,↑ W 𝑎7,↓ W 𝑎X,↓𝑎X,↑ U X U 7 This term was found to be crucial for the appearance of superconductivity21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Finally, there is a “spin swapping” term, where two opposite spins exchange orbitals: −𝐽 R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1 − δ7X/𝑎7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='↑ W 𝑎7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='↓𝑎X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='↓ W 𝑎X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='↑ U X U 7 32 When restricting ourselves to a Hilbert space where the three degenerate orbitals are populated by three electrons (as is appropriate for A3C60 in the atomic limit),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' we can use a basis given by the different possible arrangements in which the orbitals can be populated: {|↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='|↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑↓⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='|↑↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='|0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑↓⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='|↑↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='|0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' |↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='|↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='|↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↓⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='|↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' ↑⟩} as well as a second set of states created by flipping all spins in the set above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' In this basis, the Hamiltonian takes on the form: 𝐻m − (3𝑈 + 5𝐽)𝐼o = −𝐽 ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 +1 0 0 0 0 0 0 0 0 +1 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 +2 −1 +1 0 0 0 0 0 0 0 −1 +2 −1 0 0 0 0 0 0 0 +1 −1 +2 0 0 0 0 0 0 0 0 0 0 +4⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ where 𝐼o is the identity matrix, which encodes an overall energy offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' This matrix is block- diagonal, meaning that different sectors of the Hilbert space are not coupled to each other: For example, there is no term that destroys or creates pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Because of the inverted Hund’s coupling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' because J is negative, the stretched state |↑, ↑, ↑⟩ as well as its global spin-flip partner |↓, ↓, ↓⟩ are now the most energetic local eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The local ground state is 6-fold degenerate, with an exemplary instance given by: |𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='⟩ = (|↑, ↑↓ ,0⟩ +|↑ ,0, ↑↓⟩)/√2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' it is a state where one singlet pair of electrons has de- localized over two orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The first excited manifold is 10-fold degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Six of those states are of the type |𝑒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='⟩ = ((|↑, ↑↓ ,0⟩ −|↑ ,0, ↑↓⟩)/√2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' identical to the ground state except for the phase of the de-localized singlet pair (and hence corresponding to a different local angular momentum) – as illustrated in Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The energy difference between these two manifolds is given by 2J=37meV, remarkably close to the observed resonance in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' However, several questions remain in order to determine whether an excitation of this transition is responsible for the experimental observation: 33 Firstly, how does the light field of the laser couple to this excitation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' As the size of a buckyball is comparable to the distance between buckyballs both inter-site and intra-site driving terms may be comparable in terms of the associated energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Understanding possible inter-site driving terms (arising from the oscillating energy difference between neighbouring sites, given by the electric field multiplied with the charge and the lattice spacing) will require a calculation featuring multiple buckyballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Locally, because the dynamical Jahn- Teller distortion causes the populated orbitals to be superpositions of several undistorted orbitals, we may expect the electric field to lift the orbital degeneracy, for example through an orbital offset term of the type 𝐻offset = Δ(𝑛U,↑ + 𝑛U,↓), where Δ encodes the amplitude of the drive and is oscillating in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Such a term would in fact cause an excitation from |𝑒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='⟩ to |𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='⟩, but it would not populate any un-paired state (which are not affected by this driving term, as all orbitals are equally occupied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Secondly, K3C60 has an electronic bandwidth of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='5eV21, meaning that the system is far away from the atomic limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' zero inter-site tunneling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Nevertheless, because the excitation here does not require inter-site tunneling (unlike e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=', double occupancy creation in a regular one-band Hubbard model), it may remain sufficiently separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Finally, how does this excitation generate superconductivity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Indeed, the Suhl-Kondo mechanism suggests that in a multi-band system, pairs in any local superposition can contribute to superconductivity, but how the generation of excited-state pairs can lead to superconducting properties starting from a normal state remains to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content='1: Ground state and first excited state of the local Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The yellow lines indicate the phase of the pair which is de-localized over two orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' The energy spacing between these two states is given by -2J 2J 34 References 1 Mitrano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Possible light-induced superconductivity in K3C60 at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} +page_content=' Nature 530, 461-464, (2016).' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFAT4oBgHgl3EQfoR1F/content/2301.08633v1.pdf'} diff --git a/7tAzT4oBgHgl3EQfgfwd/content/tmp_files/2301.01468v1.pdf.txt b/7tAzT4oBgHgl3EQfgfwd/content/tmp_files/2301.01468v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d67e7a80a4887b6baa4ae433fb7fe011a511143c --- /dev/null +++ b/7tAzT4oBgHgl3EQfgfwd/content/tmp_files/2301.01468v1.pdf.txt @@ -0,0 +1,1582 @@ +Black hole interiors +in holographic topological semimetals +Ling-Long Gao a,b1, Yan Liu a,b2 and Hong-Da Lyu a,b3 +aCenter for Gravitational Physics, Department of Space Science +and International Research Institute of Multidisciplinary Science, +Beihang University, Beijing 100191, China +bPeng Huanwu Collaborative Center for Research and Education, +Beihang University, Beijing 100191, China +Abstract +We study the black hole interiors in holographic Weyl semimetals and holo- +graphic nodal line semimetals. +We find that the black hole singularities are of +Kasner form. In the topologically nontrivial phase at low temperature, both the +Kasner exponents of the metric fields and the proper time from the horizon to the +singularity are almost constant, likely reflecting the topological nature of the topo- +logical semimetals. We also find some specific behaviors inside the horizon in each +holographic semimetal model. +1Email: linglonggao@buaa.edu.cn +2Email: yanliu@buaa.edu.cn +3Email: hongdalyu@buaa.edu.cn +arXiv:2301.01468v1 [hep-th] 4 Jan 2023 + +Contents +1 +Introduction +1 +2 +Inside holographic Weyl semimetal +3 +2.1 +Inner structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Behaviors of Kasner exponents . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.3 +Proper time of timelike geodesics +. . . . . . . . . . . . . . . . . . . . . . +9 +3 +Inside holographic nodal line semimetal +10 +3.1 +Kasner exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +3.2 +Proper time of timelike geodesics +. . . . . . . . . . . . . . . . . . . . . . +15 +4 +Conclusion and discussion +15 +A Equations in holographic WSM +16 +B Equations in holographic NLSM +18 +1 +Introduction +The conventional classifications on the phases of matter are rooted in the Landau paradigm +of symmetry breaking theory [1]. Over the past thirty years, new states of matter have +been found which are beyond the concept of Landau paradigm. +One example is the +topological states of matter, including the quantum Hall states, topological insulators, +topological semimetals and so on [2]. Different from the conventional Landau paradigm, +there is no symmetry breaking during the topological phase transition and it attracts lots +of research attention. +In recent years, the strongly interacting topological Weyl semimetals (WSM) [3,4] and +nodal line semimetals (NLSM) [5,6] have been explicitly constructed from the holographic +duality. Both holographic WSM and NLSM are shown to possess nontrivial topological +invariants [7]. Remarkably, holographic WSM exhibits interesting effects inherited from +the boundary states [8]. These features suggest that the physical properties associated to +topology from the weakly coupled field theories persist in the strongly coupled topologi- +cal systems from the holography. Moreover, the systems could go through a topological +phase transition to a topologically trivial semimetal phase, see [9] for a review on the +1 + +developments.4 +In the holographic WSM, during the topological phase transition the +anomalous Hall conductivity could be served as an order parameter, while in the holo- +graphic NLSM it is not clear about the order parameters. Whether possible universal +“order parameter” exist for the topological phase transitions? What is the topological +nature in the topological phase from holography? These are elusive problems we aim to +explore from the holographic duality. +In holography, the thermal states are dual to black hole geometries in the bulk. The +black hole interior is expected to encode important information of the dual field the- +ory [28–30]. In the case that the thermal states are described by the black holes with +simple Kasner singularities, it has been shown recently in [31] that the order of the ther- +mal phase transition in the dual field theory is connected to the behavior of the Kasner +exponents of the black hole singularity.5 For the topological phase transitions in holo- +graphic topological semimetals at finite temperature, the systems experience a smooth +crossover from a topological phase, a critical phase to a trivial phase. Although the phase +crossover is different from thermal phase transitions, it is still interesting to explore the +interior geometries in holographic topological semimetals, in order to uncover possible +universal behavior during the topological phase transitions. +It turns out that there exist both universal and special behaviors of the singularities +in holographic topological semimetals. The universal behavior is similar to the topolog- +ical nature of topological phase and might give hints to the problems we raised for the +topological semimetals, while the special features can be understood from the fact that +the holographic WSM and the holographic NLSM share similarities and also differences +in the constructions as emphasized in [5, 7]. More precisely, in both cases, two matters +fields are added which play same role from the point of view of the boundary field theory, +while they play different roles in the bulk geometry. In the boundary field theory, one of +the two matter fields is to deform the Dirac point into two Weyl nodes or a nodal line, +while the other matter field is to gap the system. In the bulk, in the topological phase +of holographic WSM the IR geometry of Schwarzschild black hole is not deformed by the +matter fields, while the backreaction of the matter fields on the gravitational geometry is +quite strong in IR in the topological phase of holographic NLSM. We will see that these +two different situations lead to different properties of the black hole singularities in the +topological phases. +It is known that the information of the interior geometry can be probed from the +geodesics which correspond to certain correlators in the dual field theory. For example, +the proper time from the horizon to the singularity can be extracted from the thermal +one point function of certain heavy operator [30]. We will compute this quantity in the +4Other interesting developments can be found in e.g. [10–27]. +5Other studies on the geometric aspects of black hole singularities can be found in e.g. [32–53]. +2 + +bulk and study its behavior in the topological phases and trivial phases. +This paper is organized as follows. In Sec. 2, we will first review the holographic WSM +and then study its interior geometry as well as the proper time of the timelike geodesics. +In Sec. 3, we will review the holographic NLSM and then also study its interior geometry +and the proper time of the timelike geodesics. Sec. 4 is devoted to the conclusions and +open questions. The details of calculations are in the appendices. +2 +Inside holographic Weyl semimetal +In this section we first briefly review the holographic WSM which describes a topological +phase transition from topological WSM phase to a trivial semimetal phase. Then we +study the interior geometry of the black hole solutions and discuss the possible universal +behavior of the black hole singularities as well as the interior geometry. We also comment +on the possible observable as the role of “order parameter” during the topological phase +transition. +The action of the holographic WSM [3,4] is +S = +� +d5x√−g +� 1 +2κ2 +� +R + 12 +L2 +� +− 1 +4F2 − 1 +4F 2 + α +3 ϵabcdeAa +� +FbcFde + 3FbcFde +� +− (DaΦ)∗(DaΦ) − V (Φ) +� +, +(2.1) +where two gauge fields are dual to vector and axial currents respectively. A special Chern- +Simons structure is introduced to match the Wald identity for these currents. An axially +charged scalar field Φ is also introduced in the model with the source interpreted as the +mass term. Note that DaΦ = ∂aΦ − iqAaΦ where Aa is the axial U(1) gauge potential, +and V (Φ) = m2|Φ|2 + λ +2|Φ|4. We set 2κ2 = L = 1. +We focus on the finite temperature and use the following ansatz +ds2 = −udt2 + dr2 +u + f(dx2 + dy2) + hdz2 , +A = Azdz , +Φ = φ . +(2.2) +The equations of motion for the fields can be found in the appendix A. In the following we +consider m2 = −3, q = 1, λ = 1/10. Generalization to the other values of the parameters +is straightforward. +We use the following boundary conditions for the matter fields +lim +r→∞ Az = b , +lim +r→∞ rφ = M , +(2.3) +3 + +where b is the time reversal symmetry breaking parameter which play the role of split- +ting a Dirac point into two Weyl points, and M is the mass parameter which gaps the +Dirac point. The competing between these two effects leads to interesting topological +phase transitions. The system is completely determined by the dimensionless parameters +T/b, M/b. +In the weakly coupled WSM, the quantum topological phase transition could be man- +ifest from both the band structure and equivalently the behavior of the anomalous Hall +conductivity. In the strongly coupled model from holography, the anomalous Hall con- +ductivity behaves similarly to the weakly coupled case, indicating that there is a topo- +logical phase transition, as shown in Fig. 1. The lines in red, blue and purple are for +T/b = 0.05, 0.02, 0.01 respectively. The transition becomes sharp at zero temperature +and the dashed gray line is the critical value of the transition at zero temperature. +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +M +b +σAHE +8 α b +Figure 1: +Plot of anomalous Hall conductivity as a function of M/b at the temperatures +T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). The gray dashed line is the critical value of M/b +of the quantum phase transition at zero temperature. +2.1 +Inner structures +The phase transitions could be parameterized by the anomalous Hall conductivity which +is completely determined by the horizon value of the axial gauge field Az. Given the +possible connection between the physics inside and outside the horizon, it is interesting +to study the black hole inner structures during the topological phase transitions. +From the black hole solutions we have obtained, we could integrate the system further +to the singularity since the geometry is smooth at the horizon. We find that at low +temperature, the matter field φ oscillates inside the horizon only in the topological phase +4 + +(i.e. M/b < 0.744). The typical behavior is shown in Fig. 2, where the oscillation regime +of the scalar field φ (which has been rescaled according to φ/φh) as a function of r/rh +at fixed T/b (left) or M/b (right) are plotted respectively. We find that when we fix +the temperature T/b, the times of oscilation become less when we increase M/b from 0 +to (M/b)c. Furthermore, when we fix M/b < (M/b)c, the lower temperature, the more +times that φ oscillates. Note that the other fields do not show any oscillation from the +horizon to the singularity. +Different from the holographic superconductor cases, the oscillation here is not related +to the collapse of Einstein-Rosen bridge [34], since there is no inner horizon any more +for holographic WSM. Similar oscillation behavior has been found previously in neutral +helical black holes [35]. +��� +��� +��� +��� +��� +��� +-��� +-��� +-��� +��� +��� +��� +r +rh +ϕ +ϕh +��� +��� +��� +��� +��� +��� +-��� +-��� +��� +��� +��� +r +rh +ϕ +ϕh +Figure 2: The plots of φ/φh along radial direction in the oscillation region at fixed T/b = 0.02 +(left) while M/b = 0.1 (purple), 0.4 (blue), 0.6 (orange), 0.74 (red), as well as at fixed M/b = 0.1 +(right) while T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). Here φh is the horizon value of φ. +2.2 +Behaviors of Kasner exponents +The interior solution can be further integrated to the singularity. Near the singularity +rs, we assume that at the leading order the fields behave as +u ∼ −u0(r − rs)nu , +f ∼ f0(r − rs)nf , +h ∼ h0(r − rs)nh , +φ ∼ nφ ln(r − rs) , (2.4) +where u0, f0, h0 and nu, nf, nh, nφ are all constants. Here u0, f0, h0 depend on the scaling +symmetry in (A.3),(A.4),(A.5) while nu, nf, nh, nφ are not. Also note that here rs is not +necessarily to be zero since there is a shift symmetry of the system r → r + α along the +radial direction which was used to set the boundary behavior (A.8). Moreover, as we +shall see later, the axial gauge field Az is determined by the ansatz (2.4). +5 + +Near the singularity the equations of motion (A.6) can be simplified under the assump- +tion that the ignored terms are subleading which will be numerically checked afterward, +u′′ + h′ +2hu′ − +� +f ′′ + f ′h′ +2h +� u +f = 0 , +f ′′ +f + u′′ +2u − f ′2 +4f 2 + f ′u′ +fu + 1 +2φ′2 = 0 , +1 +2φ′2 − u′ +2u +�f ′ +f + h′ +2h +� +− f ′h′ +2fh − f ′2 +4f 2 = 0 , +A′′ +z + +�f ′ +f − h′ +2h + u′ +u +� +A′ +z = 0 , +φ′′ + +�f ′ +f + h′ +2h + u′ +u +� +φ′ = 0 . +(2.5) +Substituting (2.4) into (2.5), we obtain +nh = 2 (1 − nu − nf) , +nφ = ± +� +(2nf + nu)(1 − nu) − 3n2 +f +2 . +(2.6) +We can also solve the fourth equation in (2.5) to obtain at leading order Az +Az ≃ Azs0 + Azs1(r − rs)nh . +(2.7) +Note that the leading term Azs0 can be rescaled to be 1, while Azs1 could be determined +from the radial conserved quantities as will be discussed later. Thus there are only two +independent parameters in (2.4) and (2.7). +Note that in the above equations (2.5), we have assumed that the terms ignored are +subleading. More explicitly, we have assumed +nu < 2 , +nf + nu < 1 , +2nf + nu > 0 . +(2.8) +Numerically we have checked that all the above relations are satisfied for the parameters +we have considered, which indicates that the singularities are stable and of form (2.4) +and (2.7). +There are two radical conserved charge associated to the scaling symmetries of the +system, +Q1 = +√ +h(u′f − uf ′) , +(2.9) +Q2 = u′√ +hf − h′ +√ +h +uf − AzA′ +z +uf +√ +h +. +(2.10) +6 + +We have used them to check the accuracy of the numerics. Moreover, evaluate them at +the horizon and at the singularity we obtain +4πTf1 +� +h1 = Ts = u0f0 +� +h0(nf − nu) +(2.11) += u0f0 +√h0 +(nhAzs0Azs1 − h0(2nf + 3nu − 2) ) +(2.12) +where s is the density of entropy. From (2.11), we have nf > nu in addition to the con- +straints (2.8). Moreover, the above two conserved quantities give the relations nhAzs0Azs1 = +h0(3nf +2nu −2) which turns out to be zero in the topological phase at low temperature +where Azs1 = 3nf + 2nu − 2 = 0. +Starting from (2.2, 2.4) and performing the coordinate transformation +τ = − +2 +√n0(nu − 2)(r − rs)(2−nu)/2 , +(2.13) +we obtain the Kasner form for the fields +ds2 = −dτ 2 + ctτ 2ptdt2 + cxτ 2px(dx2 + dy2) + czτ 2pzdz2 , +φ = pφ log τ + cφ , +(2.14) +where +pt = +nu +2 − nu +, +px = +nf +2 − nu +, +pz = +nh +2 − nu +, +pφ = +2nφ +2 − nu +. +(2.15) +Note that Az is a constant at the leading order. Using the relations (2.6), the above +Kasner exponents can be expressed in terms of nu and nf, +pt = +nu +2 − nu +, +px = +nf +2 − nu +, +pz = 2(1 − nu − nf) +2 − nu +, +pφ = ± +� +4(2nf + nu)(1 − nu) − 6n2 +f +2 − nu +. +(2.16) +Note that the sign of pφ in (2.16) can only be determined from numerics. They satisfy +the following Kasner relations +pt + 2px + pz = 1 , +p2 +t + 2p2 +x + p2 +z + p2 +φ = 1 . +(2.17) +It indicates that only two of the four Kasner exponents are independent. +In Fig. 3, we show the Kasner exponents as a function of M/b at different tempera- +tures T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). We find that at low temperature, the +Kasner exponents in the Weyl semimetal phase take the same value of the Schwarzschild +black hole (e.g. within the difference of order less than 10−9 between M/b = 0.5 and +M/b = 0 at T/b = 0.01). This reminds us the topological feature in terms of the black +7 + +hole singularity. It is related to the fact that the matter fields do not backreact relevantly +to the Schwarzschild solution in the topological phase, i.e. the probe limit of system in +terms of matter fields in the Schwarzschild black hole background works well. We have +also checked that inside the black holes, in the topological phase the matter fields ob- +tained from the backreacted case match well with the solutions obtained from the probe +limit. In the quantum critical regime, the Kasner exponents oscillate. While in the trivial +phase, the Kasner exponent does not have any oscillate behavior. +��� +��� +��� +��� +��� +��� +��� +-���� +-���� +-���� +-���� +-���� +-���� +M +b +pt +��� +��� +��� +��� +��� +��� +��� +-��� +-��� +-��� +��� +��� +��� +M +b +pϕ +��� +��� +��� +��� +��� +��� +��� +���� +���� +���� +���� +���� +���� +M +b +px +��� +��� +��� +��� +��� +��� +��� +���� +���� +���� +���� +M +b +pz +Figure 3: Plots of Kasner exponents as a function of M/b. For all cases we have T/b = 0.05 +(red), 0.02 (blue), 0.01 (purple). The dashed gray vertical lines are the Kasner exponents of +five dimension Schwarzschild black hole. +Note that in [5], a paradigm for constructing the topological phase was proposed +and the holographic Weyl semimetal belongs to the first type, where the matter fields are +irrelevant in the IR of the Schwarzschild black hole. It seems likely that in any topological +phase of this kind, the singularities are of Kasner form taking values of Schwarzschild +black hole. +8 + +2.3 +Proper time of timelike geodesics +One of interesting connection between the interior geometry and the boundary observable +is given in [30] that the proper time of radial timelike geodesic can be encoded in the +thermal one point functions of heavy operators. It is thus interesting to study the proper +time of radial timelike geodesics to see if it has specific behavior during the topological +phase transitions. +We consider radial timelike geodesic for which gtt ˙t2 + grr ˙r2 = −1, where the dot +denotes the derivative with respect to the proper time τ. Along the geodesic there is a +conserved charge E = −gtt ˙t which can be interpreted as energy. Then the equation of +motion of the geodesic becomes +E2 +gtt ++ grr ˙r2 = −1 , +(2.18) +from which we obtain +dτ +dr = +1 +√ +E2 − u . +(2.19) +The proper time from the horizon to the singularity of a particle with E = 0 (i.e. the +longest time) is +τs = +� rh +rs +dr +√−u . +(2.20) +The plots of τs as a function of M/b for different T/b are shown in Fig. 4. +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +M +b +τs +Figure 4: Plots of the proper time τs from the horizon to the singularity as a function of M/b +at different temperatures T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). +The proper time from the horizon to the singularity in the topological phase is equal +to the case of Schwarzschild black hole τs = π/4 (e.g. within the difference of order less +9 + +than 10−4 between M/b = 0.5 and M/b = 0 at T/b = 0.01). This is expected from +the fact that in the topological phase at low temperature the interior of the black holes +match well with the Schwarzschild black hole. In the topologically trivial phase τs is +monotonically decreasing. Moreover, τs shows a jump behavior and takes a maximum +value in the critical regime. Note that τs is encoded in the thermal one point function +of heavy operators in the form of ⟨O⟩ ∝ e−imτs where the complexified mass m has +Im(m) < 0 [30]. One might use this thermal one point function as the “order” parameter +for the topological phase transition. The behavior of the proper time also reminds us the +behavior of the dimensionless information screening length in [14]. One obvious difference +is that the information screening length is determined by the quantities at the horizon, +while τs is determined by the geometry from the horizon to the singularity. +3 +Inside holographic nodal line semimetal +In the previous section, we have seen that the interior of the black hole geometries +for the holographic WSM exhibit interesting behavior. In the topological WSM phase, +the Kasner exponents of the dual geometries take the same value of the Schwarzschild +black hole at low temperature, as shown in Fig. 3. Moreover, the dual operator which +encodes the proper time from the horizon to the singularity could be served as an “order +parameter” during the topological phase transition, as shown in Fig. 4. To check if these +behaviors are universal for any topological phase transitions, in this section we study the +other topological phase transition model from holography, i.e. the holographic NLSM +model which describes a phase transition from the topological NLSM phase to a trivial +semimetal phase [5,6]. +The action for the holographic NLSM [6] is +S = +� +d5x √−g +� 1 +2κ2 +� +R + 12 +L2 +� +− 1 +4F2 − 1 +4F 2 + α +3 ϵabcdeAa +� +FbcFde + 3FbcFde +� +− (DaΦ)∗(DaΦ) − V1(Φ) − 1 +6ηϵabcde� +iBabH∗ +cde − iB∗ +abHcde +� +− V2(Bab) − λ|Φ|2B∗ +abBab +� +, +(3.1) +where Fab = ∂aVb − ∂bVa is the vector gauge field strength. Fab = ∂aAb − ∂bAa is the +axial gauge field strength. Da = ∇a −iq1Aa is the covariant derivative and q1 is the axial +charge of scalar field. α is the Chern-Simons coupling. Bab is an antisymmetric complex +two form field with the field strength +Habc = ∂aBbc + ∂bBca + ∂cBab − iq2AaBbc − iq2AbBca − iq2AcBab , +(3.2) +10 + +where q2 is the axial charge of the two form field. η is the Chern-Simons coupling strength +of the two form field. The introduction of the Chern-Simons terms while not canonical +kinetic term for the two form field follows from the self-duality condition of the two form +operator in the weakly coupled theory [6]. The potential terms are chosen as +V1 = m2 +1|Φ|2 + λ1 +2 |Φ|4 , +V2 = m2 +2B∗ +abBab , +(3.3) +where m2 +1 and m2 +2 are the mass parameters of the scalar field and the two form field. The +λ term in the action (3.1) denotes the interaction between the scalar field and the two +form field. We set 2κ2 = L = 1. +Similar to the holographic WSM, we focus on the finite temperature solution and take +the ansatz +ds2 = −udt2 + dr2 +u + f(dx2 + dy2) + hdz2 , +Φ = φ , +Bxy = −Byx = Bxy , +Btz = −Bzt = iBtz . +(3.4) +Plugging the above ansatz into the equations of motion, we could obtain the dynamical +equations of the fields, which can be found in the appendix B. In the following we choose +m2 +1 = −3, m2 +2 = 1, η = 2 and q1 = q2 = 1, λ = 1, λ1 = 0.1 for simplicity. +With the following boundary conditions, +lim +r→∞ rφ = M , +lim +r→∞ +Bxy +r += lim +r→∞ +Btz +r += b , +(3.5) +we can integrate the system from the boundary to the horizon. Different from the holo- +graphic WSM, in holographic NLSM there is no sharp “order parameter” like anomalous +Hall conductivity. Nevertheless, it was found in [6] that at zero temperature, the dual +fermionic spectral function shows multiple Fermi surfaces with the topology of nodal lines +when M/b < (M/b)c while it is gapped when M/b > (M/b)c. This indicates that the +system undergoes a topological phase transitions from topological NLSM to topologically +trivial semimetal phase. +With the regularity condition near the horizon, the system can be further integrated +to the singularity. In the following we will discuss the interior geometries and singularities +of the system. +11 + +3.1 +Kasner exponents +Close to the singularity r → rs, similar to the holographic WSM case we again take the +ansatz +u ∼ −u0(r − rs)nu , +f ∼ f0(r − rs)nf , +h ∼ h0(r − rs)nh , +φ ∼ nφ ln(r − rs) , (3.6) +where u0, f0, h0 and nu, nf, nh, nφ are all constants. The other two matter fields Btz and +Bxy will be determined by the above ansatz. +The equations of motion can be simplified close to the singularity under the assump- +tion that the ignored terms are subleading +u′′ +u − f ′′ +f + h′ +2h +�u′ +u − f ′ +f +� += 0 , +u′′ +2u + f ′′ +f − f ′2 +4f 2 + f ′u′ +fu + 1 +2φ′2 = 0 , +f ′2 +4f 2 + f ′h′ +2fh + u′ +2u +�f ′ +f + h′ +2h +� +− 1 +2φ′2 = 0 , +φ′′ + +�f ′ +f + h′ +2h + u′ +u +� +φ′ = 0 , +B′ +tz − η +√ +h +2f (λφ2)Bxy = 0 , +B′ +xy − +ηf +2 +√ +hu +(λφ2)Btz = 0 . +(3.7) +The first four equations in (3.7) are the same as the ones in holographic WSM. Similarly, +we obtain +nh = 2 (1 − nu − nf) , +nφ = ± +� +(2nf + nu)(1 − nu) − 3n2 +f +2 . +(3.8) +From the last two equations in (3.7) we have the following leading order solutions for the +two form fields near the singularity +Bxy ∼ Bxy0 + . . . , +Btz ∼ Btz0 + . . . , +(3.9) +where the dots are subleading terms of form (r − rs)2−nu−2nf(log(r − rs))2 and (r − +rs)2nf(log(r − rs))2 respectively. Here we have assumed 2 − nu − 2nf > 0 and nf > 0, +otherwise the leading solution of the two form field might be divergent. Similar to the +holographic WSM, these constants of the two form field depend on the scaling symmetry +of the system. +12 + +Note that in (3.7) we have assumed that the ignored terms are subleading. More +explicitly, we have assumed +nu < 2 , +2nu + nh < 2 , +nu + 2nf < 2 . +(3.10) +Note that the last two inequalities of above are consistent with the assumptions used in +obtaining (3.9). We have checked numerically that the inequalities (3.10) are satisfied for +the parameters we have considered. +Similar to the discussion in section 2.2, we can make a coordinate transformation +(2.13) to write the metric (3.6) into the Kasner form as (2.14) with the parameters (2.16) +and the Kasner relations (2.17). Here the leading order of the two form fields are constant +close to the singularity. +The two conserved charges of the scaling symmetries are +Q1 = 8 +ηBtzBxy + u +√ +h +(f ′h − fh′) , +(3.11) +Q2 = +f +√ +h +(u′h − uh′) . +(3.12) +Evaluate them at the horizon and at the singularity we obtain +8 +ηBxy0Btz0 = u0f0 +� +h0(nf − nh) +(3.13) +and +4πTf1 +� +h1 = Ts = u0f0 +� +h0(2 − 2nf − 3nu) +(3.14) +where s is the density of entropy. We have checked the above relations numerically. +In Fig. 5, we show the Kasner exponents for the holographic NLSM as functions of +M/b at different temperature T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). We find that +at low temperature, the Kasner exponents pt, px, pz of the metric fields in the NLSM +semimetal phase are almost constant in the topological phase (e.g. within the difference +of order less than 1% between M/b = 0.5 and M/b = 0 at T/b = 0.01), which is quite +similar to the holographic WSM, while pφ changes a lot in the topological phase. Note +that this is consistent with the Kasner relations (2.17) since pφ is small. It is expected that +at extremely low temperature, the properties of the Kasner exponents in the holographic +NLSM might be the same as those in the holographic WSM, i.e. all the Kasner exponents +are constant. Due to numerical difficulty we have not explored such a low temperature +regime. +Different from the holographic WSM where the geometry is the same as Schwarzschild +black hole with a constant nonzero Az when M/b = 0. Here when M/b = 0, in the +13 + +��� +��� +��� +��� +��� +-���� +-���� +-���� +-���� +-���� +M +b +pt +��� +��� +��� +��� +��� +-��� +-��� +-��� +��� +��� +��� +M +b +pϕ +��� +��� +��� +��� +��� +���� +���� +���� +���� +���� +���� +���� +M +b +px +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +M +b +pz +Figure 5: Plots of Kasner exponents for holographic NLSM as a function of M/b. For all cases +we have T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). The horizontal dashed gray lines represent +the Kasner exponents for M/b = 0 at T/b = 0.01. The vertical dashed gray lines represent the +quantum critical point at zero temperature. +holographic NLSM, due to the fact that the matter fields backreact to the IR geometry +and the Kasner exponents are no longer the constant exponents of Schwarzschild black +hole and instead they depend on T/b, as shown in the first three pictures in Fig. 6. +Nevertheless, at low enough temperature we see that the Kasner exponents are nearly +constant. +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +-0.298 +-0.296 +-0.294 +-0.292 +-0.290 +-0.288 +-0.286 +T +b +pt +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.182 +0.184 +0.186 +0.188 +0.190 +0.192 +T +b +px +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.916 +0.918 +0.920 +0.922 +T +b +pz +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.902 +0.904 +0.906 +0.908 +0.910 +0.912 +0.914 +T +b +τs +Figure 6: Plots of Kasner exponents and τs for holographic NLSM as a function of T/b when +M/b = 0. +14 + +3.2 +Proper time of timelike geodesics +Similar to the holographic WSM, we can also discuss the proper time from the horizon +to the singularity in holographic NLSM. In Fig. 7, we show the proper time τs as a +function of M/b at different temperatures. Again we see that at low temperature, the +proper time is almost a constant in the topological phase (e.g. within the difference of +order less than 5‰ between M/b = 0.5 and M/b = 0 at T/b = 0.01), which shows a +topological behavior under the changes of the systems. Similar to the holographic WSM, +we could take the operator which encodes the information of τs as the order parameter +for the topological phase transition in holographic NLSM. In the trivial phase, the proper +time τs is monotonically decreasing when we increase M/b. +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +��� +M +b +τs +Figure 7: Plots of the proper time τs from the horizon to the singularity as a function of M/b +at different temperatures T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). +4 +Conclusion and discussion +We have studied the interior geometries of black holes in two different holographic topo- +logical semimetals. We find that the singularities of the geometries are of simple Kasner +form, together with a constant one form gauge potential or constant two form fields. +In the topological WSM phase, all the Kasner exponents are constant taking values of +Schwarzschild black hole at low temperature. In the topological NLSM phase, the Kasner +exponents of the metric fields are also almost constant (the difference is of order less than +1% at T/b = 0.01), while the Kasner exponent of the scalar field is small and changes +a bit in the topological phase. Moreover, we find the proper times from the horizon to +the singularity are nearly constant in both holographic WSM and holographic NLSM. +These features seem to be of topological in the sense that they stay as constant during +15 + +the changes of physical parameters of the systems. The proper time in the trivial phases +of the two holographic semimetal decreases when we increase M/b. +In addition to the above universal behavior, specific behaviors inside the horizon are +also found. In the topological phase of holographic WSM, we find the oscillations of +the matter field φ inside the horizon at low temperature. In other phases we have not +found any oscillations of fields. The Kasner exponents oscillate in the critical regime +of holographic WSM. There is no oscillation of background fields in holographic NLSM. +In the trivial phases of the two holographic semimetals, the Kasner exponents behave +differently, where the details can be found in Fig. 3 and Fig. 5. +It would be interesting to connect the topological features of Kasner exponents and +the proper times in the topological phases of the two holographic semimetals to the +topological invariants. It is known that they can be extracted from the correlators of +heavy operators. It is very interesting to determine the precise observables associated +to these quantities to understand the role played by topology. This would shed light +on the universal theories describing the topological semimetals. Meanwhile, it is also +interesting to check the behavior of these physical quantities in the topological phases +of other holographic topological semimetals, e.g. [18, 25], to check if they are universal +feature of topological semimetals. +Acknowledgments +We are grateful to Matteo Baggioli, Karl Landsteiner, Ya-Wen Sun, Xin-Meng Wu, Jun- +Kun Zhao for useful discussions. This work is supported by the National Natural Science +Foundation of China grant No.11875083. +A +Equations in holographic WSM +In this appendix we list the useful equations for calculating the geometries in holographic +WSM in section 2. +16 + +The equations of motion for the action (2.1) are +Rab − 1 +2gab(R + 12) − Tab = 0 , +∇bF ba + αϵabcde(FbcFde + FbcFde) − iq (Φ∗(DaΦ) − Φ(DaΦ)∗) = 0 , +∇bFba + 2αϵabcdeFbcFde = 0 , +DaDaΦ − m2Φ − λΦ∗Φ2 = 0 , +(A.1) +where +Tab =1 +2(FacF c +b − 1 +4gabF2) + 1 +2(FacF c +b − 1 +4gabF 2) + 1 +2((DaΦ)∗DbΦ + (DbΦ)∗DaΦ) +− 1 +2gab((DcΦ)∗DcΦ + V (Φ)) +(A.2) +and DaΦ = ∂aΦ − iqAaΦ. +There are three different scaling symmetries of the system +(x, y) → a(x, y) , f → a−2f ; +(A.3) +z → az , h → a−2h , Az → a−1Az ; +(A.4) +r → ar , (t, x, y, z) → a−1(t, x, y, z) , (u, f, h) → a2(u, f, h) , Az → aAz . +(A.5) +For the ansatz (2.2), we have equations +u′′ + h′ +2hu′ − +� +f ′′ + f ′h′ +2h +�u +f = 0 , +f ′′ +f + u′′ +2u − f ′2 +4f 2 + f ′u′ +fu − 6 +u + φ2 +2u +� +m2 + λ +2φ2 − q2A2 +z +h +� +− A′2 +z +4h + 1 +2φ′2 = 0 , +1 +2φ′2 + 6 +u − u′ +2u +�f ′ +f + h′ +2h +� +− f ′h′ +2fh − f ′2 +4f 2 + A′2 +z +4h − φ2 +2u +� +m2 + λ +2φ2 − q2A2 +z +h +� += 0 , +A′′ +z + +�f ′ +f − h′ +2h + u′ +u +� +A′ +z − 2q2φ2 +u +Az = 0 , +φ′′ + +�f ′ +f + h′ +2h + u′ +u +� +φ′ − 1 +u +�q2A2 +z +h ++ m2 + λφ2� +φ = 0 . +(A.6) +Near the horizon r = rh, the fields can be expanded as follows, +u = 4πT(r − rh) + · · · , +f = f1 − f1Az2 +2m2φ2 +1 + λφ4 +1 − 24 +6Az1q2φ2 +1 +(r − rh) + · · · , +h = h1 − +� +Az1Az2 + h1Az2 +2m2φ2 +1 + λφ4 +1 − 24 +6Az1q2φ2 +1 +� +(r − rh) + · · · , +Az = Az1 + Az2(r − rh) + · · · , +φ = φ1 + Az2 +A2 +z1q2 + h1(m2 + λφ2 +1) +2Az1h1q2φ2 +1 +(r − rh) + · · · , +(A.7) +17 + +where T = +φ2 +1q2Az1 +2πAz2 . Note that there is a shift symmetry r → r + α along the radial +direction which can be used to fix rh to be any value and we choose rh = 1. There are +five free parameters T, f1, h1, Az1, φ1 and we can use the scaling symmetries (A.3, A.4) to +fix f1 = 1, h1 = 1 respectively. Then we can shoot three parameters T, Az1, φ1 to obtain +the parameters T, M, b of boundary field theory, i.e. the two dimensionless parameters +T/b, M/b according the scaling symmetry in (A.5) (we work in unit b = 1). +When r → ∞, the UV expansions are +u = r2 − M 2 +3 ++ M 4(2 + 3λ) +18 +ln r +r2 − Mb +r2 + · · · , +f = r2 − M 2 +3 ++ M 4(2 + 3λ) +18 +ln r +r2 + f3 +r2 + · · · , +h = r2 − M 2 +3 ++ M 4(2 + 3λ) + 9b2M 2q2 +18 +ln r +r2 + h3 +r2 + · · · , +Az = b − bM 2q2ln r +r2 + η +r2 + · · · , +φ = M +r − (3b2Mq2 + 2M 3 + 3λM 3)) +6 +ln r +r3 + O +r3 + · · · , +(A.8) +where h3 = +1 +72M(−72O + 9b2Mq2 + M 3(14 + 9λ)) − 2f3. +Note that in order to match the expansion (A.8) we should use the shift symmetry of +the system r → r + α which could change the location of the horizon/singularity. +B +Equations in holographic NLSM +In this appendix, we list the calculations for the geometries in holographic NLSM in +section 3. +The equations of motion for the action (3.1) are +Rab − 1 +2gab(R + 12) − Tab = 0 , +∇bFba + 2αϵabcdeFbcFde = 0 , +∇bF ba + αϵabcde(FbcFde + FbcFde) − iq1 (Φ∗DaΦ − (DaΦ)∗Φ) + q2 +η ϵabcdeBbcB∗ +de = 0 , +DaDaΦ − ∂Φ∗V1 − λΦB∗ +abBab = 0 , +i +3ηϵabcdeHcde − m2 +2Bab − λΦ∗ΦBab = 0 , +(B.1) +18 + +where +Tab = 1 +2(FacF c +b − 1 +4gabF2) + 1 +2(FacF c +b − 1 +4gabF 2) + 1 +2 +� +(DaΦ)∗DbΦ + (DbΦ)∗DaΦ +� ++ (m2 +2 + λ|Φ|2)(B∗ +acB c +b + B∗ +bcB c +a ) − 1 +2gab +� +(DcΦ)∗DcΦ + V1 + V2 + λ|Φ|2B∗ +cdBcd� +. +(B.2) +With the ansatz (3.4), the equations are +u′′ +u − f ′′ +f + h′ +2h +�u′ +u − f ′ +f +� +− 4 +u(m2 +2 + λφ2) +�B2 +tz +uh + B2 +xy +f 2 +� += 0 , +u′′ +2u + f ′′ +f − f ′2 +4f 2 + f ′u′ +fu − 6 +u + 1 +u(m2 +2 + λφ2) +�B2 +tz +uh + B2 +xy +f 2 +� ++φ2 +2u +� +m2 +1 + λ1φ2 +2 +� ++ φ′2 +2 = 0 , +f ′2 +4f 2 + f ′h′ +2fh + u′ +2u +�f ′ +f + h′ +2h +� +− 6 +u + 1 +u(m2 +2 + λφ2) +� +−B2 +tz +uh + B2 +xy +f 2 +� ++φ2 +2u +� +m2 +1 + λ1φ2 +2 +� +− 1 +2φ′2 = 0 , +B′ +tz − η +√ +h +2f (m2 +2 + λφ2)Bxy = 0 , +B′ +xy − +ηf +2 +√ +hu +(m2 +2 + λφ2)Btz = 0 , +φ′′ + φ′ +�u′ +u + f ′ +f + h′ +2h +� +− +� +m2 +1 + λ1φ2 − 2λB2 +tz +uh ++ 2λB2 +xy +f 2 +� φ +u = 0 . +(B.3) +There are three different scaling symmetries of the system +(x, y) → a(x, y) , f → a−2f , Bxy → a−2Bxy ; +(B.4) +z → az , h → a−2h , Btz → a−1Btz ; +(B.5) +r → ar , (t, x, y, z) → a−1(t, x, y, z) , (u, f, h, Bxy, Btz) → a2(u, f, h, Bxy, Btz) . +(B.6) +19 + +Near the horizon r → rh, the fields can be expanded as follows, +u = 4πT(r − rh) + · · · , +f = f1 − 4Bxy2 +� +8B2 +xy1(m2 +2 + λφ2 +1) + f 2 +1(2m2 +1φ2 +1 + λ1φ4 +1 − 24) +� +3Bxy1f1η2(m2 +2 + λφ2 +1)2 +(r − rh) + · · · , +h = h1 − 4h1Bxy2 +� +4B2 +xy1(m2 +2 + λφ2 +1) − f 2 +1(2m2 +1φ2 +1 + λ1φ4 +1 − 24) +� +3Bxy1f1η2(m2 +2 + λφ2 +1)2 +(r − rh) + · · · , +Bxy = Bxy1 + Bxy2(r − rh) + · · · , +Btz = η√h1Bxy1(m2 +2 + λφ2 +1) +2f1 +(r − rh) + · · · , +φ = φ1 + 4Bxy2φ1 +� +2λB2 +xy1 + f 2 +1(m2 +1 + λ1φ2 +1) +� +Bxy1f 2 +1η2(m2 +2 + λφ2 +1)2 +(r − rh) + · · · , +(B.7) +where T = Bxy1η2(m2 +2+λφ2 +1)2 +16πBxy2 +. The strategy of the numerics the same as the holographic +WSM. We first use the shift symmetry r → r + α to fix rh = 1. Then we also have five +free parameters T, f1, h1, Bxy1, φ1 and we can use the scaling symmetries (B.4, B.5) to +fix f1 = 1, h1 = 1 respectively. After that we have only three near horizon parameters +T, Bxy1, φ1, from which we obtain T, M, b in the dual field theory, which are equivalently +two dimensionless parameters T/b, M/b according the scaling symmetry (B.6). +Near the boundary r → ∞, we have +u = r2 − 2b2 − M 2 +3 ++ 8b4 + M 4(2 + 3λ1) +18 +ln r +r2 − Mb +r2 + · · · , +f = r2 − M 2 +3 ++ 8b4 + M 4(2 + 3λ1) +18 +ln r +r2 + f3 +r2 + · · · , +h = r2 − 2b2 − M 2 +3 ++ 8b4 + M 4(2 + 3λ1) +18 +ln r +r2 + h3 +r2 + · · · , +Bxy = br + 2b3 ln r +r ++ b2 +r + · · · , +Btz = br − 2b3 ln r +r +− b (b2 + M 2(1 + λ)) + b2 +r ++ · · · , +φ = M +r − M 3(2 + 3λ1) +6 +ln r +r3 + O +r3 + · · · , +(B.8) +where b2 = +1 +48b (−56b4 + 72(2f3 + h3) − 8b2M 2(2 + 3λ) − M 4(14 + 9λ1) + 72MO). +Note that to match the expansion (B.8) we should use the shift symmetry r → r + α +which could change the location of the horizon/singularity . +20 + +References +[1] J. McGreevy, Generalized Symmetries in Condensed Matter, [arXiv:2204.03045]. +[2] X. -G. 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Vegh, What lies beyond the horizon of a holographic p-wave super- +conductor, JHEP 12 (2022), 045 [arXiv:2210.01046]. +24 + diff --git a/7tAzT4oBgHgl3EQfgfwd/content/tmp_files/load_file.txt b/7tAzT4oBgHgl3EQfgfwd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d9f11e63be8cec1d1d8ce977db62ea333d8cbf4 --- /dev/null +++ b/7tAzT4oBgHgl3EQfgfwd/content/tmp_files/load_file.txt @@ -0,0 +1,994 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf,len=993 +page_content='Black hole interiors in holographic topological semimetals Ling-Long Gao a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Yan Liu a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='b2 and Hong-Da Lyu a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='b3 aCenter for Gravitational Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Department of Space Science and International Research Institute of Multidisciplinary Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Beijing 100191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' China bPeng Huanwu Collaborative Center for Research and Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Beijing 100191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' China Abstract We study the black hole interiors in holographic Weyl semimetals and holo- graphic nodal line semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We find that the black hole singularities are of Kasner form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the topologically nontrivial phase at low temperature, both the Kasner exponents of the metric fields and the proper time from the horizon to the singularity are almost constant, likely reflecting the topological nature of the topo- logical semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We also find some specific behaviors inside the horizon in each holographic semimetal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 1Email: linglonggao@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='cn 2Email: yanliu@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='cn 3Email: hongdalyu@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01468v1 [hep-th] 4 Jan 2023 Contents 1 Introduction 1 2 Inside holographic Weyl semimetal 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1 Inner structures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='2 Proper time of timelike geodesics .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 15 4 Conclusion and discussion 15 A Equations in holographic WSM 16 B Equations in holographic NLSM 18 1 Introduction The conventional classifications on the phases of matter are rooted in the Landau paradigm of symmetry breaking theory [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Over the past thirty years, new states of matter have been found which are beyond the concept of Landau paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' One example is the topological states of matter, including the quantum Hall states, topological insulators, topological semimetals and so on [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Different from the conventional Landau paradigm, there is no symmetry breaking during the topological phase transition and it attracts lots of research attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In recent years, the strongly interacting topological Weyl semimetals (WSM) [3,4] and nodal line semimetals (NLSM) [5,6] have been explicitly constructed from the holographic duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Both holographic WSM and NLSM are shown to possess nontrivial topological invariants [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Remarkably, holographic WSM exhibits interesting effects inherited from the boundary states [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' These features suggest that the physical properties associated to topology from the weakly coupled field theories persist in the strongly coupled topologi- cal systems from the holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Moreover, the systems could go through a topological phase transition to a topologically trivial semimetal phase, see [9] for a review on the 1 developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4 In the holographic WSM, during the topological phase transition the anomalous Hall conductivity could be served as an order parameter, while in the holo- graphic NLSM it is not clear about the order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Whether possible universal “order parameter” exist for the topological phase transitions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' What is the topological nature in the topological phase from holography?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' These are elusive problems we aim to explore from the holographic duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In holography, the thermal states are dual to black hole geometries in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The black hole interior is expected to encode important information of the dual field the- ory [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the case that the thermal states are described by the black holes with simple Kasner singularities, it has been shown recently in [31] that the order of the ther- mal phase transition in the dual field theory is connected to the behavior of the Kasner exponents of the black hole singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5 For the topological phase transitions in holo- graphic topological semimetals at finite temperature, the systems experience a smooth crossover from a topological phase, a critical phase to a trivial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Although the phase crossover is different from thermal phase transitions, it is still interesting to explore the interior geometries in holographic topological semimetals, in order to uncover possible universal behavior during the topological phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' It turns out that there exist both universal and special behaviors of the singularities in holographic topological semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The universal behavior is similar to the topolog- ical nature of topological phase and might give hints to the problems we raised for the topological semimetals, while the special features can be understood from the fact that the holographic WSM and the holographic NLSM share similarities and also differences in the constructions as emphasized in [5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' More precisely, in both cases, two matters fields are added which play same role from the point of view of the boundary field theory, while they play different roles in the bulk geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the boundary field theory, one of the two matter fields is to deform the Dirac point into two Weyl nodes or a nodal line, while the other matter field is to gap the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the bulk, in the topological phase of holographic WSM the IR geometry of Schwarzschild black hole is not deformed by the matter fields, while the backreaction of the matter fields on the gravitational geometry is quite strong in IR in the topological phase of holographic NLSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We will see that these two different situations lead to different properties of the black hole singularities in the topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' It is known that the information of the interior geometry can be probed from the geodesics which correspond to certain correlators in the dual field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' For example, the proper time from the horizon to the singularity can be extracted from the thermal one point function of certain heavy operator [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We will compute this quantity in the 4Other interesting developments can be found in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' [10–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 5Other studies on the geometric aspects of black hole singularities can be found in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' [32–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 2 bulk and study its behavior in the topological phases and trivial phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 2, we will first review the holographic WSM and then study its interior geometry as well as the proper time of the timelike geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 3, we will review the holographic NLSM and then also study its interior geometry and the proper time of the timelike geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 4 is devoted to the conclusions and open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The details of calculations are in the appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 2 Inside holographic Weyl semimetal In this section we first briefly review the holographic WSM which describes a topological phase transition from topological WSM phase to a trivial semimetal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Then we study the interior geometry of the black hole solutions and discuss the possible universal behavior of the black hole singularities as well as the interior geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We also comment on the possible observable as the role of “order parameter” during the topological phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The action of the holographic WSM [3,4] is S = � d5x√−g � 1 2κ2 � R + 12 L2 � − 1 4F2 − 1 4F 2 + α 3 ϵabcdeAa � FbcFde + 3FbcFde � − (DaΦ)∗(DaΦ) − V (Φ) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1) where two gauge fields are dual to vector and axial currents respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' A special Chern- Simons structure is introduced to match the Wald identity for these currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' An axially charged scalar field Φ is also introduced in the model with the source interpreted as the mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Note that DaΦ = ∂aΦ − iqAaΦ where Aa is the axial U(1) gauge potential, and V (Φ) = m2|Φ|2 + λ 2|Φ|4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We set 2κ2 = L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We focus on the finite temperature and use the following ansatz ds2 = −udt2 + dr2 u + f(dx2 + dy2) + hdz2 , A = Azdz , Φ = φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='2) The equations of motion for the fields can be found in the appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the following we consider m2 = −3, q = 1, λ = 1/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Generalization to the other values of the parameters is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We use the following boundary conditions for the matter fields lim r→∞ Az = b , lim r→∞ rφ = M , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='3) 3 where b is the time reversal symmetry breaking parameter which play the role of split- ting a Dirac point into two Weyl points, and M is the mass parameter which gaps the Dirac point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The competing between these two effects leads to interesting topological phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The system is completely determined by the dimensionless parameters T/b, M/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the weakly coupled WSM, the quantum topological phase transition could be man- ifest from both the band structure and equivalently the behavior of the anomalous Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the strongly coupled model from holography, the anomalous Hall con- ductivity behaves similarly to the weakly coupled case, indicating that there is a topo- logical phase transition, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The lines in red, blue and purple are for T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The transition becomes sharp at zero temperature and the dashed gray line is the critical value of the transition at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� M b σAHE 8 α b Figure 1: Plot of anomalous Hall conductivity as a function of M/b at the temperatures T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='05 (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The gray dashed line is the critical value of M/b of the quantum phase transition at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1 Inner structures The phase transitions could be parameterized by the anomalous Hall conductivity which is completely determined by the horizon value of the axial gauge field Az.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Given the possible connection between the physics inside and outside the horizon, it is interesting to study the black hole inner structures during the topological phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' From the black hole solutions we have obtained, we could integrate the system further to the singularity since the geometry is smooth at the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We find that at low temperature, the matter field φ oscillates inside the horizon only in the topological phase 4 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' M/b < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='744).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The typical behavior is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 2, where the oscillation regime of the scalar field φ (which has been rescaled according to φ/φh) as a function of r/rh at fixed T/b (left) or M/b (right) are plotted respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We find that when we fix the temperature T/b, the times of oscilation become less when we increase M/b from 0 to (M/b)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Furthermore, when we fix M/b < (M/b)c, the lower temperature, the more times that φ oscillates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Note that the other fields do not show any oscillation from the horizon to the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Different from the holographic superconductor cases, the oscillation here is not related to the collapse of Einstein-Rosen bridge [34], since there is no inner horizon any more for holographic WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Similar oscillation behavior has been found previously in neutral helical black holes [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� r rh ϕ ϕh ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� r rh ϕ ϕh Figure 2: The plots of φ/φh along radial direction in the oscillation region at fixed T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 (left) while M/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1 (purple), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='6 (orange), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='74 (red), as well as at fixed M/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1 (right) while T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='05 (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Here φh is the horizon value of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='2 Behaviors of Kasner exponents The interior solution can be further integrated to the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Near the singularity rs, we assume that at the leading order the fields behave as u ∼ −u0(r − rs)nu , f ∼ f0(r − rs)nf , h ∼ h0(r − rs)nh , φ ∼ nφ ln(r − rs) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4) where u0, f0, h0 and nu, nf, nh, nφ are all constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Here u0, f0, h0 depend on the scaling symmetry in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='3),(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4),(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5) while nu, nf, nh, nφ are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Also note that here rs is not necessarily to be zero since there is a shift symmetry of the system r → r + α along the radial direction which was used to set the boundary behavior (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Moreover, as we shall see later, the axial gauge field Az is determined by the ansatz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 5 Near the singularity the equations of motion (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='6) can be simplified under the assump- tion that the ignored terms are subleading which will be numerically checked afterward, u′′ + h′ 2hu′ − � f ′′ + f ′h′ 2h � u f = 0 , f ′′ f + u′′ 2u − f ′2 4f 2 + f ′u′ fu + 1 2φ′2 = 0 , 1 2φ′2 − u′ 2u �f ′ f + h′ 2h � − f ′h′ 2fh − f ′2 4f 2 = 0 , A′′ z + �f ′ f − h′ 2h + u′ u � A′ z = 0 , φ′′ + �f ′ f + h′ 2h + u′ u � φ′ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5) Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5), we obtain nh = 2 (1 − nu − nf) , nφ = ± � (2nf + nu)(1 − nu) − 3n2 f 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='6) We can also solve the fourth equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5) to obtain at leading order Az Az ≃ Azs0 + Azs1(r − rs)nh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='7) Note that the leading term Azs0 can be rescaled to be 1, while Azs1 could be determined from the radial conserved quantities as will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Thus there are only two independent parameters in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Note that in the above equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5), we have assumed that the terms ignored are subleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' More explicitly, we have assumed nu < 2 , nf + nu < 1 , 2nf + nu > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='8) Numerically we have checked that all the above relations are satisfied for the parameters we have considered, which indicates that the singularities are stable and of form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' There are two radical conserved charge associated to the scaling symmetries of the system, Q1 = √ h(u′f − uf ′) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='9) Q2 = u′√ hf − h′ √ h uf − AzA′ z uf √ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='10) 6 We have used them to check the accuracy of the numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Moreover, evaluate them at the horizon and at the singularity we obtain 4πTf1 � h1 = Ts = u0f0 � h0(nf − nu) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='11) = u0f0 √h0 (nhAzs0Azs1 − h0(2nf + 3nu − 2) ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='12) where s is the density of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='11), we have nf > nu in addition to the con- straints (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Moreover, the above two conserved quantities give the relations nhAzs0Azs1 = h0(3nf +2nu −2) which turns out to be zero in the topological phase at low temperature where Azs1 = 3nf + 2nu − 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Starting from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4) and performing the coordinate transformation τ = − 2 √n0(nu − 2)(r − rs)(2−nu)/2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='13) we obtain the Kasner form for the fields ds2 = −dτ 2 + ctτ 2ptdt2 + cxτ 2px(dx2 + dy2) + czτ 2pzdz2 , φ = pφ log τ + cφ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='14) where pt = nu 2 − nu , px = nf 2 − nu , pz = nh 2 − nu , pφ = 2nφ 2 − nu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='15) Note that Az is a constant at the leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Using the relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='6), the above Kasner exponents can be expressed in terms of nu and nf, pt = nu 2 − nu , px = nf 2 − nu , pz = 2(1 − nu − nf) 2 − nu , pφ = ± � 4(2nf + nu)(1 − nu) − 6n2 f 2 − nu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='16) Note that the sign of pφ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='16) can only be determined from numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' They satisfy the following Kasner relations pt + 2px + pz = 1 , p2 t + 2p2 x + p2 z + p2 φ = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='17) It indicates that only two of the four Kasner exponents are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 3, we show the Kasner exponents as a function of M/b at different tempera- tures T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='05 (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We find that at low temperature, the Kasner exponents in the Weyl semimetal phase take the same value of the Schwarzschild black hole (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' within the difference of order less than 10−9 between M/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5 and M/b = 0 at T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' This reminds us the topological feature in terms of the black 7 hole singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' It is related to the fact that the matter fields do not backreact relevantly to the Schwarzschild solution in the topological phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' the probe limit of system in terms of matter fields in the Schwarzschild black hole background works well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We have also checked that inside the black holes, in the topological phase the matter fields ob- tained from the backreacted case match well with the solutions obtained from the probe limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the quantum critical regime, the Kasner exponents oscillate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' While in the trivial phase, the Kasner exponent does not have any oscillate behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' ��� ��� ��� ��� ��� ��� ��� ���� ���� ���� ���� ���� ���� M b pt ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� M b pϕ ��� ��� ��� ��� ��� ��� ��� ���� ���� ���� ���� ���� ���� M b px ��� ��� ��� ��� ��� ��� ��� ���� ���� ���� ���� M b pz Figure 3: Plots of Kasner exponents as a function of M/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' For all cases we have T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='05 (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The dashed gray vertical lines are the Kasner exponents of five dimension Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Note that in [5], a paradigm for constructing the topological phase was proposed and the holographic Weyl semimetal belongs to the first type, where the matter fields are irrelevant in the IR of the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' It seems likely that in any topological phase of this kind, the singularities are of Kasner form taking values of Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='3 Proper time of timelike geodesics One of interesting connection between the interior geometry and the boundary observable is given in [30] that the proper time of radial timelike geodesic can be encoded in the thermal one point functions of heavy operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' It is thus interesting to study the proper time of radial timelike geodesics to see if it has specific behavior during the topological phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We consider radial timelike geodesic for which gtt ˙t2 + grr ˙r2 = −1, where the dot denotes the derivative with respect to the proper time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Along the geodesic there is a conserved charge E = −gtt ˙t which can be interpreted as energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Then the equation of motion of the geodesic becomes E2 gtt + grr ˙r2 = −1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='18) from which we obtain dτ dr = 1 √ E2 − u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='19) The proper time from the horizon to the singularity of a particle with E = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' the longest time) is τs = � rh rs dr √−u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='20) The plots of τs as a function of M/b for different T/b are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� M b τs Figure 4: Plots of the proper time τs from the horizon to the singularity as a function of M/b at different temperatures T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='05 (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The proper time from the horizon to the singularity in the topological phase is equal to the case of Schwarzschild black hole τs = π/4 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' within the difference of order less 9 than 10−4 between M/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5 and M/b = 0 at T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' This is expected from the fact that in the topological phase at low temperature the interior of the black holes match well with the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the topologically trivial phase τs is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Moreover, τs shows a jump behavior and takes a maximum value in the critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Note that τs is encoded in the thermal one point function of heavy operators in the form of ⟨O⟩ ∝ e−imτs where the complexified mass m has Im(m) < 0 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' One might use this thermal one point function as the “order” parameter for the topological phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The behavior of the proper time also reminds us the behavior of the dimensionless information screening length in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' One obvious difference is that the information screening length is determined by the quantities at the horizon, while τs is determined by the geometry from the horizon to the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 3 Inside holographic nodal line semimetal In the previous section, we have seen that the interior of the black hole geometries for the holographic WSM exhibit interesting behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the topological WSM phase, the Kasner exponents of the dual geometries take the same value of the Schwarzschild black hole at low temperature, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Moreover, the dual operator which encodes the proper time from the horizon to the singularity could be served as an “order parameter” during the topological phase transition, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' To check if these behaviors are universal for any topological phase transitions, in this section we study the other topological phase transition model from holography, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' the holographic NLSM model which describes a phase transition from the topological NLSM phase to a trivial semimetal phase [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The action for the holographic NLSM [6] is S = � d5x √−g � 1 2κ2 � R + 12 L2 � − 1 4F2 − 1 4F 2 + α 3 ϵabcdeAa � FbcFde + 3FbcFde � − (DaΦ)∗(DaΦ) − V1(Φ) − 1 6ηϵabcde� iBabH∗ cde − iB∗ abHcde � − V2(Bab) − λ|Φ|2B∗ abBab � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1) where Fab = ∂aVb − ∂bVa is the vector gauge field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Fab = ∂aAb − ∂bAa is the axial gauge field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Da = ∇a −iq1Aa is the covariant derivative and q1 is the axial charge of scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' α is the Chern-Simons coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Bab is an antisymmetric complex two form field with the field strength Habc = ∂aBbc + ∂bBca + ∂cBab − iq2AaBbc − iq2AbBca − iq2AcBab , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='2) 10 where q2 is the axial charge of the two form field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' η is the Chern-Simons coupling strength of the two form field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The introduction of the Chern-Simons terms while not canonical kinetic term for the two form field follows from the self-duality condition of the two form operator in the weakly coupled theory [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The potential terms are chosen as V1 = m2 1|Φ|2 + λ1 2 |Φ|4 , V2 = m2 2B∗ abBab , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='3) where m2 1 and m2 2 are the mass parameters of the scalar field and the two form field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The λ term in the action (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1) denotes the interaction between the scalar field and the two form field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We set 2κ2 = L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Similar to the holographic WSM, we focus on the finite temperature solution and take the ansatz ds2 = −udt2 + dr2 u + f(dx2 + dy2) + hdz2 , Φ = φ , Bxy = −Byx = Bxy , Btz = −Bzt = iBtz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4) Plugging the above ansatz into the equations of motion, we could obtain the dynamical equations of the fields, which can be found in the appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the following we choose m2 1 = −3, m2 2 = 1, η = 2 and q1 = q2 = 1, λ = 1, λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' With the following boundary conditions, lim r→∞ rφ = M , lim r→∞ Bxy r = lim r→∞ Btz r = b , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5) we can integrate the system from the boundary to the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Different from the holo- graphic WSM, in holographic NLSM there is no sharp “order parameter” like anomalous Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Nevertheless, it was found in [6] that at zero temperature, the dual fermionic spectral function shows multiple Fermi surfaces with the topology of nodal lines when M/b < (M/b)c while it is gapped when M/b > (M/b)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' This indicates that the system undergoes a topological phase transitions from topological NLSM to topologically trivial semimetal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' With the regularity condition near the horizon, the system can be further integrated to the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the following we will discuss the interior geometries and singularities of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1 Kasner exponents Close to the singularity r → rs, similar to the holographic WSM case we again take the ansatz u ∼ −u0(r − rs)nu , f ∼ f0(r − rs)nf , h ∼ h0(r − rs)nh , φ ∼ nφ ln(r − rs) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='6) where u0, f0, h0 and nu, nf, nh, nφ are all constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The other two matter fields Btz and Bxy will be determined by the above ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The equations of motion can be simplified close to the singularity under the assump- tion that the ignored terms are subleading u′′ u − f ′′ f + h′ 2h �u′ u − f ′ f � = 0 , u′′ 2u + f ′′ f − f ′2 4f 2 + f ′u′ fu + 1 2φ′2 = 0 , f ′2 4f 2 + f ′h′ 2fh + u′ 2u �f ′ f + h′ 2h � − 1 2φ′2 = 0 , φ′′ + �f ′ f + h′ 2h + u′ u � φ′ = 0 , B′ tz − η √ h 2f (λφ2)Bxy = 0 , B′ xy − ηf 2 √ hu (λφ2)Btz = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='7) The first four equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='7) are the same as the ones in holographic WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Similarly, we obtain nh = 2 (1 − nu − nf) , nφ = ± � (2nf + nu)(1 − nu) − 3n2 f 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='8) From the last two equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='7) we have the following leading order solutions for the two form fields near the singularity Bxy ∼ Bxy0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' , Btz ∼ Btz0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='9) where the dots are subleading terms of form (r − rs)2−nu−2nf(log(r − rs))2 and (r − rs)2nf(log(r − rs))2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Here we have assumed 2 − nu − 2nf > 0 and nf > 0, otherwise the leading solution of the two form field might be divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Similar to the holographic WSM, these constants of the two form field depend on the scaling symmetry of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 12 Note that in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='7) we have assumed that the ignored terms are subleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' More explicitly, we have assumed nu < 2 , 2nu + nh < 2 , nu + 2nf < 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='10) Note that the last two inequalities of above are consistent with the assumptions used in obtaining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We have checked numerically that the inequalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='10) are satisfied for the parameters we have considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Similar to the discussion in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='2, we can make a coordinate transformation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='13) to write the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='6) into the Kasner form as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='14) with the parameters (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='16) and the Kasner relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Here the leading order of the two form fields are constant close to the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The two conserved charges of the scaling symmetries are Q1 = 8 ηBtzBxy + u √ h (f ′h − fh′) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='11) Q2 = f √ h (u′h − uh′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='12) Evaluate them at the horizon and at the singularity we obtain 8 ηBxy0Btz0 = u0f0 � h0(nf − nh) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='13) and 4πTf1 � h1 = Ts = u0f0 � h0(2 − 2nf − 3nu) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='14) where s is the density of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We have checked the above relations numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 5, we show the Kasner exponents for the holographic NLSM as functions of M/b at different temperature T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='05 (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We find that at low temperature, the Kasner exponents pt, px, pz of the metric fields in the NLSM semimetal phase are almost constant in the topological phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' within the difference of order less than 1% between M/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5 and M/b = 0 at T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01), which is quite similar to the holographic WSM, while pφ changes a lot in the topological phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Note that this is consistent with the Kasner relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='17) since pφ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' It is expected that at extremely low temperature, the properties of the Kasner exponents in the holographic NLSM might be the same as those in the holographic WSM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' all the Kasner exponents are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Due to numerical difficulty we have not explored such a low temperature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Different from the holographic WSM where the geometry is the same as Schwarzschild black hole with a constant nonzero Az when M/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Here when M/b = 0, in the 13 ��� ��� ��� ��� ��� ���� ���� ���� ���� ���� M b pt ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� M b pϕ ��� ��� ��� ��� ��� ���� ���� ���� ���� ���� ���� ���� M b px ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� M b pz Figure 5: Plots of Kasner exponents for holographic NLSM as a function of M/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' For all cases we have T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='05 (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The horizontal dashed gray lines represent the Kasner exponents for M/b = 0 at T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The vertical dashed gray lines represent the quantum critical point at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' holographic NLSM, due to the fact that the matter fields backreact to the IR geometry and the Kasner exponents are no longer the constant exponents of Schwarzschild black hole and instead they depend on T/b, as shown in the first three pictures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Nevertheless, at low enough temperature we see that the Kasner exponents are nearly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='286 T b pt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='192 T b px 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='922 T b pz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='914 T b τs Figure 6: Plots of Kasner exponents and τs for holographic NLSM as a function of T/b when M/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='2 Proper time of timelike geodesics Similar to the holographic WSM, we can also discuss the proper time from the horizon to the singularity in holographic NLSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 7, we show the proper time τs as a function of M/b at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Again we see that at low temperature, the proper time is almost a constant in the topological phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' within the difference of order less than 5‰ between M/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5 and M/b = 0 at T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01), which shows a topological behavior under the changes of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Similar to the holographic WSM, we could take the operator which encodes the information of τs as the order parameter for the topological phase transition in holographic NLSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the trivial phase, the proper time τs is monotonically decreasing when we increase M/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� M b τs Figure 7: Plots of the proper time τs from the horizon to the singularity as a function of M/b at different temperatures T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='05 (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='02 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 4 Conclusion and discussion We have studied the interior geometries of black holes in two different holographic topo- logical semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We find that the singularities of the geometries are of simple Kasner form, together with a constant one form gauge potential or constant two form fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the topological WSM phase, all the Kasner exponents are constant taking values of Schwarzschild black hole at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the topological NLSM phase, the Kasner exponents of the metric fields are also almost constant (the difference is of order less than 1% at T/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='01), while the Kasner exponent of the scalar field is small and changes a bit in the topological phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Moreover, we find the proper times from the horizon to the singularity are nearly constant in both holographic WSM and holographic NLSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' These features seem to be of topological in the sense that they stay as constant during 15 the changes of physical parameters of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The proper time in the trivial phases of the two holographic semimetal decreases when we increase M/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In addition to the above universal behavior, specific behaviors inside the horizon are also found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the topological phase of holographic WSM, we find the oscillations of the matter field φ inside the horizon at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In other phases we have not found any oscillations of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The Kasner exponents oscillate in the critical regime of holographic WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' There is no oscillation of background fields in holographic NLSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' In the trivial phases of the two holographic semimetals, the Kasner exponents behave differently, where the details can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' It would be interesting to connect the topological features of Kasner exponents and the proper times in the topological phases of the two holographic semimetals to the topological invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' It is known that they can be extracted from the correlators of heavy operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' It is very interesting to determine the precise observables associated to these quantities to understand the role played by topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' This would shed light on the universal theories describing the topological semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Meanwhile, it is also interesting to check the behavior of these physical quantities in the topological phases of other holographic topological semimetals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' [18, 25], to check if they are universal feature of topological semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Acknowledgments We are grateful to Matteo Baggioli, Karl Landsteiner, Ya-Wen Sun, Xin-Meng Wu, Jun- Kun Zhao for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' This work is supported by the National Natural Science Foundation of China grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='11875083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' A Equations in holographic WSM In this appendix we list the useful equations for calculating the geometries in holographic WSM in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 16 The equations of motion for the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1) are Rab − 1 2gab(R + 12) − Tab = 0 , ∇bF ba + αϵabcde(FbcFde + FbcFde) − iq (Φ∗(DaΦ) − Φ(DaΦ)∗) = 0 , ∇bFba + 2αϵabcdeFbcFde = 0 , DaDaΦ − m2Φ − λΦ∗Φ2 = 0 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1) where Tab =1 2(FacF c b − 1 4gabF2) + 1 2(FacF c b − 1 4gabF 2) + 1 2((DaΦ)∗DbΦ + (DbΦ)∗DaΦ) − 1 2gab((DcΦ)∗DcΦ + V (Φ)) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='2) and DaΦ = ∂aΦ − iqAaΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' There are three different scaling symmetries of the system (x, y) → a(x, y) , f → a−2f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='3) z → az , h → a−2h , Az → a−1Az ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4) r → ar , (t, x, y, z) → a−1(t, x, y, z) , (u, f, h) → a2(u, f, h) , Az → aAz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5) For the ansatz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='2), we have equations u′′ + h′ 2hu′ − � f ′′ + f ′h′ 2h �u f = 0 , f ′′ f + u′′ 2u − f ′2 4f 2 + f ′u′ fu − 6 u + φ2 2u � m2 + λ 2φ2 − q2A2 z h � − A′2 z 4h + 1 2φ′2 = 0 , 1 2φ′2 + 6 u − u′ 2u �f ′ f + h′ 2h � − f ′h′ 2fh − f ′2 4f 2 + A′2 z 4h − φ2 2u � m2 + λ 2φ2 − q2A2 z h � = 0 , A′′ z + �f ′ f − h′ 2h + u′ u � A′ z − 2q2φ2 u Az = 0 , φ′′ + �f ′ f + h′ 2h + u′ u � φ′ − 1 u �q2A2 z h + m2 + λφ2� φ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='6) Near the horizon r = rh, the fields can be expanded as follows, u = 4πT(r − rh) + · · · , f = f1 − f1Az2 2m2φ2 1 + λφ4 1 − 24 6Az1q2φ2 1 (r − rh) + · · · , h = h1 − � Az1Az2 + h1Az2 2m2φ2 1 + λφ4 1 − 24 6Az1q2φ2 1 � (r − rh) + · · · , Az = Az1 + Az2(r − rh) + · · · , φ = φ1 + Az2 A2 z1q2 + h1(m2 + λφ2 1) 2Az1h1q2φ2 1 (r − rh) + · · · , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='7) 17 where T = φ2 1q2Az1 2πAz2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Note that there is a shift symmetry r → r + α along the radial direction which can be used to fix rh to be any value and we choose rh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' There are five free parameters T, f1, h1, Az1, φ1 and we can use the scaling symmetries (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4) to fix f1 = 1, h1 = 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Then we can shoot three parameters T, Az1, φ1 to obtain the parameters T, M, b of boundary field theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' the two dimensionless parameters T/b, M/b according the scaling symmetry in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5) (we work in unit b = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' When r → ∞, the UV expansions are u = r2 − M 2 3 + M 4(2 + 3λ) 18 ln r r2 − Mb r2 + · · · , f = r2 − M 2 3 + M 4(2 + 3λ) 18 ln r r2 + f3 r2 + · · · , h = r2 − M 2 3 + M 4(2 + 3λ) + 9b2M 2q2 18 ln r r2 + h3 r2 + · · · , Az = b − bM 2q2ln r r2 + η r2 + · · · , φ = M r − (3b2Mq2 + 2M 3 + 3λM 3)) 6 ln r r3 + O r3 + · · · , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='8) where h3 = 1 72M(−72O + 9b2Mq2 + M 3(14 + 9λ)) − 2f3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Note that in order to match the expansion (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='8) we should use the shift symmetry of the system r → r + α which could change the location of the horizon/singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' B Equations in holographic NLSM In this appendix, we list the calculations for the geometries in holographic NLSM in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The equations of motion for the action (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1) are Rab − 1 2gab(R + 12) − Tab = 0 , ∇bFba + 2αϵabcdeFbcFde = 0 , ∇bF ba + αϵabcde(FbcFde + FbcFde) − iq1 (Φ∗DaΦ − (DaΦ)∗Φ) + q2 η ϵabcdeBbcB∗ de = 0 , DaDaΦ − ∂Φ∗V1 − λΦB∗ abBab = 0 , i 3ηϵabcdeHcde − m2 2Bab − λΦ∗ΦBab = 0 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='1) 18 where Tab = 1 2(FacF c b − 1 4gabF2) + 1 2(FacF c b − 1 4gabF 2) + 1 2 � (DaΦ)∗DbΦ + (DbΦ)∗DaΦ � + (m2 2 + λ|Φ|2)(B∗ acB c b + B∗ bcB c a ) − 1 2gab � (DcΦ)∗DcΦ + V1 + V2 + λ|Φ|2B∗ cdBcd� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='2) With the ansatz (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' the equations are u′′ u − f ′′ f + h′ 2h �u′ u − f ′ f � − 4 u(m2 2 + λφ2) �B2 tz uh + B2 xy f 2 � = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' u′′ 2u + f ′′ f − f ′2 4f 2 + f ′u′ fu − 6 u + 1 u(m2 2 + λφ2) �B2 tz uh + B2 xy f 2 � +φ2 2u � m2 1 + λ1φ2 2 � + φ′2 2 = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' f ′2 4f 2 + f ′h′ 2fh + u′ 2u �f ′ f + h′ 2h � − 6 u + 1 u(m2 2 + λφ2) � −B2 tz uh + B2 xy f 2 � +φ2 2u � m2 1 + λ1φ2 2 � − 1 2φ′2 = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' B′ tz − η √ h 2f (m2 2 + λφ2)Bxy = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' B′ xy − ηf 2 √ hu (m2 2 + λφ2)Btz = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' φ′′ + φ′ �u′ u + f ′ f + h′ 2h � − � m2 1 + λ1φ2 − 2λB2 tz uh + 2λB2 xy f 2 � φ u = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='3) There are three different scaling symmetries of the system (x, y) → a(x, y) , f → a−2f , Bxy → a−2Bxy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4) z → az , h → a−2h , Btz → a−1Btz ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5) r → ar , (t, x, y, z) → a−1(t, x, y, z) , (u, f, h, Bxy, Btz) → a2(u, f, h, Bxy, Btz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='6) 19 Near the horizon r → rh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' the fields can be expanded as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' u = 4πT(r − rh) + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' f = f1 − 4Bxy2 � 8B2 xy1(m2 2 + λφ2 1) + f 2 1(2m2 1φ2 1 + λ1φ4 1 − 24) � 3Bxy1f1η2(m2 2 + λφ2 1)2 (r − rh) + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' h = h1 − 4h1Bxy2 � 4B2 xy1(m2 2 + λφ2 1) − f 2 1(2m2 1φ2 1 + λ1φ4 1 − 24) � 3Bxy1f1η2(m2 2 + λφ2 1)2 (r − rh) + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Bxy = Bxy1 + Bxy2(r − rh) + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Btz = η√h1Bxy1(m2 2 + λφ2 1) 2f1 (r − rh) + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' φ = φ1 + 4Bxy2φ1 � 2λB2 xy1 + f 2 1(m2 1 + λ1φ2 1) � Bxy1f 2 1η2(m2 2 + λφ2 1)2 (r − rh) + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='7) where T = Bxy1η2(m2 2+λφ2 1)2 16πBxy2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' The strategy of the numerics the same as the holographic WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' We first use the shift symmetry r → r + α to fix rh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Then we also have five free parameters T, f1, h1, Bxy1, φ1 and we can use the scaling symmetries (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='4, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='5) to fix f1 = 1, h1 = 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' After that we have only three near horizon parameters T, Bxy1, φ1, from which we obtain T, M, b in the dual field theory, which are equivalently two dimensionless parameters T/b, M/b according the scaling symmetry (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Near the boundary r → ∞, we have u = r2 − 2b2 − M 2 3 + 8b4 + M 4(2 + 3λ1) 18 ln r r2 − Mb r2 + · · · , f = r2 − M 2 3 + 8b4 + M 4(2 + 3λ1) 18 ln r r2 + f3 r2 + · · · , h = r2 − 2b2 − M 2 3 + 8b4 + M 4(2 + 3λ1) 18 ln r r2 + h3 r2 + · · · , Bxy = br + 2b3 ln r r + b2 r + · · · , Btz = br − 2b3 ln r r − b (b2 + M 2(1 + λ)) + b2 r + · · · , φ = M r − M 3(2 + 3λ1) 6 ln r r3 + O r3 + · · · , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='8) where b2 = 1 48b (−56b4 + 72(2f3 + h3) − 8b2M 2(2 + 3λ) − M 4(14 + 9λ1) + 72MO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' Note that to match the expansion (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='8) we should use the shift symmetry r → r + α which could change the location of the horizon/singularity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' 20 References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' McGreevy, Generalized Symmetries in Condensed Matter, [arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content='03045].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' [2] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfgfwd/content/2301.01468v1.pdf'} +page_content=' -G.' 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Patra +National Institute of Technology(NIT), Rourkela, Odisha, +India +Indian Institute of Technology (IIT), Varanasi, Uttar Pradesh, +India + + +Abstract +Machine Translation (MT) system generally aims at automatic +representation of source language into target language retaining the +originality of context using various Natural Language Processing (NLP) +techniques. Among various NLP methods, Statistical Machine Trans- +lation (SMT) is a very popular and successful architecture used for +both low as well as high-resource languages. SMT uses probabilis- +tic and statistical techniques to analyze information and conversion. +This paper canvasses about the development of bilingual SMT mod- +els for translating English to fifteen low-resource Indian Languages +(ILs) and vice versa. At the outset, all 15 languages are briefed with +a short description related to our experimental need. Further, a de- +tailed analysis of Samanantar and OPUS dataset for model building, +along with standard benchmark dataset (Flores-200) for fine-tuning +and testing, is done as a part of our experiment. Different preprocess- +ing approaches are proposed in this paper to handle the noise of the +dataset. To create the system, MOSES open-source SMT toolkit is +explored. “Distance” reordering is utilized with the aim to understand +the rules of grammar and context-dependent adjustments through a + +2 + + + + + +phrase reordering categorization framework. In our experiment, the +quality of the translation is evaluated using standard metrics such as +BLEU, METEOR, and RIBES. + +1 Introduction +Technology reaches new heights through its journey from the origins of ideas +to their full-scale practical implementation. One such journey is heading to- +wards elimination of language barrier in order to establish a seamless social +communication in every domain. In this regard, advancement on relevant +fields such as Natural Language Processing (NLP), Machine Learning (ML) +and Artificial Intelligence (AI) based Language Modelling (LM) significantly +contributes for evolving a flawless automatic Machine Translation (MT) sys- +tem (Dorr et al ( 2004)). Irrespective of various heuristic approaches to +maintain both lexical and contextual interpretation of source language(s) +onto the translated target language(s), it is still challenging to cope with +required fluency, adequacy, accent, and overall accuracy (Chapelle et al ( +2010)). However, it is feasible with the advent of modern NLP (AI-based) +approaches wherein a high-quality and high resource (i.e. large quantity +of corpora available) parallel corpus (translation pairs in source and target +languages) is required to train a good translation system. Hence, for high- +resource languages having massive digital footprint across the globe, MT sys- +tems prove to be quite efficient with adequate training. On the other hand, +it becomes very complicated for low-resource languages suffering from uni- +versal recognition and scanty digital presence. Such imbalance often leads to +poor-quality translation in presence of low-resource language(s) in the form +of either target or source. Therefore, MT systems need to understand the +syntax (rules to combine words), semantics (meaning of words and combi- +nations), and morphology (rules to cover morphemes - smallest meaningful +units - into words) of such low-resource languages (Somers (2011)). +Based on the heuristic paradigms, MT models are classified into rule-based +(RBMT), example-based (EBMT), statistical (SMT), and neural (NMT) sys- +tems (Tripathi et al (2010)). Each has its own advantages and disadvantages. +RBMT models follow a set of rules to define a language and the interaction +between different linguistic devices (words, phrases, sentences) in the lan- +guage (Jussà et al (2012), Michael et al (2000)). These sets of rules and +systems defined for a translation in a language pair are hard-coded on the + +3 + + + + + +machine. The linguistic information used in an RBMT model is mainly +the target and source languages collected from unilingual (one language), +bilingual (two languages), or multilingual (more than two languages) dic- +tionaries. In addition, the model also uses grammar covering the syntactic, +semantic, and morphological regularities of each language. However, a well- +built RBMT model requires highly skilled and expert human labour due to +its complexity making it hard to build. In addition, the ambiguous proper- +ties of languages make them prone to take more time and efforts to resolve, +especially in large and complex models. RBMT models require a lot of effort +to be made functional in day-to-day life. Hence, the need for more efficient +translation systems than RBMT still persists. EBMT methods make use of a +large number of translation examples (John (2005)). Notably, EBMT mod- +els make use of bilingual corpora manipulation, i.e. the breaking down of +a bilingual corpus into smaller parts, translating those parts into the target +language, and recompiling it to form whole translated sentences. They do +not account much for the syntax, semantic and morphological analysis of the +target and source language (like RBMT models). In contrast, SMT is better +when compared to RBMT and EBMT models, as it does not require human +intervention (Adam (2008)). It is a way of translation wherein a statistical- +based learning algorithm is applied to a large bilingual corpus that helps +the machine learn the translation. This method also enables the machine to +translate sentences not encountered by the machine during its training and +testing. The objective of SMT is to convert an input word sequence from the +source language into the target language. It has dominated academic MT +research and a portion of the commercial MT sector in less than two decades. +On the other hand, neural machine translation (NMT) is performed using a +neural network (NN) (Stasimioti (2020)). Unlike SMT, NMT does not have a +distinct translation model, language model, or model for reordering. Instead, +it has a single sequence model that determines one word at a time. The pre- +diction is based on the source sentence effort previously generated sequence +in the target language. NMT is a deep learning-based method of machine +learning that utilizes a large NN that relies on word vector representations. +Even though the NMT has achieved remarkable results in a few trans- +lation experiments using high-resource language, researchers are unsure if +the NMT could actually replace SMT and if its success would extend to +other tasks. Eventually, the experiment of (Michał (2016)) on the corpus +of the United Nations (consisting of 15 low-resource languages) brings the +fact. From the result of his experiment, it is evident that the performance of + +4 + + + + + +SMT is better than that of NMT for the majority of cases, as measured by +BLEU score. Many researchers (Lohar et al (2019), Zhou et al (2017), Wang +et al (2017), Castilho et al (2017)) have pointed out various disadvantages +of NMT over SMT using low resource language, such as the fact that NMT +requires more corpus and resources than SMT. In comparison with SMT, +NMT training typically takes longer. Additionally, research has shown that +when there is a domain incompatibility between testing and training data, +SMT performance is superior to that of NMT (Xing et al (2018), Mahata et +al (2018)). Long sentences are another area where SMT excels. +English and ILs are languages with less parallel text data, which motivates us +to work with ILs. This research examines the effectiveness of SMT systems on +low-resource language pairs, of which many are rarely worked on. The dataset +used in our experiment for all fifteen Indian languages is tested for the first +time for all languages using SMT. Hence, the objective of this work is to build +an MT system using SMT for languages such as Assamese (AS), Malayalam +(ML), Bengali (BN), Marathi (MR), Gujarati (GU), Kannada (KN), Hindi +(HI), Oriya (OR), Punjabi (PA), Telugu (TE), Sindhi (SD), Sinhala (SI), +Nepali (NE), Tamil (TA), and Urdu (UR) to English (EN) and vice versa and +to check the effectiveness of SMT with low-resource language pairs. +Our main goal is to develop an MT system for low-resource languages, i.e., ILs, +that can serve as a baseline system. The following is a summary of our +work’s main contributions: +• To the best of our knowledge, this work is the first attempt to use SMT +with the Samanantar and OPUS Dataset to investigate the MT for all +fifteen IL-EN and EN-IL pairs (both directions), including both the +Dravidian and Indo-Aryan groups. +• To bring forth the linguistic approach of ILs in terms of translation. +Scripts, writing style, and grammar with proper examples are also dis- +cussed. +• Various data filtration methods are investigated in order to clean the +data and improve translation quality. +• Distance-based reordering is utilized to check the translation quality of +ILs. + +5 + +| +| + + + + +• Better realistic assessment of translation quality is possible from the +presentation of results, as obtained using different automated metrics +like BLEU, METEOR, and RIBES. +This paper is arranged as follows. Subsections 1.1 and 1.2 give some +insight into SMT and cover the ILs used for our experiments. In Section 2, +some prominent works on SMT and NMT using ILs are described. The +experimental framework, including an overview of the dataset and method- +ology, is explained in Section 3. Section 4 narrates some of the prominent +metrics used for MT evaluation. Results are presented in Section 5 followed by +the conclusion and future direction in Section 6. + +1.1 SMT +Statistical Machine Tramslation (SMT) is dependent on statistical methods +(Philipp et al (2007), Richard et al (2002), Mary et al (2011) ). It is a data- +driven technique that makes use of parallel-aligned corpora. It utilizes +mathematical equations to calculate the likelihood of source-to-target lan- +guage translation. Probability P (Tl Si) is assigned by SMT. Here Tl is the target +language and Si is the source input. It utilizes Bayes’ theorem to +determine the maximum probability P (Tl|Si), which is as follows: +P (Tl | Si) ∝ P (Tl)P (Si | Tl) +(1) +SMT consists of three phases: the language model(LM) P (Tl) for target +language probability calculation, the translation model(TM) P (Si Tl) for +conditional probability estimation of the target to the source language, and the +decoder model (DM), which searches among possible source sentences the +one which maximizes probabilities (Kumawat et al (2014)). +To calculate the probability of a sentence, the LM utilizes the n-gram model. It +assigns the probability of a single word to the last n words that come before it +in the sentence and estimates the translation’s likelihood. The chain rule aids +in breaking down the sentence into conditional probability products. + +P (s) = P (w1, w2, w3, ..., wn) += P (w1)P (w2|w1)P (w3|w1w2)P (w4|w1w2w3)...P (wn|w1w2...wn−1) += P (w1)P (w2|w1)P (w3|w1w2)P (w4|w1w2w3)...P (wn|w1w2...wn−k) +(2) +Where, P (s) is the probability of the sentence s, consisting of words w1, w2, +..., wn, assuming a k-gram model. It utilizes the bilingual parallel corpus + +6 + + + + + +of the desired language pair. This is accomplished by calculating the like- +lihood of words or phrases extracted from sentences. The DM is the final +and most crucial phase of SMT. It assists in the selection of words with the +highest probability to be translated by maximizing the likelihood, i.e. +P (Tl)P (Si | Tl). +1.2 Language preference +India is a multilingual nation where people from various states use a variety of +regional tongues. Such diversity of language brings difficulty in commu- +nicating with one another for information exchange. Further, limitations in +public communication also bring inconvenience to share feelings, thoughts, +opinions and facts, as well as to deal with business purposes. Moreover, there +are many helpful resources available on the internet in English but many In- +dians struggle to take benefit of those due to language barriers. Hence, it is +crucial to have an easy translation solution for regional languages to support +effective communication and to help utilising global resources. To make it +possible, technological innovation are continuing to find out efficient methods +for a flawless translation using machines, because it is impractical to have hu- +man translators everywhere. For machine translation, an enormous amount +of resources is required for training with a proper knowledge-base (rules) for +better efficiency so as to fulfill the demand of a flawless translation solu- +tion. For translation, understanding the meaning of words is important, but +words are not enough to constitute a language as a whole. They must be +used in sentence construction that adheres to strict grammar rules and every +language is having its own writing style. In our work, 15 commonly spoken +languages (over various regions of India) are chosen. Table 1 describes the +languages used in our experiments with their linguistic features (ethnologue +(2022)). A short introduction about them in terms of translation is given +below. +English(EN) +English language is the primary language of roughly 45 countries and +is spoken by nearly 1,132 million people. It is written in Roman script, +which uses both uppercase and lowercase characters. English uses the +subject-verb-object structure. For example (expl1), “The poor man +took food”, and (expl2) “food took the poor man”. When the position + +7 + + + + + +of the subject changes in the preceding sentences, the significance and +meaning of the English sentence change. +Assamese(AS) +Over 15 million native Assamese speakers live in the state of Assam in +the northeastern region of India. It is one of Assam’s official languages. +Additionally, it is spoken in various regions of other northeastern In- +dian states. It uses the Bengali-Assamese script and is written left to +right. It also follows the SOV format. “Gita is eating mango” is an +English sentence that when translated into Assamese became গীতাই আম +খাই আআছ which follows subject object verb format. গীতাই (Gita, +subject), আম(mango, object) and খাই আআছ(is eating, verb). +Malayalam(ML) +People in Kerala and a few societies in Karnataka and Tamil Nadu use +Malayalam for communication. This language is spoken by about 35 +million citizens. It uses the SOV style of writing and a nominative- +accusative case marking sequence. It is written in Malayalam script in +left-to-right fashion. Sentence like സീതയ്ക്ക് ചിത്തരചന ഇഷ്ട മാണ് +which in English became “Sita loves drawing”. Here the word +സീതയ്ക്ക് +(Sita,Subject), ഇഷ്ടമാണ് +(loves,Verb) and ചിത്തര +ചന(drawing,Object). +Bengali(BN) +It is the primary language of Bangladesh and the second most spoken +language in India. Over 265 million people use it as their primary +or second language. Approximately 11 million Bengali speakers exist +in Bangladesh. In India, states such as Assam, Tripura, and West +Bengal use this language. It is a member of the Indo-Aryan family. In +Bengali sentences, the standard word order is Subject-Object-Verb. For +example, in sentence আ রাি জ ভাত খায় which in English is “Rosy eats +rice ”. Here আ রাি জ (Rosy, Subject), ভাত (Rice, object) and খায় +(Eats,Verb). +Marathi(MR) +Marathi is associated with the Sanskrit-derived group of Indian lan- +guages and is used by 95 million people in India for communication, +primarily in the central and western regions. The fourth most widely +spoken language in India is Marathi, which has a sizable native-speaker + +8 + + + + + +population. Similar to Hindi and Nepali, Marathi is written in the De- +vanagari script in left-to-right order. It follows the Subject-Object-Verb +order. For example, the sentence तो दध ि पतो, which means “He drinks +milk.” in English, has तो दध +ject), and ि पतो(drinks, +verb). +Gujarati(GU) +ि पतो where तो(He, subject), +दध +(Milk, ob- +Gujarati is spoken by 45 million citizens in Gujarat and is associated +with the Indo-Aryan group. It uses the SOV writing style and is drafted +from left to right in Gujarati script. For example, in the sentence તે +આઈસ્ક્રીમ ખાય છે. which in English is “He is eating ice cream.” where તે +is subject, આઈસ્ક્રીમ is an object and verb is ખાય છે. +Kannada (KN) +Karnataka’s official language is Kannada, which is also widely used +in other parts of India. In India, about 36 million people speak and +write Kannada. Despite being a Dravidian language with extensive +historical literature, Kannada has few computational linguistics re- +sources, making it challenging to study the language’s literature due +to its semantic and syntactic diversity. Subject-Object-Verb is the way +the Kannada language is structured. Kannada is a highly agglutina- +tive language. It uses the left-to-right Kannada script For example, +ರಾಮ ಶಾಲೆಗೆ ಹೋದ(SOV) is in English is “Rama went to school”. Here, +ರಾಮ(Rama, Subject), ಶಾಲೆ(school, object), and ಹೋದರು(went, verb). +Hindi(HI) +Hindi is one of the official and national languages of India. There are +more than 615 million people who use Hindi as their primary language, +and even more than 341 million who speak it as a second language. +However, the sentence structure is Subject Object Verb as shown in +the example: गीता स्क ल जाती है। is in English is “Geeta goes to school”. In +this sentence, गीता (Gita ,Subject), स्क ल (School, Object) and जाती +है(Goes, Verb). The Indian Constitution mandates that Hindi written +in Devanagari be used as the Union’s official language. +Oriya(OR) +The Oriya language is the primary language of Odisha, a state in east- +ern India. Oriya belongs to the Eastern Indo-Aryan group of languages. + +9 + +Its standard format is subject-object-verb (SOV) and is written in the + +Odia script from left to right. +Punjabi (PA) +Punjabi text is written in a subject-object-verb format and is spoken +in India and Pakistan, and a few small groups in the United Kingdom, +United Arab Emirates, Malaysia, the United States, South Africa, and +Canada. It is written in two scripts: the western Perso-Arabic Shah- +mukhi script and the eastern Gurmukhi script. Gurmukhi is drafted +from left to right, whereas Shahmukhi is written in the opposite direc- +tion. ਅਸ ੀਂ ਭਾਰਤ ਹਾੀਂ is in English “We are Indians” where ਅਸ ੀਂ(We,Sub- +ject), ਭਾਰਤ (Are,Verb) and ਹਾੀਂ (Indians,Object). +Telugu(TE) +Telugu is the official language of two Indian states in the south: Andhra +Pradesh and Telangana. It is also spoken by the Telugu-speaking im- +migrant communities in the United States, Canada, and the United +Kingdom. Text structure in Telugu takes the form of a subject-object- +verb and from left to right.ఆమె నన్ను కొటి్టంది in English “she beat +me” where ఆమె(she, Subject), నన్ను (Me, Object) and కొటి్టంది(beat, +Verb). +Sindhi(SD) +Sindhi is a language spoken by 25 million speakers in Pakistan and 5 +million in India. It is written in a modified Perso-Arabic script in Pak- +istan (right-to-left), whereas it is written in a variety of scripts in India, +like Devanagari, Khudabadi, and Gurmukhi (left-to-right). It follows the +Subject-Object-Verb format. For example, the sentence “Partha +loves books” is رٿﭘﺎ ﮐﻲ ﺘﺎﺑﻦڪ ﻦﺳﺎ رﭘﻴﺎ ﻫﻲآ where ٿﺎرﭘﺎ(Partha, subject), +ﮐﻲ ﺘﺎﺑﻦڪ (books, object) and ﻦﺳﺎ رﭘﻴﺎ ﻫﻲآ (loves, verb). +Nepali(NE) +It is official language and the lingua franca in Nepal, and also spoken +by some communities in India. Nepali is written in left-to-right De- +vanagari script. It is a language written in Subject-Object-Verb order For +example, “Sita ate apples” when converted to the Nepali language + +10 + + + + + +becomes सीताले स्याउ खाइन्. Here, सीताले(Sita, subject), स्याउ(apples, +object) and खाइन्(ate, verb). +Sinhala(SI) +The majority of Sri Lankans speak Sinhala as their first language. Sin- +hala is an Indo-Aryan language that differs from English in terms of +grammatical structure, morphological variation, and subject-object- +verb (SOV) word order. It is written in right-to-left Sinhala script. +A sentence like “Pavan writes a letter” is in Sinhala is පවන් ලිපියක් +ලියයි where පවන්(Pavan, subject), ලිපියක්(a letter, object) and +ලියයි(writes, verb). +Tamil(TA) +Tamil is a language spoken primarily in Tamil Nadu, a state in southern +India, as well as in countries with a large Tamil diaspora, which includes +Sri Lanka, Malaysia, and Singapore, to name a few. The phonological +differences exist within Tamil Nadu between southern, western, and +northern speech. Tamil is a Dravidian language of the southern branch, +with a rich literary tradition dating back over 2000 years. Tamil spoken +in India and Sri Lanka are two different dialects. It uses the Subject +Object Verb format. For example sentence: “I like paintings” in Tamil +becomes எனக்கு ஓவியங்கள் பிடிக்கும் where the Iஎனக்கு (I, +Subject), விலங்குகள் (Paintings, Object) and பிடிக்கும் (Like, +Verb). +Urdu(UR) +It is Pakistan’s national language and is also spoken widely in India. +In Pakistan and India, Urdu is spoken by over 170 million citizens and is +also spoken in some communities in the United Kingdom, the United +States, and the United Arab Emirates. Script for Urdu is a modified +and revised version of the Perso-Arabic script. Urdu writing structure is +Subject Object Verb. For example “she reads a book” which in Urdu is +وہ ﯾﮏا بﮐﺘﺎ ﭘﮍﻫﺘﯽ ۔ہﮯ where وہ(she, Subject), ﯾﮏا بﮐﺘﺎ(book, object) and +ﭘﮍﻫﺘﺎ ہﮯ(reads, verb). + +11 + + + + + +Table 1: Linguistic Features of Languages Used in MT Experiments + + +Languages +Script +Word +Order +Family +Number +of Speakers +(in millions) +Writing +Direction +Assamese (AS) +Bengali +SOV +Indo-European +15 +left to right +Malayalam (ML) +Malayalam +SOV +Dravidian +38 +left to right +Bengali (BN) +Bengali +SOV +Indo-European +265 +left to right +Marathi (MR) +Devanagari +SOV +Indo-European +95 +left to right +Gujarati (GU) +Gujarati +SOV +Indo-European +60 +left to right +Kannada (KN) +Kannada +SOV +Dravidian +36 +left to right +Hindi (HI) +Devanagari +SOV +Indo-European +615 +left to right +Oriya (OR) +Oriya +SOV +Indo-European +38 +left to right +Punjabi (PA) +Perso-Arabic, +Gurmukhi +SOV +Indo-European +125 +right to left +left to right +Telugu (TE) +Telugu +SOV +Dravidian +93 +left-to-right +Sindhi (SD) +Devanagari +Perso -Arabic +SOV +Indo-European +25 +left to right +right to left +Sinhala (SI) +Sinhala +SOV +Indo-European +17 +left to right +Nepali (NE) +Devanagari +SOV +Indo-European +24 +left to right +Tamil(TA) +Tamil +SOV +Dravidian +81 +left to right +Urdu (UR) +Urdu +SOV +Indo-European +170 +right to left +English (EN) +Roman +SVO +Indo-European +1,132 +left to right + +2 Related Work +A few works on SMT using some Indic Languages are discussed in this sec- +tion. +(Dasgupta et al (2004)) has discussed a technique for English (EN) to Bengali +(BN) MT that utilizes the syntax of EN sentences to BN while minimizing +the time of translation. In the process to create the target sentences, a dic- +tionary is used to know the object and subject, as well as other entities like +person and number in their work. +English-to-Hindi (EN-HI) SMT system has been created by (Ananthakrish- +nan et al (2009)) using morphological and syntactic pre-processing in SMT + +12 + + + + + +model. In their work, the suffixes in HI language are segmented for mor- +phological processing before rearranging the EN source sentences as per HI +syntax. +In 2010, research has been conducted by (Zbib et al (2010)) at MIT, USA, +using the grammatical structures in statistical machine translation with the +Newswire corpus for Arabic to EN language to give better translation results. +Work on Kannada-to-English MTS with SMT, by (Kumar et al (2015)), using +Bible corpus on 20,000 sentences shows a remarkable feat with 14.5 BLEU +score which is even supported by (Papineni et al (2002)).(Kaur et al (2011)) +has presented a translation model based on SMT for English (EN) to Punjabi +(PA) with their own corpus containing 3844 names in both languages with +BLEU and word accuracy as 0.4123 (with range 0-1) and 50.22%, respec- +tively. +(Nalluri et al (2011)) has created “enTel,” an SMT-based EN to Telugu(TE) +MT system, using the Johns Hopkins University Open Source Architecture (Li +et al (2009)). For the purpose of training the translation system, TE par- allel +dataset from the Enabling Minority Language Engineering (EMILLE) is +used for their work. +In the year 2014, an SMT Framework for Sinhala(SI)-Tamil(TA) MT Sys- +tem has been created by (Randil et al (2014)). In their work, the result +of SMT-dependent translation between language pairs, including TA-SI and +SI-TA has been shown. Outcomes of the experiments using the SMT model +give more noticeable results for the SI-TA than the TA-SI language pair. For +languages closely related, SMT shows remarkable results. +In 2017, a survey has been conducted by (Khan et al (2017)) on the IL-EN +language MT models reveal the importance of SMT over 8 languages i.e. +Hindi (HI), Bengali (BN), Gujarati (GU), Urdu (UR), Telugu (TE), Pun- +jabi (PA), Tamil (TA), and Malayalam (ML). In their work, EMILLE corpus +(Nalluri et al (2011)) is used and Moses SMT model is preferred to make +the translation models, with out-of-vocabulary (OOV) words transliterated +to EN. In their work, the evaluation using BLEU, NIST and UNK counts as +metrics reveals the overall SMT performance as satisfactory (PA-EN and UR- +EN models as the best and the HI-EN and GU-EN models as the worst). An +EN-BN SMT system has been presented by (Islam et al (2010)). In their work, +to handle OOV (out-of-vocabulary) words, a transliteration module is +presented. In order to address the systematic grammatical distinctions be- +tween EN and BN, a preposition handling module has been added. BLEU, +NIST and TER scores has been used to check the effectiveness of their sys- + +13 + + + + + +tem. +Nowadays, NMT is widely appreciated for its advancement in the develop- +ment of machine translation with remarkable improvement in quality. Hence, +many researchers have compared both techniques for low and high-resource +languages. +(Antonio et al (2017)) has performed a thorough evaluation using statistical- +based and neural machine translation systems for nine language directions +along a variety of dimensions. In their experiment, for long sentences, SMT +systems perform better than the NMT. Recently, (Castilho et al (2017)) has +used automatic metrics and expert translators to conduct a thorough quan- +titative and qualitative comparison of NMT and SMT. SMT shows better +according to their experiments. +The comparison of NMT and SMT for the Nepali (NE) using the Nepali +National Corpus (NNC) with 6535 sentences has been shown by (Acharya et al +(2018)). The researchers have proved in their experiments that the SMT model +performs better than the NMT-based system with a small corpus with a 5.27 +BLEU score. +In 2021, Long Short-Term Memory networks (LSTMs) integrated with atten- +tion mechanism using WAT corpus have been used in experiments by (Singh +et al (2003)) to achieve a 15.7 BLEU score as opposed to a baseline of 14.5 +BLEU score. +(Abujar et al (2021)) has developed a BN-EN MT model on AmaderCAT cor- +pus using Sequence-to-Sequence (seq2seq) architecture, a special class of Re- +current Neural Networks to develop the translation system and has achieved +a BLEU score of 22.3. +In the year 2021, translation of English and Hindi-to-Tamil languages us- ing +both SMT and NMT has been presented by (Akshai et al (2021)). The +disadvantages of NMT have been shown in their experiments such as the +occurrence of numerous errors by NMT when interpreting domain terms and +OOV (Out of vocabulary) phrases. NMT frequently constructs inaccurate +lexical choices for polysemous words and occasionally counters reordering +mistakes while translating words and domain terms. The translations that +have been generated by the NMT models mostly include repetitions of pre- +viously transcribed words, odd translations, and many unexpected sentences +having no correlation with the original sentence. + +14 + + + + +3 Experimental Framework +3.1 Dataset +Samanantar and OPUS datasets for model building and standard benchmark +dataset i.e. Flores 200 for testing are utilized. Samanantar is the largest cor- +pus collection for ILs (Gowtham et al (2022)). The collection includes more +than 45 million sentence pairs in English and 11 ILs. The Samanantar Corpus +has been used for Assamese (AS), Malayalam (ML), Bengali (BN), Marathi +(MR), Gujarati (GU), Kannada (KN), Hindi (HI), Oriya (OR), Punjabi (PA), +Telugu (TE), and Tamil (TA) for the experiments. OPUS is a large resource +with freely available parallel corpora. The corpus includes data from many +domains and covers over 90 languages (Tiedemann (2012)). The OPUS cor- +pus is used for Sinhala (SI), Sindhi (SD), Urdu (UR), and Nepali (NE). Table 2 +gives statistics of the dataset used in our experiments. +FLORES-200 (Marta et al (2022)) dataset is a multilingual parallel dataset +with 200 languages, that are used as human-translated benchmarks. It con- +sists of two corpora, labeled “dev” (997 lines) and “devtest” (1013 lines). +The “dev” dataset has been used for fine-tuning, and the “devtest” dataset +has been used for testing. + +3.2 Methodology +Our proposed process comprises of following major steps: +1. Setting up SMT System Moses SMT Toolkit is used to build our +SMT systems. It is written in C++ and Perl. At the moment, this is +one of the best SMT tools available. First, Moses, GIZA++ (Och +(2003)), CMPH (for binarization) and SRILM in Ubuntu are installed. +For training, fine-tuning and testing processes, the system needs a par- +allel corpus of the language pair in addition to configurable phases +according to developer’s choice to follow. +2. Data Preprocessing A qualitative corpus plays a major role in any MT +task. While obtaining corpora from various sources, data qual- ity i.e. +critical for the effectiveness of an MT system, can never be +ascertained. So, removing unnecessary noise is an important task be- +fore using the data to train our statistical machine translation model. +Following processes are used to preprocess and clean it: + +15 + + + + + +Table 2: Parallel corpus statistics +English to Indic +Parallel Corpus(Sentences) +Assamese (AS) + +0.14M +Malayalam (ML) +5.85M +Bengali(BN) +8.52M +Marathi(MR) +3.32M +Gujarati(GU) +3.05M +Kannada(KN) +4.07M +Hindi(HI) +8.56M +Oriya(OR) +1.00M +Punjabi(PA) +2.42M +Telugu(TE) +4.82M +Sindhi(SD) +1.95M +Sinhala(SI) +8.68M +Nepali(NE) +3.35M +Tamil(TA) +5.16M +Urdu(UR) +8.95M + +• Data Cleaning and Formatting The goal of data cleaning is +either to find and fix or to delete erroneous data from the corpus. +Here, characters those are used neither in ILs nor in English are +removed. Some of the punctuation in extended Unicode is con- +verted to its standard counterpart. Numbers in the IL corpus are +converted from English to IL scripts. Characters outside the stan- +dard alphabets of the language pair, extra spaces, and unprintable +characters are also removed from the corpus. The preprocessing +techniques used in our work have been summarized as follows: +– Removing unprintable characters +– Removing characters outside the language pair +– Removing extra spaces +– Deaccenting accented characters +– Changing non-standard Unicode punctuation characters in +both corpora to their standard counterparts +– Changing uncommon punctuations to more common ones +– Changing numbers to a uniform numbering system and script + +16 + + + + + +3. Tokenization: It is the process of dividing a character sequence into +smaller units known as tokens based on a given character sequence and +a specified document unit. Words, punctuation, and numerals serve as +these tokens in our instance. The corpus is tokenized using a modified +Moses tokenizer (Koehn et al (2007)). Redundant punctuations (quo- +tation marks, apostrophes, and commas) are also removed from the +corpus. +4. Training Truecasing Model: This is the procedure for adding case +information to text that has been incorrectly cased or is not cased (Lita +et al (2003)). Data sparsity is lessened with the use of true casing. A +truecaser model (a model which changes the words at the beginning +of the sentence to the most common casing) is trained on the training +dataset. The Moses truecasing is used for the same. +5. Training Language and Translation Models: In MOSES, the +training procedure utilizes word and segment occurrences to draw con- +nections between the target and source languages. The language and +translation models are trained on the training dataset and binarized. +GIZA++ grow-diag-final-and alignment is used for word alignments, +which start with the intersection of the two alignments and then add +the additional alignment points. +The grow-diag-final-and model starts with the intersection of the align- +ments from source to target and target to source, then two steps are +used to add additional alignment points (Och (2003)): +grow-diag: For every neighboring point to the alignments measured, +if either source or target word is not aligned already but is present +in the union of the alignment, then the neighboring point is in- +cluded in the alignment. +final: If any phrase pairs are unaligned but present in the union, add +the point to the alignment. +• Word Alignment Model: After preprocessing the words, the +next step is word alignment. The proposed work employs the +GIZA++ (Och (2003)) incorporation of the IBM models to ac- +complish the word procedure. The GIZA++ model assesses the +likelihood of word-to-word alignment for each source and target +word in each sentence. To produce a good-quality word alignment, + +17 + +− +− +− +− + + + + +the alignment is produced using a series of successive estimations. +To process a corpus with a larger quantity of sentences, the process +takes several hours. The alignment method’s outcomes establish +a connection between the target and source words. +• Reordering It is the process of restructuring the word order of +one natural language sentence to make it more similar to the word +order of another natural language sentence. It is a critical task +in transcription for languages with different syntactic structures. +The Moses system learns different reordering possibilities for each +phrase during the training process. Instead of default reordering, +the model uses the distance reordering model (Kumawat et al +(2014)). +– Distance-Based Reordering: The reordering of the tar- get +output phrases is represented by the relative distortion +probability distribution re (St, Et 1). Here, St refers to the +starting position of the source phrase that is interpreted into +the t 1 th target phrase. The reordering distance (St - Et +1) is calculated as follows: When taking source words out of +sequence, the reordering distance is the number of words ig- +nored (either forward or backward). If two phrases are trans- +lated in sequence, then t = Et 1 +1; that is, the first word of +the phrase immediately follows the last word of the pre- +vious phrase. A reordering cost of re(0) is used in this case. +The distance-based model assigns a linear cost to reordering +distance, implying that the movement of phrases over long +distances is more expensive. +6. Fine tuning: It is the process of determining the best configuration file +settings for a translation model when it is used for a specific pur- pose. +It uses a translation model to translate all 15 ILs source language phrases +in the tuning set. Then, it compares the model’s output to a set of +reference (human translations) and adjusts the settings to improve +translation quality. This procedure is repeated several times. The +tuning process repeats the steps with each iteration until the transla- +tion quality is optimized. The model is fine-tuned on the preprocessed +Flores-200 dev dataset. +7. Translation: The final model is used to translate the preprocessed + +18 + + + + + +Flores-200 devtest dataset from the source to the target language. +8. Postprocessing and Detokenization: Redundant punctuation marks +(quotation marks, apostrophes, and commas) are removed, and the +translation file is detokenized using the Moses detokenizer. +9. Evaluation: The evaluation metrics use for our experiments are ME- +TEOR (Banerjee et al (2005)), RIBES (Wołk et al (2016)), and BLEU +(Papineni et al (2002)). + +4 Essential metrics for MT translation evalu- +ation +The most crucial phase of any MT system is MT evaluation. Both automatic +and manual methods can be applied to analyze MT tasks. The effective- +ness of a system’s output can be evaluated either directly through human +assessments, or indirectly using reading cases, other downstream activities, +and even through estimating the amount of effort necessary to rectify the +output. A better outcome is obtained through manual evaluation, which +includes task-based evaluations, fluency and adequacy scores, human vot- ing +for translations task, post-editing measures, etc. However, the major +challenges of manual evaluation are time-intensiveness, absence of repeata- +bility and high cost. In order to evaluate the effectiveness of MT output, +different automated approaches are there such as Metric for Evaluation of +Translation with Explicit Ordering (METEOR), Bilingual Evaluation Un- +derstudy(BLEU), Levenshtein, Rank-based Intuitive Bilingual Evaluation +Score(RIBES), Word Error Rate (WER) and NIST exist. Several intuitive +advantages exist for automated metrics that can give points for synonyms or +paraphrases. A few of the evaluation metrics which are used in our work are +discussed below +1. Bilingual Evaluation Understudy (BLEU): The most widely used +method for evaluating machine translation (MT) is known as BLEU. +This method, first introduced in 2002 (Papineni et al (2002)) exam- ines +one or more reference translations to the hypothetical translation. When +the hypothetical translation matches numerous strings with the +reference translation, the MT evaluation gives it a higher score. The +BLEU system assigns a translation a score from 0 to 1. However, it is + + +19 + +1 + + + +usually represented as a percentage value. The nearer the translation is +to 1, the more it corresponds to the reference translation. This match- +ing of translation is conducted word-by-word in the same word order +in both datasets. SacreBLEU is used to calculate the BLEU scores of +baseline models. +2. Rank-based Intuitive Bilingual Evaluation Score (RIBES): It +is calculated by incorporating a rank correlation coefficient before uni- +gram matches, eliminating the necessity for higher-order n-gram matches. +This metric is concerned with word order. To compare SMT and ref- +erence translations, it employs Kendall’s tau coefficient (τ ) based on +word order to indicate rank differences (Wołk et al (2016)). To assure +positive values, the coefficient is normalized as shown below: + + +Normalized Kendall’s τ (NKT) = τ + 1 +2 +(5) +This coefficient can be paired with unigram-precision p1 and Brevity +Penalty BP and changed to prevent overestimation of the correlation +between only relevant words in SMT and reference translations. + + +RIBES = NKT.(pα).(BPβ ) +(6) +Here, α and β are parameters between 0 and 1. + +20 + + + + + +3. Metric for Evaluation of Translation with Explicit Ordering +(METEOR): Meteor scores a translation depending on explicit word- +by-word similarities between both the translation and a provided ref- +erence translation (Banerjee et al (2005)). It is specifically created to +generate sentence-level scores that are highly correlated with human +evaluations of translation quality. Meteor utilizes and highlights recall +in combination with precision, a feature that numerous measures have +verified as crucial for a strong correlation with human judgments. It also +intends to address the problem of imprecise reference translations by +utilizing adaptable word matching in consideration with synonyms and +morphological variances. To achieve a score of 1, the words of the +machine-generated output should be present in the reference and each +of the words of the reference is in the machine-generated output. + +5 Results and Discussion +In this work, the evaluation metrics used are METEOR (Banerjee et al +(2005)), RIBES (Wołk et al (2016)), and BLEU (Papineni et al (2002)). All +the evaluation metrics used in our work are prominent metrics for deter- +mining the quality of the machine-translated text. +Table 3 displays the translation of all the 15 ILs to English and vice versa +using SMT without fine-tuning. Evaluation metrics of SMT with finetuning +using the Flores-200 dev dataset are shown in Table 4. RIBES and METEOR +range is 0-1. For EN-IL and IL-EN language using SMT, the BLEU score +lies between 0.46 to 13.09 and 0.49 to 15.41 respectively. The RIBES score for +EN-IL and IL-EN is between 0.04 to 0.63 and 0.14 to 0.61 respectively. +METEOR scores lie between 0.01 to 0.28 for EN-IL and 0.02 to 0.28 for +IL-EN. SMT models using distance reordering techniques are giving better +BLEU Scores for languages BN, PA, UR, HI, and GU than the rest. With- +out fine-tuning, SI performs the worst in terms of all three metrics of all +languages in both directions, whereas with fine-tuning EN-SI and TA-EN +perform worse than all other EN-IL and IL-EN models respectively with all + +21 + + + + + + + + + + + + + + +Table 3: Evaluation Metrics Result of SMT without Finetuning + +Languages +Pairs +BLEU +RIBES +METEOR +AS +EN-AS +1.90 +0.50 +0.09 +AS-EN +3.21 +0.46 +0.11 +ML +EN-ML +3.79 +0.27 +0.08 +ML-EN +4.59 +0.43 +0.12 +BN +EN-BN +6.41 +0.62 +0.17 +BN-EN +3.06 +0.45 +012 +MR +EN-MR +3.17 +0.43 +0.09 +MR-EN +3.62 +0.43 +0.09 +GU +EN-GU +7.62 +0.56 +0.16 +GU-EN +10.14 +0.59 +0.21 +KN +EN-KN +5.06 +0.40 +0.11 +KN-EN +7.17 +0.51 +0.16 +HI +EN-HI +13.09 +0.63 +0.28 +HI-EN +15.41 +0.64 +0.28 +OR +EN-OR +3.92 +0.59 +0.14 +OR-EN +6.41 +0.52 +0.17 +PA +EN-PA +7.22 +0.63 +0.18 +PA-EN +11.7 +0.61 +0.24 +TE +EN-TE +8.16 +0.42 +0.12 +TE-EN +5.77 +0.52 +0.18 +SD +EN-SD +1.29 +0.39 +0.08 +SD-EN +2.48 +0.35 +0.09 +SI +EN-SI +0.93 +0.05 +0.02 +SI-EN +0.49 +0.14 +0.05 +NE +EN-NE +6.00 +0.58 +0.16 +NE-EN +8.29 +0.53 +0.19 +TA +EN-TA +2.78 +0.16 +0.05 +TA-EN +2.64 +0.31 +0.07 +UR +EN-UR +9.43 +0.62 +0.24 +UR-EN +11.35 +0.61 +0.23 + + +22 + + + + + + + + + + + + + + +Table 4: Evaluation Metrics Result of SMT with Finetuning + +Languages +Pairs +BLEU +RIBES +METEOR +AS +EN-AS +2.17 +0.50 +0.08 +AS-EN +3.21 +0.42 +0.10 +ML +EN-ML +2.05 +0.23 +0.06 +ML-EN +1.84 +0.27 +0.06 +BN +EN-BN +8.26 +0.63 +0.19 +BN-EN +12.16 +0.60 +0.23 +MR +EN-MR +2.43 +0.39 +0.08 +MR-EN +2.49 +0.36 +0.07 +GU +EN-GU +5.82 +0.52 +0.14 +GU-EN +3.56 +0.45 +0.01 +KN +EN-KN +3.35 +0.09 +0.14 +KN-EN +3.67 +0.41 +0.10 +HI +EN-HI +8.64 +0.57 +0.22 +HI-EN +5.38 +0.49 +0.14 +OR +EN-OR +5.25 +0.58 +0.15 +OR-EN +2.22 +0.39 +0.11 +PA +EN-PA +5.71 +0.60 +0.15 +PA-EN +7.75 +0.55 +0.19 +TE +EN-TE +4.4 +0.38 +0.10 +TE-EN +3.34 +0.44 +0.12 +SD +EN-SD +1.59 +0.41 +0.09 +SD-EN +2.53 +0.38 +0.09 +SI +EN-SI +0.46 +0.04 +0.01 +SI-EN +3.11 +0.37 +0.11 +NE +EN-NE +4.00 +0.55 +0.14 +NE-EN +5.25 +0.49 +0.13 +TA +EN-TA +1.86 +0.16 +0.05 +TA-EN +1.03 +0.08 +0.02 +UR +EN-UR +6.34 +0.56 +0.19 +UR-EN +7.07 +0.54 +0.18 + +23 + + + +three metrics. HI and BN languages have qualitative, large, and less noisy +datasets compared to other languages. Hence, HI performs the best among all +languages without fine-tuning in all three metrics in both directions, and BN +performs the best among all languages with fine-tuning in both direc- tions +with respect to BLEU and RIBES. In addition, UR and PA also produce good +RIBES metrics than other languages. RIBES score for PA is 0.63(for EN- +PA) and 0.61(PA-EN), and for UR, RIBES score is 0.62(EN-UR) and +0.61(UR-EN). +Even though SI has a good amount of corpus, the corpus does not have reli- +able translations compared to other languages. For example, the sentence in +English “Heb. 11:32-34; Judg. 16:18-21, 28-30 Jehovah’s spirit operated on +Samson in a unique way because of unusual circumstances” has been trans- +lated to Sinhala in the corpus as “11:32-34; Gනි.”, which only translates +“Heb. 11:32-34;”. Hence, SI does not perform well compared to other lan- +guages. Similarly, in the EN-TA corpus, the sentence “He’s my boss” has +been translated to “அவர் எனது ே மலாளர் மட்டும்தான் .” which ac- +tually means “He is only my manager”. From the example, it is clear that +EN-TA corpus also has ambiguity. Additionally, even though the ILs-English +and English-ILs systems are trained using the same corpus, a significant dis- +crepancy in the BLEU scores is observed. This is due to the significant +morphological diversity of ILs and the relative difficulty of translating from +English to ILs. It has been observed that SI has a high number of lines (8.68 +M) but performs poorly as compared to languages like PA (2.42 M) and GU +(3.05 M). It is also observed that languages with very steep slopes tend to +have low scores. For example, EN-TA and EN-ML have 60% sentences with +less than 4 tokens, and they have not-so-good scores as shown in Figure 1. +In contrast, languages with good scores, like HI and BN have more gentle +slopes. So, length of sentences is a contributing factor. EN-SD is an ex- +ception which has a gentle slope but does not give good scores, because the +corpus does not have good translation quality. Therefore, the quality of the +corpus matters more than the size of the dataset. + +6 Conclusion and Future Work +This paper has presented the MT work for 15 ILs to English and vice versa +using SMT. It also describes the linguistic features of all 15 ILs. A tailor- +made preprocessing approach has been incorporated into this work. The + +24 + + + + + + + + + + + + + + + + + + +Figure 1: Less than ogive for number of tokens in a sentence for all fifteen +language corpora + +LessthanOgiveforNumberoflokens +1.2 +given +numberoftokenslessthan +1 +0.8 +0.6 +ofsentences +0.4 +Ratio +0.2 +0 +0 +1 +2 +m +4 +5 +6 +7 +8 +9 +Numberoftokens +EN-AS(AS) +EN-SI(SI) +EN-ML(ML) +EN-MR(MR) +EN-OR(OR) +EN-SD(SD) +EN-UR(UR) +—EN-BN(BN) +EN-GU(GU) +EN-HI(HI) +EN-KN(KN) +EN-NE(NE) +EN-PA(PA) +EN-TA(TA) +EN-TE(TE)25 + + + + + +model has utilized the grow-diag-final-and alignment model and distance re- +ordering model. For checking the quality of translation, different evaluation +Metrics such as BLEU, RIBES, and METEOR are utilized in this work. +From the result, it is observed that the proposed SMT model is quite satis- +factory for some of the ILs. However, the level of performance is not at par +with the rest of the ILs and there lies the need of improvement to be made. +Due to the scarcity and quality of parallel corpus, the metrics obtained are +quite low. +It has been observed that the translations of some of the languages are not +sufficiently accurate. Measures of validating corpus quality shall be explored +in order to observe the corpus quality and remove inaccurate lines. Dravid- ian +languages are in general agglutinative languages (words are made up of +morphemes, with each morpheme contributing to the meaning of the word). +In future, means to infer translations from the breakdown of words in these +languages shall also be explored. +Interestingly, in some of the ILs, finetuning schemes are hampering the qual- +ity. The causes of this phenomenon shall be analyzed and mitigated via +techniques such as noise reduction, corpus cleaning, and finetuning schemes +for those languages to ensure better quality. 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No language left +behind: Scaling human-centered machine translation. arXiv preprint +arXiv:2207.04672, 2214-2218 + +What are the top 200 most spoken languages., 2022. Ethnologue: Languages of +the World https://www.ethnologue.com/guides/ethnologue200 + +Costa-jussà, Marta R., James Cross, Onur Çelebi, Maha Elbayad, Kenneth +Heafield, Kevin Heffernan, Elahe Kalbassi et al., 2003. Minimum error rate +training in statistical machine translation. Proceedings of the 41st Annual +Meeting on Association for Computational Linguistics, 160–167 + +Khan, N. J., Anwar, W., and Durrani, N., 2017. Machine translation ap- +proaches and survey for Indian languages. arXiv preprint arXiv:1701.04290 + + + +32 + + + +Lita, L. V., Ittycheriah, A., Roukos, S., and Kambhatla, N. 2003. Trucasing. +In Proceedings of the 41st Annual Meeting on Association for Computa- +tional Linguistics, 152-159 + +Kumawat, Sudhakar and Nitish Chandra., 2014. Distance-based Reordering in +English to Hindi Statistical Machine Translation. International Journal of +Computer Applications, 27-40 + +Koehn, P., Hoang, H., Birch, A., Callison-Burch, C., Federico, M., Bertoldi, +N., Cowan, B., Shen, W., Moran, C., Zens, R., Dyer, C., Bojar, O., +Constantin, A., and Herbst, E. 2007. Moses: Open source toolkit for +statistical machine translation. In Proceedings of the Annual Meeting of the +Association for Computational Linguistics, 177–180 + +33 + + + diff --git a/9NAyT4oBgHgl3EQfqPh6/content/tmp_files/load_file.txt b/9NAyT4oBgHgl3EQfqPh6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d4bbd568404ab29258b488b5638cd1793b14762 --- /dev/null +++ b/9NAyT4oBgHgl3EQfqPh6/content/tmp_files/load_file.txt @@ -0,0 +1,935 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf,len=934 +page_content='1 Statistical Machine Translation for Indic Languages Sudhansu Bala Das, Divyajoti Panda, Tapas Kumar Mishra, and Bidyut Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Patra National Institute of Technology(NIT), Rourkela, Odisha, India Indian Institute of Technology (IIT), Varanasi, Uttar Pradesh, India Abstract Machine Translation (MT) system generally aims at automatic representation of source language into target language retaining the originality of context using various Natural Language Processing (NLP) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Among various NLP methods, Statistical Machine Trans- lation (SMT) is a very popular and successful architecture used for both low as well as high-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' SMT uses probabilis- tic and statistical techniques to analyze information and conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This paper canvasses about the development of bilingual SMT mod- els for translating English to fifteen low-resource Indian Languages (ILs) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' At the outset, all 15 languages are briefed with a short description related to our experimental need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Further, a de- tailed analysis of Samanantar and OPUS dataset for model building, along with standard benchmark dataset (Flores-200) for fine-tuning and testing, is done as a part of our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Different preprocess- ing approaches are proposed in this paper to handle the noise of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' To create the system, MOSES open-source SMT toolkit is explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' “Distance” reordering is utilized with the aim to understand the rules of grammar and context-dependent adjustments through a 2 phrase reordering categorization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In our experiment, the quality of the translation is evaluated using standard metrics such as BLEU, METEOR, and RIBES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 1 Introduction Technology reaches new heights through its journey from the origins of ideas to their full-scale practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' One such journey is heading to- wards elimination of language barrier in order to establish a seamless social communication in every domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In this regard, advancement on relevant fields such as Natural Language Processing (NLP), Machine Learning (ML) and Artificial Intelligence (AI) based Language Modelling (LM) significantly contributes for evolving a flawless automatic Machine Translation (MT) sys- tem (Dorr et al ( 2004)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Irrespective of various heuristic approaches to maintain both lexical and contextual interpretation of source language(s) onto the translated target language(s), it is still challenging to cope with required fluency, adequacy, accent, and overall accuracy (Chapelle et al ( 2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' However, it is feasible with the advent of modern NLP (AI-based) approaches wherein a high-quality and high resource (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' large quantity of corpora available) parallel corpus (translation pairs in source and target languages) is required to train a good translation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Hence, for high- resource languages having massive digital footprint across the globe, MT sys- tems prove to be quite efficient with adequate training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' On the other hand, it becomes very complicated for low-resource languages suffering from uni- versal recognition and scanty digital presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Such imbalance often leads to poor-quality translation in presence of low-resource language(s) in the form of either target or source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Therefore, MT systems need to understand the syntax (rules to combine words), semantics (meaning of words and combi- nations), and morphology (rules to cover morphemes - smallest meaningful units - into words) of such low-resource languages (Somers (2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Based on the heuristic paradigms, MT models are classified into rule-based (RBMT), example-based (EBMT), statistical (SMT), and neural (NMT) sys- tems (Tripathi et al (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Each has its own advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' RBMT models follow a set of rules to define a language and the interaction between different linguistic devices (words, phrases, sentences) in the lan- guage (Jussà et al (2012), Michael et al (2000)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' These sets of rules and systems defined for a translation in a language pair are hard-coded on the 3 machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The linguistic information used in an RBMT model is mainly the target and source languages collected from unilingual (one language), bilingual (two languages), or multilingual (more than two languages) dic- tionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In addition, the model also uses grammar covering the syntactic, semantic, and morphological regularities of each language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' However, a well- built RBMT model requires highly skilled and expert human labour due to its complexity making it hard to build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In addition, the ambiguous proper- ties of languages make them prone to take more time and efforts to resolve, especially in large and complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' RBMT models require a lot of effort to be made functional in day-to-day life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Hence, the need for more efficient translation systems than RBMT still persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' EBMT methods make use of a large number of translation examples (John (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Notably, EBMT mod- els make use of bilingual corpora manipulation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' the breaking down of a bilingual corpus into smaller parts, translating those parts into the target language, and recompiling it to form whole translated sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' They do not account much for the syntax, semantic and morphological analysis of the target and source language (like RBMT models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In contrast, SMT is better when compared to RBMT and EBMT models, as it does not require human intervention (Adam (2008)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is a way of translation wherein a statistical- based learning algorithm is applied to a large bilingual corpus that helps the machine learn the translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This method also enables the machine to translate sentences not encountered by the machine during its training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The objective of SMT is to convert an input word sequence from the source language into the target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It has dominated academic MT research and a portion of the commercial MT sector in less than two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' On the other hand, neural machine translation (NMT) is performed using a neural network (NN) (Stasimioti (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Unlike SMT, NMT does not have a distinct translation model, language model, or model for reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Instead, it has a single sequence model that determines one word at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The pre- diction is based on the source sentence effort previously generated sequence in the target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' NMT is a deep learning-based method of machine learning that utilizes a large NN that relies on word vector representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Even though the NMT has achieved remarkable results in a few trans- lation experiments using high-resource language, researchers are unsure if the NMT could actually replace SMT and if its success would extend to other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Eventually, the experiment of (Michał (2016)) on the corpus of the United Nations (consisting of 15 low-resource languages) brings the fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' From the result of his experiment, it is evident that the performance of 4 SMT is better than that of NMT for the majority of cases, as measured by BLEU score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Many researchers (Lohar et al (2019), Zhou et al (2017), Wang et al (2017), Castilho et al (2017)) have pointed out various disadvantages of NMT over SMT using low resource language, such as the fact that NMT requires more corpus and resources than SMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In comparison with SMT, NMT training typically takes longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Additionally, research has shown that when there is a domain incompatibility between testing and training data, SMT performance is superior to that of NMT (Xing et al (2018), Mahata et al (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Long sentences are another area where SMT excels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' English and ILs are languages with less parallel text data, which motivates us to work with ILs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This research examines the effectiveness of SMT systems on low-resource language pairs, of which many are rarely worked on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The dataset used in our experiment for all fifteen Indian languages is tested for the first time for all languages using SMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Hence, the objective of this work is to build an MT system using SMT for languages such as Assamese (AS), Malayalam (ML), Bengali (BN), Marathi (MR), Gujarati (GU), Kannada (KN), Hindi (HI), Oriya (OR), Punjabi (PA), Telugu (TE), Sindhi (SD), Sinhala (SI), Nepali (NE), Tamil (TA), and Urdu (UR) to English (EN) and vice versa and to check the effectiveness of SMT with low-resource language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Our main goal is to develop an MT system for low-resource languages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=', ILs, that can serve as a baseline system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The following is a summary of our work’s main contributions: • To the best of our knowledge, this work is the first attempt to use SMT with the Samanantar and OPUS Dataset to investigate the MT for all fifteen IL-EN and EN-IL pairs (both directions), including both the Dravidian and Indo-Aryan groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' • To bring forth the linguistic approach of ILs in terms of translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Scripts, writing style, and grammar with proper examples are also dis- cussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' • Various data filtration methods are investigated in order to clean the data and improve translation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' • Distance-based reordering is utilized to check the translation quality of ILs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 5 | | Better realistic assessment of translation quality is possible from the presentation of results, as obtained using different automated metrics like BLEU, METEOR, and RIBES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Subsections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='2 give some insight into SMT and cover the ILs used for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In Section 2, some prominent works on SMT and NMT using ILs are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The experimental framework, including an overview of the dataset and method ology, is explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Section 4 narrates some of the prominent metrics used for MT evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Results are presented in Section 5 followed by the conclusion and future direction in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='1 SMT Statistical Machine Tramslation (SMT) is dependent on statistical methods (Philipp et al (2007), Richard et al (2002), Mary et al (2011) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is a data- driven technique that makes use of parallel-aligned corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It utilizes mathematical equations to calculate the likelihood of source-to-target lan- guage translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Probability P (Tl Si) is assigned by SMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Here Tl is the target language and Si is the source input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It utilizes Bayes’ theorem to determine the maximum probability P (Tl|Si), which is as follows: P (Tl | Si) ∝ P (Tl)P (Si | Tl) (1) SMT consists of three phases: the language model(LM) P (Tl) for target language probability calculation, the translation model(TM) P (Si Tl) for conditional probability estimation of the target to the source language, and the decoder model (DM), which searches among possible source sentences the one which maximizes probabilities (Kumawat et al (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' To calculate the probability of a sentence, the LM utilizes the n-gram model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It assigns the probability of a single word to the last n words that come before it in the sentence and estimates the translation’s likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The chain rule aids in breaking down the sentence into conditional probability products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' P (s) = P (w1, w2, w3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=', wn) = P (w1)P (w2|w1)P (w3|w1w2)P (w4|w1w2w3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='P (wn|w1w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='wn−1) = P (w1)P (w2|w1)P (w3|w1w2)P (w4|w1w2w3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='P (wn|w1w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='wn−k) (2) Where, P (s) is the probability of the sentence s, consisting of words w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=', wn, assuming a k-gram model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It utilizes the bilingual parallel corpus 6 of the desired language pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This is accomplished by calculating the like- lihood of words or phrases extracted from sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The DM is the final and most crucial phase of SMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It assists in the selection of words with the highest probability to be translated by maximizing the likelihood, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' P (Tl)P (Si | Tl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='2 Language preference India is a multilingual nation where people from various states use a variety of regional tongues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Such diversity of language brings difficulty in commu- nicating with one another for information exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Further, limitations in public communication also bring inconvenience to share feelings, thoughts, opinions and facts, as well as to deal with business purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Moreover, there are many helpful resources available on the internet in English but many In- dians struggle to take benefit of those due to language barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Hence, it is crucial to have an easy translation solution for regional languages to support effective communication and to help utilising global resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' To make it possible, technological innovation are continuing to find out efficient methods for a flawless translation using machines, because it is impractical to have hu- man translators everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For machine translation, an enormous amount of resources is required for training with a proper knowledge-base (rules) for better efficiency so as to fulfill the demand of a flawless translation solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For translation, understanding the meaning of words is important, but words are not enough to constitute a language as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' They must be used in sentence construction that adheres to strict grammar rules and every language is having its own writing style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In our work, 15 commonly spoken languages (over various regions of India) are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Table 1 describes the languages used in our experiments with their linguistic features (ethnologue (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' A short introduction about them in terms of translation is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' English(EN) English language is the primary language of roughly 45 countries and is spoken by nearly 1,132 million people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is written in Roman script, which uses both uppercase and lowercase characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' English uses the subject-verb-object structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For example (expl1), “The poor man took food”, and (expl2) “food took the poor man”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' When the position 7 of the subject changes in the preceding sentences, the significance and meaning of the English sentence change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Assamese(AS) Over 15 million native Assamese speakers live in the state of Assam in the northeastern region of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is one of Assam’s official languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Additionally, it is spoken in various regions of other northeastern In- dian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It uses the Bengali-Assamese script and is written left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It also follows the SOV format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' “Gita is eating mango” is an English sentence that when translated into Assamese became গীতাই আম খাই আআছ which follows subject object verb format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' গীতাই (Gita, subject), আম(mango, object) and খাই আআছ(is eating, verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Malayalam(ML) People in Kerala and a few societies in Karnataka and Tamil Nadu use Malayalam for communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This language is spoken by about 35 million citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It uses the SOV style of writing and a nominative- accusative case marking sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is written in Malayalam script in left-to-right fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Sentence like സീതയ്ക്ക് ചിത്തരചന ഇഷ്ട മാണ് which in English became “Sita loves drawing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Here the word സീതയ്ക്ക് (Sita,Subject), ഇഷ്ടമാണ് (loves,Verb) and ചിത്തര ചന(drawing,Object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Bengali(BN) It is the primary language of Bangladesh and the second most spoken language in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Over 265 million people use it as their primary or second language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Approximately 11 million Bengali speakers exist in Bangladesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In India, states such as Assam, Tripura, and West Bengal use this language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is a member of the Indo-Aryan family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In Bengali sentences, the standard word order is Subject-Object-Verb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For example, in sentence আ রাি জ ভাত খায় which in English is “Rosy eats rice ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Here আ রাি জ (Rosy, Subject), ভাত (Rice, object) and খায় (Eats,Verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Marathi(MR) Marathi is associated with the Sanskrit-derived group of Indian lan- guages and is used by 95 million people in India for communication, primarily in the central and western regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The fourth most widely spoken language in India is Marathi, which has a sizable native-speaker 8 population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Similar to Hindi and Nepali, Marathi is written in the De- vanagari script in left-to-right order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It follows the Subject-Object-Verb order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For example, the sentence तो दध ि पतो, which means “He drinks milk.” in English, has तो दध ject), and ि पतो(drinks, verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Gujarati(GU) ि पतो where तो(He, subject), दध (Milk, ob- Gujarati is spoken by 45 million citizens in Gujarat and is associated with the Indo-Aryan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It uses the SOV writing style and is drafted from left to right in Gujarati script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For example, in the sentence તે આઈસ્ક્રીમ ખાય છે.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' which in English is “He is eating ice cream.” where તે is subject, આઈસ્ક્રીમ is an object and verb is ખાય છે.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Kannada (KN) Karnataka’s official language is Kannada, which is also widely used in other parts of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In India, about 36 million people speak and write Kannada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Despite being a Dravidian language with extensive historical literature, Kannada has few computational linguistics re- sources, making it challenging to study the language’s literature due to its semantic and syntactic diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Subject-Object-Verb is the way the Kannada language is structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Kannada is a highly agglutina- tive language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It uses the left-to-right Kannada script For example, ರಾಮ ಶಾಲೆಗೆ ಹೋದ(SOV) is in English is “Rama went to school”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Here, ರಾಮ(Rama, Subject), ಶಾಲೆ(school, object), and ಹೋದರು(went, verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Hindi(HI) Hindi is one of the official and national languages of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' There are more than 615 million people who use Hindi as their primary language, and even more than 341 million who speak it as a second language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' However, the sentence structure is Subject Object Verb as shown in the example: गीता स्क ल जाती है। is in English is “Geeta goes to school”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In this sentence, गीता (Gita ,Subject), स्क ल (School, Object) and जाती है(Goes, Verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The Indian Constitution mandates that Hindi written in Devanagari be used as the Union’s official language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Oriya(OR) The Oriya language is the primary language of Odisha, a state in east- ern India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Oriya belongs to the Eastern Indo-Aryan group of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 9 Its standard format is subject-object-verb (SOV) and is written in the Odia script from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Punjabi (PA) Punjabi text is written in a subject-object-verb format and is spoken in India and Pakistan, and a few small groups in the United Kingdom, United Arab Emirates, Malaysia, the United States, South Africa, and Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is written in two scripts: the western Perso-Arabic Shah- mukhi script and the eastern Gurmukhi script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Gurmukhi is drafted from left to right, whereas Shahmukhi is written in the opposite direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' ਅਸ ੀਂ ਭਾਰਤ ਹਾੀਂ is in English “We are Indians” where ਅਸ ੀਂ(We,Sub- ject), ਭਾਰਤ (Are,Verb) and ਹਾੀਂ (Indians,Object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Telugu(TE) Telugu is the official language of two Indian states in the south: Andhra Pradesh and Telangana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is also spoken by the Telugu-speaking im- migrant communities in the United States, Canada, and the United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Text structure in Telugu takes the form of a subject-object- verb and from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='ఆమె నన్ను కొటి్టంది in English “she beat me” where ఆమె(she, Subject), నన్ను (Me, Object) and కొటి్టంది(beat, Verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Sindhi(SD) Sindhi is a language spoken by 25 million speakers in Pakistan and 5 million in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is written in a modified Perso-Arabic script in Pak- istan (right-to-left), whereas it is written in a variety of scripts in India, like Devanagari, Khudabadi, and Gurmukhi (left-to-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It follows the Subject-Object-Verb format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For example, the sentence “Partha loves books” is رٿﭘﺎ ﮐﻲ ﺘﺎﺑﻦڪ ﻦﺳﺎ رﭘﻴﺎ ﻫﻲآ where ٿﺎرﭘﺎ(Partha, subject), ﮐﻲ ﺘﺎﺑﻦڪ (books, object) and ﻦﺳﺎ رﭘﻴﺎ ﻫﻲآ (loves, verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Nepali(NE) It is official language and the lingua franca in Nepal, and also spoken by some communities in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Nepali is written in left-to-right De- vanagari script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is a language written in Subject-Object-Verb order For example, “Sita ate apples” when converted to the Nepali language 10 becomes सीताले स्याउ खाइन्.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Here, सीताले(Sita, subject), स्याउ(apples, object) and खाइन्(ate, verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Sinhala(SI) The majority of Sri Lankans speak Sinhala as their first language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Sin- hala is an Indo-Aryan language that differs from English in terms of grammatical structure, morphological variation, and subject-object- verb (SOV) word order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is written in right-to-left Sinhala script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' A sentence like “Pavan writes a letter” is in Sinhala is පවන් ලිපියක් ලියයි where පවන්(Pavan, subject), ලිපියක්(a letter, object) and ලියයි(writes, verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Tamil(TA) Tamil is a language spoken primarily in Tamil Nadu, a state in southern India, as well as in countries with a large Tamil diaspora, which includes Sri Lanka, Malaysia, and Singapore, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The phonological differences exist within Tamil Nadu between southern, western, and northern speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Tamil is a Dravidian language of the southern branch, with a rich literary tradition dating back over 2000 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Tamil spoken in India and Sri Lanka are two different dialects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It uses the Subject Object Verb format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For example sentence: “I like paintings” in Tamil becomes எனக்கு ஓவியங்கள் பிடிக்கும் where the Iஎனக்கு (I, Subject), விலங்குகள் (Paintings, Object) and பிடிக்கும் (Like, Verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Urdu(UR) It is Pakistan’s national language and is also spoken widely in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In Pakistan and India, Urdu is spoken by over 170 million citizens and is also spoken in some communities in the United Kingdom, the United States, and the United Arab Emirates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Script for Urdu is a modified and revised version of the Perso-Arabic script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Urdu writing structure is Subject Object Verb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For example “she reads a book” which in Urdu is وہ ﯾﮏا بﮐﺘﺎ ﭘﮍﻫﺘﯽ ۔ہﮯ where وہ(she, Subject), ﯾﮏا بﮐﺘﺎ(book, object) and ﭘﮍﻫﺘﺎ ہﮯ(reads, verb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Table 1: Linguistic Features of Languages Used in MT Experiments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Languages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Script ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Word ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Order ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Family ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='of Speakers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='(in millions) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Writing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Direction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Assamese (AS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Bengali ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='SOV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Indo European ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='left to right ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Malayalam (ML) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Malayalam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='SOV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Dravidian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='left to right ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Bengali (BN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Bengali ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='SOV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Indo European ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='265 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='left to right ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Marathi (MR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Devanagari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='SOV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Indo European ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='95 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='left to right ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Gujarati (GU) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Gujarati ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='SOV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Indo European ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='left to right ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Kannada (KN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Kannada ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='SOV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Dravidian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='left to right ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Hindi (HI) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Devanagari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='SOV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Indo European ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='615 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='left to right ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Oriya (OR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Oriya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='SOV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Indo European ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='left to right ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Punjabi (PA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='Perso Arabic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Gurmukhi SOV Indo European 125 right to left left to right Telugu (TE) Telugu SOV Dravidian 93 left to right Sindhi (SD) Devanagari Perso Arabic SOV Indo European 25 left to right right to left Sinhala (SI) Sinhala SOV Indo European 17 left to right Nepali (NE) Devanagari SOV Indo European 24 left to right Tamil(TA) Tamil SOV Dravidian 81 left to right Urdu (UR) Urdu SOV Indo European 170 right to left English (EN) Roman SVO Indo European 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='132 left to right 2 Related Work A few works on SMT using some Indic Languages are discussed in this sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' (Dasgupta et al (2004)) has discussed a technique for English (EN) to Bengali (BN) MT that utilizes the syntax of EN sentences to BN while minimizing the time of translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In the process to create the target sentences, a dic- tionary is used to know the object and subject, as well as other entities like person and number in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' English-to-Hindi (EN-HI) SMT system has been created by (Ananthakrish- nan et al (2009)) using morphological and syntactic pre-processing in SMT 12 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In their work, the suffixes in HI language are segmented for mor- phological processing before rearranging the EN source sentences as per HI syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In 2010, research has been conducted by (Zbib et al (2010)) at MIT, USA, using the grammatical structures in statistical machine translation with the Newswire corpus for Arabic to EN language to give better translation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Work on Kannada-to-English MTS with SMT, by (Kumar et al (2015)), using Bible corpus on 20,000 sentences shows a remarkable feat with 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='5 BLEU score which is even supported by (Papineni et al (2002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' (Kaur et al (2011)) has presented a translation model based on SMT for English (EN) to Punjabi (PA) with their own corpus containing 3844 names in both languages with BLEU and word accuracy as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='4123 (with range 0-1) and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='22%, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' (Nalluri et al (2011)) has created “enTel,” an SMT-based EN to Telugu(TE) MT system, using the Johns Hopkins University Open Source Architecture (Li et al (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For the purpose of training the translation system, TE par- allel dataset from the Enabling Minority Language Engineering (EMILLE) is used for their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In the year 2014, an SMT Framework for Sinhala(SI)-Tamil(TA) MT Sys- tem has been created by (Randil et al (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In their work, the result of SMT-dependent translation between language pairs, including TA-SI and SI-TA has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Outcomes of the experiments using the SMT model give more noticeable results for the SI-TA than the TA-SI language pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For languages closely related, SMT shows remarkable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In 2017, a survey has been conducted by (Khan et al (2017)) on the IL-EN language MT models reveal the importance of SMT over 8 languages i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Hindi (HI), Bengali (BN), Gujarati (GU), Urdu (UR), Telugu (TE), Pun- jabi (PA), Tamil (TA), and Malayalam (ML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In their work, EMILLE corpus (Nalluri et al (2011)) is used and Moses SMT model is preferred to make the translation models, with out-of-vocabulary (OOV) words transliterated to EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In their work, the evaluation using BLEU, NIST and UNK counts as metrics reveals the overall SMT performance as satisfactory (PA-EN and UR- EN models as the best and the HI-EN and GU-EN models as the worst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' An EN-BN SMT system has been presented by (Islam et al (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In their work, to handle OOV (out-of-vocabulary) words, a transliteration module is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In order to address the systematic grammatical distinctions be- tween EN and BN, a preposition handling module has been added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' BLEU, NIST and TER scores has been used to check the effectiveness of their sys- 13 tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Nowadays, NMT is widely appreciated for its advancement in the develop- ment of machine translation with remarkable improvement in quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Hence, many researchers have compared both techniques for low and high-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' (Antonio et al (2017)) has performed a thorough evaluation using statistical- based and neural machine translation systems for nine language directions along a variety of dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In their experiment, for long sentences, SMT systems perform better than the NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Recently, (Castilho et al (2017)) has used automatic metrics and expert translators to conduct a thorough quan- titative and qualitative comparison of NMT and SMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' SMT shows better according to their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The comparison of NMT and SMT for the Nepali (NE) using the Nepali National Corpus (NNC) with 6535 sentences has been shown by (Acharya et al (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The researchers have proved in their experiments that the SMT model performs better than the NMT-based system with a small corpus with a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='27 BLEU score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In 2021, Long Short-Term Memory networks (LSTMs) integrated with atten- tion mechanism using WAT corpus have been used in experiments by (Singh et al (2003)) to achieve a 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='7 BLEU score as opposed to a baseline of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='5 BLEU score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' (Abujar et al (2021)) has developed a BN-EN MT model on AmaderCAT cor- pus using Sequence-to-Sequence (seq2seq) architecture, a special class of Re- current Neural Networks to develop the translation system and has achieved a BLEU score of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In the year 2021, translation of English and Hindi-to-Tamil languages us- ing both SMT and NMT has been presented by (Akshai et al (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The disadvantages of NMT have been shown in their experiments such as the occurrence of numerous errors by NMT when interpreting domain terms and OOV (Out of vocabulary) phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' NMT frequently constructs inaccurate lexical choices for polysemous words and occasionally counters reordering mistakes while translating words and domain terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The translations that have been generated by the NMT models mostly include repetitions of pre- viously transcribed words, odd translations, and many unexpected sentences having no correlation with the original sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 14 3 Experimental Framework 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='1 Dataset Samanantar and OPUS datasets for model building and standard benchmark dataset i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Flores 200 for testing are utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Samanantar is the largest cor- pus collection for ILs (Gowtham et al (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The collection includes more than 45 million sentence pairs in English and 11 ILs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The Samanantar Corpus has been used for Assamese (AS), Malayalam (ML), Bengali (BN), Marathi (MR), Gujarati (GU), Kannada (KN), Hindi (HI), Oriya (OR), Punjabi (PA), Telugu (TE), and Tamil (TA) for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' OPUS is a large resource with freely available parallel corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The corpus includes data from many domains and covers over 90 languages (Tiedemann (2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The OPUS cor- pus is used for Sinhala (SI), Sindhi (SD), Urdu (UR), and Nepali (NE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Table 2 gives statistics of the dataset used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' FLORES-200 (Marta et al (2022)) dataset is a multilingual parallel dataset with 200 languages, that are used as human-translated benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It con- sists of two corpora, labeled “dev” (997 lines) and “devtest” (1013 lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The “dev” dataset has been used for fine-tuning, and the “devtest” dataset has been used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='2 Methodology Our proposed process comprises of following major steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Setting up SMT System Moses SMT Toolkit is used to build our SMT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is written in C++ and Perl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' At the moment, this is one of the best SMT tools available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' First, Moses, GIZA++ (Och (2003)), CMPH (for binarization) and SRILM in Ubuntu are installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For training, fine-tuning and testing processes, the system needs a par- allel corpus of the language pair in addition to configurable phases according to developer’s choice to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Data Preprocessing A qualitative corpus plays a major role in any MT task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' While obtaining corpora from various sources, data qual- ity i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' critical for the effectiveness of an MT system, can never be ascertained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' So, removing unnecessary noise is an important task be- fore using the data to train our statistical machine translation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Following processes are used to preprocess and clean it: 15 Table 2: Parallel corpus statistics English to Indic Parallel Corpus(Sentences) Assamese (AS) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='14M Malayalam (ML) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='85M Bengali(BN) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='52M Marathi(MR) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='32M Gujarati(GU) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='05M Kannada(KN) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='07M Hindi(HI) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='56M Oriya(OR) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='00M Punjabi(PA) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='42M Telugu(TE) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='82M Sindhi(SD) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='95M Sinhala(SI) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='68M Nepali(NE) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='35M Tamil(TA) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='16M Urdu(UR) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='95M Data Cleaning and Formatting The goal of data cleaning is either to find and fix or to delete erroneous data from the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Here, characters those are used neither in ILs nor in English are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Some of the punctuation in extended Unicode is con verted to its standard counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Numbers in the IL corpus are converted from English to IL scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Characters outside the stan dard alphabets of the language pair, extra spaces, and unprintable characters are also removed from the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The preprocessing techniques used in our work have been summarized as follows: – Removing unprintable characters – Removing characters outside the language pair – Removing extra spaces – Deaccenting accented characters – Changing non standard Unicode punctuation characters in both corpora to their standard counterparts – Changing uncommon punctuations to more common ones – Changing numbers to a uniform numbering system and script 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Tokenization: It is the process of dividing a character sequence into smaller units known as tokens based on a given character sequence and a specified document unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Words, punctuation, and numerals serve as these tokens in our instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The corpus is tokenized using a modified Moses tokenizer (Koehn et al (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Redundant punctuations (quo- tation marks, apostrophes, and commas) are also removed from the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Training Truecasing Model: This is the procedure for adding case information to text that has been incorrectly cased or is not cased (Lita et al (2003)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Data sparsity is lessened with the use of true casing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' A truecaser model (a model which changes the words at the beginning of the sentence to the most common casing) is trained on the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The Moses truecasing is used for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Training Language and Translation Models: In MOSES, the training procedure utilizes word and segment occurrences to draw con- nections between the target and source languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The language and translation models are trained on the training dataset and binarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' GIZA++ grow-diag-final-and alignment is used for word alignments, which start with the intersection of the two alignments and then add the additional alignment points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The grow-diag-final-and model starts with the intersection of the align- ments from source to target and target to source, then two steps are used to add additional alignment points (Och (2003)): grow-diag: For every neighboring point to the alignments measured, if either source or target word is not aligned already but is present in the union of the alignment, then the neighboring point is in- cluded in the alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' final: If any phrase pairs are unaligned but present in the union, add the point to the alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' • Word Alignment Model: After preprocessing the words, the next step is word alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The proposed work employs the GIZA++ (Och (2003)) incorporation of the IBM models to ac- complish the word procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The GIZA++ model assesses the likelihood of word-to-word alignment for each source and target word in each sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' To produce a good-quality word alignment, 17 − − − − the alignment is produced using a series of successive estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' To process a corpus with a larger quantity of sentences, the process takes several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The alignment method’s outcomes establish a connection between the target and source words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' • Reordering It is the process of restructuring the word order of one natural language sentence to make it more similar to the word order of another natural language sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is a critical task in transcription for languages with different syntactic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The Moses system learns different reordering possibilities for each phrase during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Instead of default reordering, the model uses the distance reordering model (Kumawat et al (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' – Distance-Based Reordering: The reordering of the tar- get output phrases is represented by the relative distortion probability distribution re (St, Et 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Here, St refers to the starting position of the source phrase that is interpreted into the t 1 th target phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The reordering distance (St - Et 1) is calculated as follows: When taking source words out of sequence, the reordering distance is the number of words ig- nored (either forward or backward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' If two phrases are trans- lated in sequence, then t = Et 1 +1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' that is, the first word of the phrase immediately follows the last word of the pre- vious phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' A reordering cost of re(0) is used in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The distance-based model assigns a linear cost to reordering distance, implying that the movement of phrases over long distances is more expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Fine tuning: It is the process of determining the best configuration file settings for a translation model when it is used for a specific pur- pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It uses a translation model to translate all 15 ILs source language phrases in the tuning set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Then, it compares the model’s output to a set of reference (human translations) and adjusts the settings to improve translation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This procedure is repeated several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The tuning process repeats the steps with each iteration until the transla- tion quality is optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The model is fine-tuned on the preprocessed Flores-200 dev dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Translation: The final model is used to translate the preprocessed 18 Flores-200 devtest dataset from the source to the target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Postprocessing and Detokenization: Redundant punctuation marks (quotation marks, apostrophes, and commas) are removed, and the translation file is detokenized using the Moses detokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Evaluation: The evaluation metrics use for our experiments are ME- TEOR (Banerjee et al (2005)), RIBES (Wołk et al (2016)), and BLEU (Papineni et al (2002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 4 Essential metrics for MT translation evalu- ation The most crucial phase of any MT system is MT evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Both automatic and manual methods can be applied to analyze MT tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The effective- ness of a system’s output can be evaluated either directly through human assessments, or indirectly using reading cases, other downstream activities, and even through estimating the amount of effort necessary to rectify the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' A better outcome is obtained through manual evaluation, which includes task-based evaluations, fluency and adequacy scores, human vot- ing for translations task, post-editing measures, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' However, the major challenges of manual evaluation are time-intensiveness, absence of repeata- bility and high cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In order to evaluate the effectiveness of MT output, different automated approaches are there such as Metric for Evaluation of Translation with Explicit Ordering (METEOR), Bilingual Evaluation Un- derstudy(BLEU), Levenshtein, Rank-based Intuitive Bilingual Evaluation Score(RIBES), Word Error Rate (WER) and NIST exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Several intuitive advantages exist for automated metrics that can give points for synonyms or paraphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' A few of the evaluation metrics which are used in our work are discussed below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Bilingual Evaluation Understudy (BLEU): The most widely used method for evaluating machine translation (MT) is known as BLEU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This method, first introduced in 2002 (Papineni et al (2002)) exam- ines one or more reference translations to the hypothetical translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' When the hypothetical translation matches numerous strings with the reference translation, the MT evaluation gives it a higher score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The BLEU system assigns a translation a score from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' However, it is 19 1 usually represented as a percentage value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The nearer the translation is to 1, the more it corresponds to the reference translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This match- ing of translation is conducted word-by-word in the same word order in both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' SacreBLEU is used to calculate the BLEU scores of baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Rank-based Intuitive Bilingual Evaluation Score (RIBES): It is calculated by incorporating a rank correlation coefficient before uni- gram matches, eliminating the necessity for higher-order n-gram matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This metric is concerned with word order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' To compare SMT and ref- erence translations, it employs Kendall’s tau coefficient (τ ) based on word order to indicate rank differences (Wołk et al (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' To assure positive values, the coefficient is normalized as shown below: Normalized Kendall’s τ (NKT) = τ + 1 2 (5) This coefficient can be paired with unigram-precision p1 and Brevity Penalty BP and changed to prevent overestimation of the correlation between only relevant words in SMT and reference translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' RIBES = NKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='(pα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' (BPβ ) (6) Here, α and β are parameters between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Metric for Evaluation of Translation with Explicit Ordering (METEOR): Meteor scores a translation depending on explicit word- by-word similarities between both the translation and a provided ref- erence translation (Banerjee et al (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is specifically created to generate sentence-level scores that are highly correlated with human evaluations of translation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Meteor utilizes and highlights recall in combination with precision, a feature that numerous measures have verified as crucial for a strong correlation with human judgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It also intends to address the problem of imprecise reference translations by utilizing adaptable word matching in consideration with synonyms and morphological variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' To achieve a score of 1, the words of the machine-generated output should be present in the reference and each of the words of the reference is in the machine-generated output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 5 Results and Discussion In this work, the evaluation metrics used are METEOR (Banerjee et al (2005)), RIBES (Wołk et al (2016)), and BLEU (Papineni et al (2002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' All the evaluation metrics used in our work are prominent metrics for deter- mining the quality of the machine-translated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Table 3 displays the translation of all the 15 ILs to English and vice versa using SMT without fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Evaluation metrics of SMT with finetuning using the Flores-200 dev dataset are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' RIBES and METEOR range is 0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For EN-IL and IL-EN language using SMT, the BLEU score lies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='46 to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='09 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='49 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='41 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The RIBES score for EN-IL and IL-EN is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='04 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='63 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='14 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='61 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' METEOR scores lie between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='28 for EN-IL and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='02 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='28 for IL-EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' SMT models using distance reordering techniques are giving better BLEU Scores for languages BN, PA, UR, HI, and GU than the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' With- out fine-tuning, SI performs the worst in terms of all three metrics of all languages in both directions, whereas with fine-tuning EN-SI and TA-EN perform worse than all other EN-IL and IL-EN models respectively with all 21 Table 3: Evaluation Metrics Result of SMT without Finetuning Languages Pairs BLEU RIBES METEOR AS EN AS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='09 AS EN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='11 ML EN ML 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='08 ML EN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='12 BN EN BN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='17 BN EN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='45 012 MR EN MR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='09 MR EN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='09 GU EN GU 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='16 GU EN 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='21 KN EN KN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='11 KN EN 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='16 HI EN HI 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='28 HI EN 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='28 OR EN OR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='14 OR EN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='17 PA EN PA 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='18 PA EN 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='24 TE EN TE 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='12 TE EN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='18 SD EN SD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='08 SD EN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='09 SI EN SI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='02 SI EN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='05 NE EN NE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='16 NE EN 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='19 TA EN TA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='05 TA EN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='07 UR EN UR 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='24 UR EN 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='23 22 Table 4: Evaluation Metrics Result of SMT with Finetuning Languages Pairs BLEU RIBES METEOR AS EN AS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='08 AS EN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='10 ML EN ML 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='06 ML EN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='06 BN EN BN 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='19 BN EN 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='23 MR EN MR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='08 MR EN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='07 GU EN GU 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='14 GU EN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='01 KN EN KN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='14 KN EN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='10 HI EN HI 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='22 HI EN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='14 OR EN OR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='15 OR EN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='11 PA EN PA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='15 PA EN 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='19 TE EN TE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='10 TE EN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='12 SD EN SD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='09 SD EN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='09 SI EN SI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='01 SI EN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='11 NE EN NE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='14 NE EN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='13 TA EN TA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='05 TA EN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='02 UR EN UR 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='19 UR EN 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='18 23 three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' HI and BN languages have qualitative, large, and less noisy datasets compared to other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Hence, HI performs the best among all languages without fine-tuning in all three metrics in both directions, and BN performs the best among all languages with fine-tuning in both direc- tions with respect to BLEU and RIBES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In addition, UR and PA also produce good RIBES metrics than other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' RIBES score for PA is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='63(for EN- PA) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='61(PA-EN), and for UR, RIBES score is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='62(EN-UR) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='61(UR-EN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Even though SI has a good amount of corpus, the corpus does not have reli- able translations compared to other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For example, the sentence in English “Heb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 11:32-34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Judg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 16:18-21, 28-30 Jehovah’s spirit operated on Samson in a unique way because of unusual circumstances” has been trans- lated to Sinhala in the corpus as “11:32-34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Gනි.”, which only translates “Heb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 11:32-34;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Hence, SI does not perform well compared to other lan- guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Similarly, in the EN-TA corpus, the sentence “He’s my boss” has been translated to “அவர் எனது ே மலாளர் மட்டும்தான் .” which ac- tually means “He is only my manager”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' From the example, it is clear that EN-TA corpus also has ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Additionally, even though the ILs-English and English-ILs systems are trained using the same corpus, a significant dis- crepancy in the BLEU scores is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' This is due to the significant morphological diversity of ILs and the relative difficulty of translating from English to ILs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It has been observed that SI has a high number of lines (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='68 M) but performs poorly as compared to languages like PA (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='42 M) and GU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='05 M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It is also observed that languages with very steep slopes tend to have low scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For example, EN-TA and EN-ML have 60% sentences with less than 4 tokens, and they have not-so-good scores as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In contrast, languages with good scores, like HI and BN have more gentle slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' So, length of sentences is a contributing factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' EN-SD is an ex- ception which has a gentle slope but does not give good scores, because the corpus does not have good translation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Therefore, the quality of the corpus matters more than the size of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' 6 Conclusion and Future Work This paper has presented the MT work for 15 ILs to English and vice versa using SMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It also describes the linguistic features of all 15 ILs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' A tailor- made preprocessing approach has been incorporated into this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The 24 Figure 1: Less than ogive for number of tokens in a sentence for all fifteen language corpora LessthanOgiveforNumberoflokens 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='2 given numberoftokenslessthan 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='6 ofsentences 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='4 Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content='2 0 0 1 2 m 4 5 6 7 8 9 Numberoftokens EN AS(AS) EN SI(SI) EN ML(ML) EN MR(MR) EN OR(OR) EN SD(SD) EN UR(UR) —EN BN(BN) EN GU(GU) EN HI(HI) EN KN(KN) EN NE(NE) EN PA(PA) EN TA(TA) EN TE(TE)25 model has utilized the grow-diag-final-and alignment model and distance re- ordering model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' For checking the quality of translation, different evaluation Metrics such as BLEU, RIBES, and METEOR are utilized in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' From the result, it is observed that the proposed SMT model is quite satis- factory for some of the ILs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' However, the level of performance is not at par with the rest of the ILs and there lies the need of improvement to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Due to the scarcity and quality of parallel corpus, the metrics obtained are quite low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' It has been observed that the translations of some of the languages are not sufficiently accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Measures of validating corpus quality shall be explored in order to observe the corpus quality and remove inaccurate lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Dravid- ian languages are in general agglutinative languages (words are made up of morphemes, with each morpheme contributing to the meaning of the word).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In future, means to infer translations from the breakdown of words in these languages shall also be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' Interestingly, in some of the ILs, finetuning schemes are hampering the qual- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' The causes of this phenomenon shall be analyzed and mitigated via techniques such as noise reduction, corpus cleaning, and finetuning schemes for those languages to ensure better quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} +page_content=' In addition, more 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for Computational Linguistics, 177–180 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfqPh6/content/2301.00539v1.pdf'} diff --git a/9tE2T4oBgHgl3EQfmAcG/content/tmp_files/2301.03993v1.pdf.txt b/9tE2T4oBgHgl3EQfmAcG/content/tmp_files/2301.03993v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8008730a83ab1396024d3a8f00cc1a3a61edf0e2 --- /dev/null +++ b/9tE2T4oBgHgl3EQfmAcG/content/tmp_files/2301.03993v1.pdf.txt @@ -0,0 +1,6461 @@ +arXiv:2301.03993v1 [math.AP] 10 Jan 2023 +A PERTURBATIVE APPROACH TO H¨OLDER CONTINUITY OF SOLUTIONS TO A +NONLOCAL p-PARABOLIC EQUATION +ALIREZA TAVAKOLI +Abstract. We study local boundedness and H¨older continuity of a parabolic equation involving the fractional +p-Laplacian of order s, with 0 < s < 1, 2 ≤ p < ∞, with a general right hand side. We focus on obtaining +precise H¨older continuity estimates. The proof is based on a perturbative argument using the already known +H¨older continuity estimate for solutions to the equation with zero right hand side. +Date: January 11, 2023. +1. Introduction +In this paper, we study the local boundedness and H¨older regularity of solutions to the inhomogeneous +equation +ut + (−∆p)su = f(x, t), +(1.1) +where f ∈ Lr +loc(I; Lq +loc(Ω)) with q ≥ 1, r ≥ 1, p ≥ 2 and s ∈ (0, 1). Here, (−∆p)s is the fractional p-Laplacian, +arising as the first variation of the Sobolev-Slobodecki˘ı seminorm +(−∆p)su(x) := 2 P.V. +� +Rn +|u(x) − u(y)|p−2(u(x) − u(y)) +|x − y|n+s p +dy. +Nonlocal equations involving operators of the type above, with a singular kernel, were first considered in [IN10] +to the best of our knowledge. +In this study, continuing the work in [BLS21], we perform a perturbative argument to obtain H¨older continuity +estimates, with explicit exponents for the equations with a right hand side. Our approach closely follows the +arguments in [TU14] and [BLS18]. +In such perturbative arguments it is often possible to establish H¨older +regularity results for bounded solutions using only L∞ estimates for the equations with zero right hand side. +This is not the case here. Due to the presence of a suprimim in time in the tail (see section 3), we are led to +proving a L∞ bound for equations with right hand sides, this is Theorem 1.1. The proof is inspired by the +work [AS67]. +Below, we state the main results. For the definition of the tail and relevant function spaces, see Section 2. +We use the following notation of parabolic cylinders +QR,r(x, T ) := BR(x0) × (T − r, T ] . +The exponent p⋆ +s = +np +n−sp is the critical exponent for the Sobolev embedding therem, see Proposition 2.5. We +denote by p′, the H¨older conjugate of p, that is p′ = +p +p−1. +Theorem 1.1. +Let Ω ⊂ Rn be a bounded and open set, I = (t0, t1], p ≥ 2, 0 < s < 1. Consider q and r such +that r ≥ p′, +1 +r + n +spq < 1 +and q ≥ (p⋆ +s)′ +in the case sp < n, +and +1 +r + 1 +q < 1 +and q > 1 +in the case sp ≥ n. +2010 Mathematics Subject Classification. 35K55, 35K65, 35R11. +Key words and phrases. Fractional p-Laplacian, Local H¨older regularity, Nonlocal diffusion. +1 + +2 +ALIREZA TAVAKOLI +Suppose u is a local weak solution of +ut + (−∆p)su = f +in Ω × I, +such that +u ∈ L∞ +loc(I; Lp−1 +sp (Rn)) +and +f ∈ Lr +loc(I; Lq +loc(Ω)). +then u is locally bounded in Ω. More specifically, if Q2R,(2Rsp)(x0,T0) ⊂ Ω × I, u bounded in QR/2,(R/2)sp(x0, T0) +and in the case sp ̸= n, the estimate reads +∥u∥L∞(Q R +2 ,( R +2 )sp) ≤ 2 +sup +T0−Rsp n. +In the case sp = n, given any l such that +p +r′ (1 − 1 +r − 1 +q )−1 < l < ∞ we get +∥u∥L∞(Q R +2 ,( R +2 )sp) ≤ 2 +sup +T0−Rsp 1 +in the case sp ≥ n. +Define the exponent +Θ(s, p) := + + + + + + + + + +s p +p − 1, +if s < p − 1 +p +, +1, +if s ≥ p − 1 +p +, +(1.2) +Suppose u is a local weak solution of +ut + (−∆p)su = f +in Ω × I, +such that +u ∈ L∞ +loc(I; L∞ +loc(Ω)) ∩ L∞ +loc(I; Lp−1 +sp (Rn)), +and +f ∈ Lr +loc(I; Lq +loc(Ω)). +Then +u ∈ Cα +x,loc(Ω × I) ∩ C +α +sp−(p−2)α +t,loc +(Ω × I), +for every 0 < α < min +� +Θ, +r(spq − n) − spq +q(r(p − 1) − (p − 2)) +� +. +More precisely, for every 0 < α < min +� +Θ, +r(spq−n)−spq +q(r(p−1)−(p−2)) +� +, R > 0, x0 ∈ Ω and T0 such that +QR,2Rs p(x0, T0) ⋐ Ω × (t0, t1], + +H ¨OLDER CONTINUITY +3 +there exists a constant C = C(n, s, p, q, r, α) > 0 such that +|u(x1, t1)−u(x2, t2)| ≤ C +� +M +�|x2 − x1| +R +�α ++ Mp−1�|t2 − t1| +Rs p +� +α +sp−(p−2)α � +(1.3) +for any (x1, t1), (x2, t2) ∈ QR/2,(R/2)s p(x0, T0), with +M = 1+∥u∥L∞(QR,2Rsp(x0,T0))+ +sup +T0−2Rsp≤t≤T0 +Tailp−1,sp(u( q, t); x0, R)+ +� +Rsp− n +q − sp +r ∥f∥Lq,r(QR,2Rsp(x0,T0)) +� +1 +1+ p−2 +r′ . +1.1. Known results. Recently, there has been a growing interest in nonlocal problems of both elliptic and +parabolic types. For studies of fractional p-Laplace operators with different (continuous) kernels see [AMRT10]. +Parabolic equations of the type (1.1) were first considered in [P15] with a slightly different diffusion operator. +See also [AABP18], [MRT16], [Va16] and [Va20] for studies of the existence, uniqueness, and long time behavior +of solutions. Here we seize the opportunity to mention [CV10], [ChD14], [ChD14(2)], and [W16] which contain +regularity results for parabolic nonlocal equations. +The local boundedness of the solutions to equations modeled on (1.1) with zero right hand side was obtained +in [S19]. The results concern operators of the form +LK = P.V. +� +Rn K(x, y, t)|u(x) − u(y)|p−2(u(x) − u(y)) dy, +where K is symmetric in the space variables and satisfies the ellipticity condition +Λ−1 +|x − y|n+sp ≤ K(x, y, t) ≤ +Λ +|x − y|n+sp . +Later in [DZZ21] local boundedness for certain right hand sides of the form f(x, t, u) was established. [S19(2)] +contains a Harnack inequality for equations with zero right hand side. H¨older regularity has also been established +in [APT22] and [L22] for all 1 < p < ∞ for equations with zero right hand sides. In [BLS21] they prove H¨older +continuity of the solutions with explicit exponents (for f = 0 and K = |x − y|−n−sp). Recently in [GDS22], the +same type of result has been established for nonlocal equations with double phase, that is for diffusion operators +involving two different degrees of homogeneity and differentiability. +In this study, continuing the work in [BLS21], we perform a perturbative argument to obtain H¨older continuity +estimates with explicit exponents for equations with a right hand side. +1.1.1. Comparison of the results to some previous works. +Local boundedness and continuity. We compare our boundedness result to [DZZ21]. Their result concerns more +general right hand sides depending on the solution as well. In the limiting case of s → 1, they reproduce the +local boundedness result contained in [DiB93] for the evolution p-Laplacian equation. To compare the results, if +we restrict their result to right hand sides that are u-independent, their assumption on the integrability becomes +q, r > n+sp +sp ( +p(n+2s) +2sp+(p−1)n). Their analysis is done with the same integrability assumption in time and space; Our +local boundedness result, Theorem 1.1, contains this range of exponents. +In the limiting case when s goes to 1, our assumptions become q ≥ (p⋆)′, r ≥ p′, and 1 − 1 +r − n +pq > 0. This is +in accordance with the classical condition for boundedness of the evolution p-Laplace equation, see for example +Remark 1 in [LSS13]. If we assume the same integrability in time and space, the condition 1 − 1 +r − n +pq > 0 +reduces to f ∈ Lˆq with ˆq > n+p +p . This matches the condition in [Ve93]. +Now we turn our attention to the nonlocal elliptic (time independent) case. For r = ∞ the condition for +boundedness and basic H¨older continuity becomes +q > n +sp , +if sp < n +and +q > 1 , +if sp ≥ n . +In the case sp < n, this is the same condition for local boundedness and continuity contained in [BP16]. When +sp > n and q ≥ 1, the boundedness and H¨older continuity for the time independent equation is automatic using + +4 +ALIREZA TAVAKOLI +Morrey’s inequality. Our result does not cover the case of r = ∞ , q = 1 , which one would expect in comparison +to the time independent case. +H¨older continuity exponent: In the case r = ∞, the critical H¨older continuity exponent +min +� +Θ, +r(spq − n) − spq +q(r(p − 1) − (p − 2)) +� += min +� +Θ, sp +1 − 1 +r − +n +spq +p − 1 − p−2 +r +� +, +(1.4) +reduces to min {Θ, +sp +p−1(1 − +n +spq )} which matches the results in [BLS18]. +Let us also compare our results to the local p-parabolic equation studied in [TU14] where precise H¨older +continuity exponents are obtained. If we send s to 1, (1.4) becomes +min {1, +r(pq − n) − pq +q(r(p − 1) − (p − 2))} , +which is in accordance with the result in [TU14]. +In [GDS22] explicit H¨older continuity exponents for the more general case of double phase nonlocal diffusion +operators were obtained. The ideas explored there are similar to the ones in [BLS21], but their result allows for +a bounded right hand side instead of just zero. Their result implies the H¨older continuity exponent that we get +in the case of f ∈ L∞, although with a slightly different estimate of the H¨older constants. +1.2. Plan of the paper. In Section 2 we introduce some notations and preliminary lemmas. We also restate +and adapt a result on the existence of solutions to our setting. +In Section 3, we establish basic local H¨older regularity and boundedness for local weak solutions. +Section 4 is devoted to proving Theorem 1.2. A so called tangential analysis is performed to get specific +H¨older continuity exponents in terms of q, r, s, and p. +The article is also accompanied by two appendices. In the first one, Appendix A, we work out the details +for a modified version of [BLS21, Theorem 1.1]. The aim is to establish a H¨older estimate in terms of the tail +quantity. +In Appendix B we justify using certain test functions in the weak formulation of (1.1). +1.3. Acknowledgements. The author warmly thanks Erik Lindgren for introducing the problem, proofreading +this paper, for his helpful comments, and long hours of fruitful discussions. The author has partially been +supported by the Swedish Research Council, grant no. 2017- 03736. +During the development of this paper, I have been a PhD student at Uppsala University. In particular, I wish +to express my gratitude to the Department of Mathematics at Uppsala University for its warm and hospitable +research environment +This paper was finalized while I was participating in the program geometric aspects of nonlinear partial +differential equations at Mittag-Leffler institute in Djursholm, Sweden during the fall of 2022. The research +program is supported by Swedish Research Council grant no. 2016-06596 +2. Preliminaries +2.1. Notation. We define the monotone function Jp : R → R by +Jp(t) = |t|p−2t . +We use the notation BR(x0) for the open ball of radius R centered at x0. If the center is the origin, we +simply write BR. We use the notation of ωn for the surface area of the unit n-dimensional ball. For parabolic +cylinders, we use the notation Qr,rθ(x0, t0) := Br(x0) × (t0 − rθ, t0]. If the center is the origin, we write Qr,rθ. +We will work with the fractional Sobolev space extensively: +W β,q(Rn) := {ψ ∈ Lq(Rn) : [ψ]W β,q(Rn) < ∞}, +0 < β < 1, +1 ≤ q < ∞, + +H ¨OLDER CONTINUITY +5 +where the seminorm [ψ]W s,p(Rn) is defined as below +[ψ]q +W β,q(Rn) = +�� +Rn×Rn +|ψ(x) − ψ(y)|q +|x − y|n+βq +dx dy. +We also need the space W β,q(Ω) for a subset Ω ⊂ Rn, defined by +W β,q(Ω) := {ψ ∈ Lq(Ω) : [ψ]W β,q(Ω) < ∞}, +0 < β < 1, +1 ≤ q < ∞, +where +[ψ]q +W β,q(Ω) = +�� +Ω×Ω +|ψ(x) − ψ(y)|q +|x − y|n+βq +dx dy. +In the following, we assume that Ω ⊂ Rn is a bounded open set in Rn. We define the space of Sobolev functions +taking boundary values g ∈ Lp−1 +sp (Rn) by +Xβ,q +g +(Ω, Ω′) = {ψ ∈ W α,q(Ω′) ∩ Lp−1 +sp (Rn) : ψ = g on Rn \ Ω }, +where Ω′ is an open set such that Ω ⋐ Ω′. +We recall the definition of tail space +Lq +α(Rn) = +� +u ∈ Lq +loc(Rn) : +� +Rn +|u|q +1 + |x|n+α dx < +∞ +� +, +q ≥ 1 and α > 0, +which is endowed with the norm +∥u∥Lq +α(Rn) = +�� +Rn +|u|q +1 + |x|n+α dx +� 1 +q +. +For every x0 ∈ Rn, R > 0 and u ∈ Lq +α(Rn), the following quantity +Tailq,α(u; x0, R) = +� +Rα +� +Rn\BR(x0) +|u|q +|x − x0|n+α dx +� 1 +q +, +plays an important role in regularity estimates for solutions to fractional problems. +Let I ⊂ R be an interval and let V be a separable, reflexive, Banach space endowed with a norm ∥ q∥V . We +denote by V ⋆ its topological dual space. Let us suppose that v is a mapping such that for almost every t ∈ I, +v(t) belongs to V . If the function t → ∥v(t)∥V is measurable on I and 1 ≤ p ≤ ∞, then v is an element of the +Banach space Lp(I; V ) if and only if +� +I +∥v(t)∥V dt < ∞ +By [Sh97, Theorem 1.5], the dual space of Lp(I; V ) can be characterized according to (Lp(I; V ))⋆ = Lp′(I; V ⋆). +We write v ∈ C(I; V ) if the mapping t → v(t) is continuous with respect to the norm on V . +2.2. Pointwise inequalities. We will need the following pointwise inequality: Let p ≥ 2, then for every +A, B ∈ R we have +|A − B|p ≤ C +� +Jp(A) − Jp(B) +� +(A − B). +(2.1) +For a proof look at [BLS21, Remark A.4], a close inspection of the proof reveals that the constant can be taken +as C = 3 · 2p−1. Before stating the next inequality, we recall [BP16, Lemma A.2]. +Lemma 2.1. +Let 1 < p < ∞ and g : R → R be an increasing function, and define +G(t) = +� t +0 +g′(τ) +1 +p dτ, +t ∈ R. +Then +Jp(a − b) +� +g(a) − g(b) +� +≥ +��G(a) − G(b) +��p. + +6 +ALIREZA TAVAKOLI +Lemma 2.2. +For p ≥ 2 and β ≥ 1 +� +Jp(a − b) − Jp(c − d) +�� +((a − c)+ +M + δ)β − ((b − d)+ +M + δ)β� +≥ +1 +3 · 2p−1 +βpp +(β + p − 1)p +���((a − c)+ +M + δ) +β+p−1 +p +− ((b − d)+ +M + δ) +β+p−1 +p +��� +p +, +(2.2) +where (t)+ +M := min {max {t, 0}, M}. +Proof. First notice that using (2.1) for a − b − c + d ̸= 0: +3 · 2p−1(Jp(a − b) − Jp(c − d)) ≥ |a − b − c + d|p +a − b − c + d += Jp((a − c) − (b − d)), +after verifying the trivial case a − b − c + d = 0, we get the inequality +(Jp(a − b) − Jp(c − d)) ≥ +1 +3 · 2p−1 Jp((a − c) − (b − d)). +(2.3) +Now we use Lemma 2.1 with g(t) = ((t)+ +M + δ)β. Then with G = � t +0 g′(τ) +1 +p dτ, +G(t) = +pβ +1 +p +β + p − 1 +� +(t+ +M + δ) +β+p−1 +p +− δ +β+p−1 +p +� +. +By Lemma 2.1 +Jp +� +(a − c) − (b − d) +�� +g(a − c) − g(b − d) +� +≥ |G(a − c) − G(b − d)|p. +Hence +Jp +� +(a − c) − (b − d) +�� +((a − c)+ +M + δ)β − ((b − d)+ +M + δ)β� +≥ +βpp +(β + p − 1) +���((a − c)+ +M + δ) +β+p−1 +p +− ((b − d)+ +M + δ) +β+p−1 +p +��� +p +. +Using (2.3) in the inequality above concludes the proof. +□ +2.3. Functional inequalities. We need the following basic inequalities for the tail. +Lemma 2.3. +Let α > 0, 1 < q < ∞, and u, v ∈ Lq +α(Rn) such that u = v on Rn \BR(x0). Then for any σ < 1, +Tailα,q(v; x0, σR) ≤ 2Tailα,q(u; x0, σR) + 2σ +−n +q +� +− +� +BR(x0) +|u − v|q dx +� 1 +q . +Proof. +Tailα,q(v; x0, σR)q = (σR)α +� +Rn\BσR(x0) +|v|q +|x − x0|n+α dx += (σR)α�� +Rn\BR(x0) +|v|q +|x − x0|n+α dx + +� +BR(x0)\BσR(x0) +|v|q +|x − x0|n+α dx +� += (σR)α�� +Rn\BR(x0) +|u|q +|x − x0|n+α dx + +� +BR(x0)\BσR(x0) +|v|q +|x − x0|n+α dx +� +≤ (σR)α�� +Rn\BR(x0) +|u|q +|x − x0|n+α dx + 2q−1 +� +BR(x0)\BσR(x0) +|u|q + |u − v|q +|x − x0|n+α +dx +� +≤ 2q−1(σR)α�� +Rn\BσR(x0) +|u|q +|x − x0|n+α dx + +� +BR(x0)\BσR(x0) +|u − v|q +|x − x0|n+α dx +� +≤ 2q−1Tailα,q(u; x0, σR)q + 2q−1σ−n − +� +BR(x0) +|u − v|q dx +□ +For a proof of the following result, see [BLS18, Lemma 2.3]. + +H ¨OLDER CONTINUITY +7 +Lemma 2.4. +Let α > 0, 0 < q < ∞. Suppose that Br(x0) ⊂ BR(x1). Then for every u ∈ Lq +α(Rn) we have +Tailq,α(u; x0, r)q ≤ +� r +R +�α� +R +R − |x − x0| +�n+α +Tailq,α(u; x1, R)q + r−n∥u∥q +Lq +If in addition u ∈ Lm +loc(Rn) for some q < m ≤ ∞, then +Tailq,α(u; x0, r)q ≤ +� r +R +�α� +R +R − |x − x0| +�n+α +Tailq,α(u; x1, R)q + +�(nωn)m − q +αm + nq +� m−q +m r− qn +m ∥u∥Lm(BR(x1)), +where ωn is the measure of the n-dimensional open ball of radius 1. +We also recall the following Sobolev and Morrey type inequalities: +Proposition 2.5. +Suppose 1 < p < ∞ and 0 < s < 1. Let Ω ⊂ Rn be an open and bounded set. Define p⋆ +s as +p⋆ +s := +np +n − sp. +(2.4) +For every u ∈ W s,p(Rn) vanishing almost everywhere in Rn \ Ω we have +∥u∥p +Lp⋆s (Ω) ≤ C1(n, s, p) [u]p +W s,p(Rn), +if sp < n +(2.5) +∥u∥p +L∞(Ω) ≤ C2(n, s, p)|Ω| +sp +n −1[u]p +W s,p(Rn), +if sp > n +(2.6) +∥u∥p +Ll(Ω) ≤ C3(n, s, p, l)|Ω| +p +l [u]p +W s,p(Rn), +for every 1 ≤ l < ∞, if sp = n +(2.7) +In particular the following Poincar´e inequality holds true +∥u∥p +Lp(Ω) ≤ C |Ω| +sp +n [u]W s,p(Rn), +(2.8) +for some C = C(n, s, p). +Remark 2.6. +The Sobolev type inequalities above are also valid for functions u ∈ Xs,p +0 (Ω, Ω′), where Ω is a +bounded open set and Ω′ is an open set such that Ω ⋐ Ω′. This can be seen using the fact that there is an +extension domain containing Ω and included in Ω′. +We will often use the following special application of H¨older’s inequality +∥u(x, t)∥Lq1,r1 (Ω×J) ≤ ∥|Ω| +1 +q1 − 1 +q2 ∥u( q, t)∥Lq2(Ω)∥Lr1(J) ≤ |Ω| +1 +q1 − 1 +q2 |J| +1 +r1 − 1 +r2 ∥u∥Lq2,r2 (Ω×J) +(2.9) +Where q1 < q2 , r1 ≤ r2. The following interpolation inequality (see e.g. [AS67]) will be useful. +Lemma 2.7. +If w is contained in Lq1,r1(Ω × J) ∩ Lq2,r2(Ω × J), then w is contained in L˜q,˜r(Ω × J), where +1 +˜r = λ +r1 ++ 1 − λ +r2 +, +1 +˜q = λ +q1 ++ 1 − λ +q2 +, +(0 ≤ λ ≤ 1). +Moreover, +∥w∥L˜ +q,˜r(Ω×J) ≤ ∥w∥λ +Lq1,r1(Ω×J)∥w∥1−λ +Lq2,r2(Ω×J). +The following three lemmas will be needed in the proof of our local boundedness result, Proposition 3.3. +Lemma 2.8. +Let sp < n and assume that w is in Lp⋆ +s,p(QR,Rsp) ∩ Lp,∞(QR,Rsp), with p⋆ +s = +np +n−sp being the +Sobolev exponent. Then w is in Lpq′,pr′(QR,Rsp) as long as q, r satisfy +1 − 1 +r − n +spq ≥ 0. +Moreover, +∥w∥p +Lpq′,pr′ (QR,Rsp) ≤ Rsp(1− 1 +r − +n +spq )� +∥w∥p +Lp,∞(QR,Rsp) + ∥w∥p +Lp⋆s,p(QR,Rsp) +� +. + +8 +ALIREZA TAVAKOLI +In particular, in the case of 1 +r + +n +spq = 1 we have +∥w∥p +Lpq′,pr′ (QR,Rsp) ≤ ∥w∥p +Lp,∞(QR,Rsp) + ∥w∥p +Lp⋆s,p(QR,Rsp) . +Proof. Consider a pair of exponents ˜r = ( 1 +r′ − (1 − 1 +r − +n +spq ))−1 = spq +n , and ˜q = q′ such that +1 +˜r ′ + +n +sp˜q ′ = 1. Using +H¨older’s inequality (2.9), we obtain +∥w∥p +Lpq′,pr′ (QR,Rsp) ≤ (Rsp) +1 +r′ − 1 +˜r ∥w∥p +Lp˜ +q,p˜r(QR,Rsp) = Rsp(1− 1 +r − +n +spq )∥w∥p +Lp˜ +q,p˜r(QR,Rsp). +Now we use Lemma 2.7 with the choice +1 +p˜r = λ +p +and +1 +p˜q = λ +p⋆s ++ 1 − λ +p +, +(0 ≤ λ ≤ 1). +This yields +∥w∥Lp˜ +q,p˜r(QR,Rsp) ≤ ∥w∥λ +Lp⋆s,p(QR,Rsp)∥w∥1−λ +Lp,∞(QR,Rsp). +The relations above hold for λ = 1 +˜r = +n +sp˜q ′ and using Young’s inequality we get +∥w∥p +Lp˜ +q,p˜r(QR,Rsp) ≤ ∥w∥pλ +Lp⋆s,p(QR,Rsp)∥w∥p(1−λ) +Lp,∞(QR,Rsp) ≤ ∥w∥p +Lp,∞(QR,Rsp) + ∥w∥p +Lp⋆s,p(QR,Rsp). +This concludes the desired result. +□ +Lemma 2.9. +Let sp > n and assume that w ∈ L∞,p(QR,Rsp) ∩ Lp,∞(QR,Rsp). Consider a pair of exponents +q ≥ 1, r ≥ 1, such that +1 − 1 +r − 1 +q = 0. +Then w belongs to Lpq′,pr′(QR,Rsp) and +∥w∥p +Lpq′,pr′ (QR,Rsp) ≤ R +sp−n +q +� +∥w∥p +Lp,∞(QR,Rsp) + Rn−sp∥w∥p +L∞,p(QR,Rsp) +� +. +Proof. We use Lemma 2.7, with the choice +1 +pr′ = λ +p +and +1 +pq′ = 1 − λ +p +, +(0 ≤ λ ≤ 1), +which holds for λ = 1 +r′ = 1 − 1 +q′ . This yields +∥w∥Lpq′,pr′ (QR,Rsp) ≤ ∥w∥λ +L∞,p(QR,Rsp)∥w∥1−λ +Lp,∞(QR,Rsp). +Therefore, using λ = 1 +r′ = 1 +q we arrive at +R +n−sp +q +∥w∥p +Lpq′,pr′ (QR,Rsp) = Rλ(n−sp)∥w∥p +Lpq′,pr′ (QR,Rsp) ≤ +� +Rn−sp∥w∥p +L∞,p(QR,Rsp) +�λ� +∥w∥p +Lp,∞(QR,Rsp) +�(1−λ) +. +Using Young’s inequality, we conclude +∥w∥p +Lpq′,pr′ (QR,Rsp) ≤ R +sp−n +q +� +∥w∥p +Lp,∞(QR,Rsp) + Rn−sp∥w∥p +L∞,p(QR,Rsp) +� +. +□ +Lemma 2.10. +Let sp = n, q ≥ 1, and r ≥ 1 such that +1 − 1 +r − 1 +q > 0. +Assume that w ∈ Ll,p(QR,Rsp) ∩ Lp,∞(QR,Rsp) for some l such that +l = p +r′ (1 − 1 +r − 1 +q )−1. +Then w belongs to Lpq′,pr′(QR,Rsp) and +∥w∥Lpq′,pr′(QR,Rsp) ≤ R +np +lr′ +� +∥w∥Lp,∞(QR,Rsp) + R +−np +l ∥w∥p +Ll,p(QR,Rsp) +� +. + +H ¨OLDER CONTINUITY +9 +Proof. We use Lemma 2.7 with the choice +1 +pr′ = λ +p +and +1 +pq′ = λ +l + 1 − λ +p +, +(0 ≤ λ ≤ 1). +Due to the assumption 1 +l = r′ +p (1 − 1 +r − 1 +q ), the above equalities hold for λ = 1 +r′ . Hence we get +∥w∥Lpq′,pr′(QR,Rsp) ≤ ∥w∥λ +Ll,p(QR,Rsp)∥w∥1−λ +Lp,∞(QR,Rsp). +Therefore, recalling that λ = 1 +r′ +R +−np +lr′ ∥w∥p +Lpq′,pr′(QR,Rsp) = R +−λnp +l +∥w∥p +Lpq′,pr′ (QR,Rsp) ≤ +� +R +−np +l ∥w∥p +Ll,p(QR,Rsp) +�λ� +∥w∥p +Lp,∞(QR,Rsp) +�1−λ +. +Using Young’s inequality for the right hand side, we can conclude +∥w∥p +Lpq′,pr′(QR,Rsp) ≤ R +np +lr′ +� +∥w∥p +Lp,∞(QR,Rsp) + R +−np +l ∥w∥p +Ll,p(QR,Rsp) +� +. +□ +2.4. Weak solutions. +Definition 2.11. +For any t0, t1 ∈ R with t0 < t1, we define I = (t0, t1]. Let +f ∈ Lp′(I; (W s,p(Ω))∗). +We say that u is a local weak solution to the equation +∂tu + (−∆p)su = f, +in Ω × I, +if for any closed interval J = [T0, T1] ⊂ I, the function u is such that +u ∈ Lp(J; W s,p +loc (Ω)) ∩ Lp−1(J; Lp−1 +s p (Rn)) ∩ C(J; L2 +loc(Ω)), +and it satisfies +− +� +J +� +Ω +u(x, t) ∂tϕ(x, t) dx dt + +� +J +�� +Rn×Rn +Jp(u(x, t) − u(y, t)) (ϕ(x, t) − ϕ(y, t)) +|x − y|n+s p +dx dy dt += +� +Ω +u(x, T0) ϕ(x, T0) dx − +� +Ω +u(x, T1) ϕ(x, T1) dx ++ +� +J +⟨f(·, t), ϕ(·, t)⟩ dt, +(2.10) +for any ϕ ∈ Lp(J; W s,p(Ω)) ∩ C1(J; L2(Ω)) which has spatial support compactly contained in Ω. In equation +(2.10), the symbol ⟨·, ·⟩ stands for the duality pairing between W s,p(Ω) and its dual space (W s,p(Ω))∗. +Now, we define the notion of a weak solution to an initial boundary value problem. +Definition 2.12. +Let I = [t0, t1], p ≥ 2, and Ω ⋐ Ω′, where Ω′ is a bounded open set in Rn. Assume that the +functions u0, f and g satisfy +u0 ∈ L2(Ω), +f ∈ Lp′(I; (Xs,p +0 (Ω, Ω′))∗), +g ∈ Lp(I; W s,p(Ω′)) ∩ Lp−1(I; Lp−1 +s p (Rn)). +We say that u is a weak solution of the initial boundary value problem + + + + + +∂tu + (−∆p)su += +f, +in Ω × I, +u += +g, +on (RN \ Ω) × I, +u(·, t0) += +u0, +on Ω, +(2.11) +if the following properties are verified: +• u ∈ Lp(I; W s,p(Ω′)) ∩ Lp−1(I; Lp−1 +sp (Rn)) ∩ C(I; L2(Ω)); + +10 +ALIREZA TAVAKOLI +• u ∈ Xg(t)(Ω, Ω′) for almost every t ∈ I, where (g(t))(x) = g(x, t); +• limt→t0 ∥u(·, t) − u0∥L2(Ω) = 0; +• for every J = [T0, T1] ⊂ I and every ϕ ∈ Lp(J; Xs,p +0 (Ω, Ω′)) ∩ C1(J; L2(Ω)) +− +� +J +� +Ω +u(x, t) ∂tϕ(x, t) dx dt + +� +J +�� +Rn×Rn +Jp(u(x, t) − u(y, t)) (ϕ(x, t) − ϕ(y, t)) +|x − y|n+sp +dx dy dt += +� +Ω +u(x, T0) ϕ(x, T0) dx − +� +Ω +u(x, T1) ϕ(x, T1) dx ++ +� +J +⟨f(·, t), ϕ(·, t)⟩ dt. +Theorem 2.13. +Let p ≥ 2, let I = (T0, T1] and suppose that g satisfies +g ∈ Lp(I; W s,p(Ω′)) ∩ Lp(I; Lp−1 +s p (Rn)) ∩ C(I; L2(Ω)), +∂tg ∈ Lp′(I; (Xs,p +0 (Ω, Ω′))∗), +lim +t→t0 ∥g(·, t) − g0∥L2(Ω) = 0, +for some g0 ∈ L2(Ω). +Suppose also that +f ∈ Lp′(I; (Xs,p +0 (Ω, Ω′))∗). +Then for any initial datum g0 ∈ L2(Ω), there exists a unique weak solution u to problem + + + + + + + +ut + (−∆p)su = f +in Ω × I +u = g +in (Rn \ Ω) × I +u(x, T0) = g(x, T0) +in Ω +(2.12) +Proof. In [BLS21, Theorem A.3] the same result is proven with a stronger condition gt ∈ Lp′(I; W s,p(Ω′)⋆). The +stronger condition, is not needed in the proof. This condition can be replaced with gt ∈ Lp′(I, Xs,p +0 (Ω; Ω′)⋆) in +all of the steps in the proof, except that the construction gives us a C(I; L2(Ω)) solution. There, the stronger +assumption is used only to show that the boundary condition is in C(I; L2(Ω)), which we assume here. +□ +3. Basic H¨older regularity and stability +Throughout the rest of the article, we assume 0 < s < 1 and 2 ≤ p < ∞. +Here, we argue that the norm of the (s, p)-caloric replacement of u is close to u if f is small enough. By the +(s, p)-caloric replacement of u in a cylinder Bρ(x0) × I we mean the solution to the following + + + + + + + +vt + (−∆p)sv = 0 +in Bρ(x0) × I +v = u +in (Rn \ Bρ(x0)) × I +v(x, τ0) = u(x, τ0) +in Bρ(x0) +(3.1) +Here τ0 is the initial point of the interval I. First we show the existence of a (s, p)-caloric replacement using +Theorem 2.13 +Proposition 3.1. +Let u be a local weak solution of ut +(−∆p)su = f in the cylinder Bσ ×J, for some interval +J = (t1, t2] with f ∈ Lq,r +loc(Bσ × J), for q > (p⋆ +s)′ and r > p′. In addition, we assume that u ∈ Lp(J; Lp−1 +sp (Rn)). +Then for any 0 < ρ < σ, and closed interval I ⋐ J, the (s, p)-caloric replacement of u in Bρ(x0) × I (weak +solution to (3.1)) exists. +Proof. We shall check the conditions in Theorem 2.13. If they are satisfied there exists a unique weak solution +v ∈ Lp(I, W s,p(Bσ)) ∩ Lp−1(I; Lp−1 +sp (Rn)) ∩ C(I; L2(Bρ)) to the problem (3.1). The only condition on u that is + +H ¨OLDER CONTINUITY +11 +not immediate from the fact that u is weak solution is ∂tu ∈ Lp′(I; Xs,p +0 (Bρ , Bσ)⋆). We have to show that for +every function ψ ∈ Lp(I; Xs,p +0 (Bρ , Bσ)) +��� +� +I +⟨ut, ψ⟩ dx dt +��� ≤ C +� +I +∥ψ∥p +W s,p(Bσ) dt. +We shall verify this for test functions belonging to the dense subspace, ψ ∈ Lp(I; Xs,p +0 (Bρ , Bσ))∩C1 +0(I; L2(B)). +We use the equation to do so. We have +� +I +⟨ut, ψ⟩ dx dt = +� +I +� +Bρ +uψt dx dt += − +� +I +�� +Rn×Rn +Jp(u(x, t) − u(y, t))(ψ(x, t) − ψ(y, t)) +|x − y|n+sp +dx dy dt + +� +I +� +Br +f(x, t)ψ(x, t) dx dt += − +� +I +�� +Bσ×Bσ +Jp(u(x, t) − u(y, t))(ψ(x, t) − ψ(y, t)) +|x − y|n+sp +dx dy dt +− 2 +� +I +� +Rn\Bσ +� +Bρ +Jp(u(x, t) − u(y, t))ψ(x, t) +|x − y|n+sp +dx dy dt + +� +I +� +Bρ +f(x, t)ψ(x, t) dx dt. +By H¨older’s inequality, we have +� +I +�� +Bσ×Bσ +|Jp(u(x, t) − u(y, t))(ψ(x, t) − ψ(y, t))| +|x − y|n+sp +dx dy dt +≤ +� +I +���Jp(u(x, t) − u(y, t)) +|x − y| +n +p′ +s(p−1) +��� +Lp′(Bσ×Bσ) +���ψ(x, t) − ψ(y, t) +|x − y| +n +p +s +��� +Lp(Bσ×Bσ) dt +≤ [u]p−1 +Lp(I;W s,p(Bσ))[ψ]Lp(I;W s,p(Bσ)). +(3.2) +For the other nonlocal term, we note that for every x ∈ Bρ and y ∈ Rn \ Bσ we have |y| ≤ +σ +σ−ρ|x − y|. Hence, +� +Rn\Bσ +|Jp(u(x, t) − u(y, t))| +|x − y|n+sp +dy ≤ ( +σ +σ − ρ)n+spC(p) +� +Rn\Bσ +|u(x, t)|p−1 + |u(y, t)|p−1 +|y|n+sp +dy +≤ C(σ, ρ, s, p, n) +� +|u(x, t)|p−1 + ∥u( q, t)∥p−1 +Lp−1 +sp +� +. +Therefore, +� +I +� +Bρ +� +Rn\Bσ +|Jp(u(x, t) − u(y, t))ψ(x, t)| +|x − y|n+sp +dy dx dt ≤ C(σ, ρ, s, p, n) +�� +I +� +Bρ +|ψ(x, t)||u(x, t)|p−1 dx dt ++ +� +I +∥ψ( q, t)∥L1(Bρ)∥u( q, t)∥p−1 +Lp−1 +sp +(Rn) dt +� +. +Using H¨older’s inequality, we have +� +I +� +Bρ +|ϕ(x, t)||u(x, t)|p−1 dx dt ≤ +� +I +∥ψ( q, t)∥Lp(Bρ)∥u( q, t)∥p−1 +Lp(Bρ) ≤ ∥ψ∥Lp(I;Lp(Bρ))∥u∥p−1 +Lp(I;Lp(Bρ)). +(3.3) +For the other term, +� +I +∥ψ( q, t)∥L1(Bρ)∥u( q, t)∥p−1 +Lp−1 +sp +(Rn) dt ≤ ∥ψ∥Lp(I;L1(Bρ))∥u( q, t)∥p−1 +Lp(I;Lp−1 +sp +(Rn)). +(3.4) +Since f ∈ Lp′(I; L(p⋆ +s)′(Bρ)), we get by H¨older’s inequality +� +I +� +Bρ +|fψ| dx dt ≤ +� +I +∥f∥L(p⋆s)′ (Bρ)∥ψ∥Lp⋆s (Bρ) dt ≤ +� +I +∥f∥L(p⋆s)′ (Bρ)∥ψ∥W s,p(Bσ) dt +≤ ∥f∥L((p⋆s)′,p′)(Bρ×I)∥ψ∥Lp(I;W s,p(Bσ)). +(3.5) +Therefore, combining with (3.2) , (3.3), and (3.4) we obtain +��� +� +I +⟨vt, ψ⟩ dt +��� ≤ C(σ, ρ, s, p, n, u, f)∥ψ∥Lp(I;W s,p(Bσ)). +□ + +12 +ALIREZA TAVAKOLI +Lemma 3.2. +Assume that f ∈ Lq,r +loc(Qσ,σsp(x0, T0)) with r ≥ p′ and +q ≥ (p⋆ +s)′ +if sp < n, +q ≥ 1 +if sp > n, and +q > 1 +if sp = n. +Let u be a local weak solution of ∂tu + (−∆p)su = f in Qσ,σsp(x0, T0) and for ρ < σ consider v to be the +(s, p)-caloric replacement of u in Qρ,ρsp(x0, T0). Then we have +− +� +Qρ,ρs p(x0,T0) +|u − v|p dx dt ≤ Cρξ ∥f∥p′ +Lq,r(Qρ,ρs p(x0,T0)) +(3.6) +and +∥u − v∥Lq′,r′ (Qρ,ρsp) ≤ Cρξ+n∥f∥ +1 +p−1 +Lq,r(Qρ,ρsp(x0,T0)), +(3.7) +with ξ = spp′(1 − 1 +r − +n +spq ) and C = C(n, s, p), in the case sp ̸= n. In the case sp = n, we can take ξ = +spp′(1 − 1 +r − 1 +q ) , with C = C(n, s, p, q) also depending on q. +Proof. Let J := [T0 − ρsp, T0], throughout the proof, we drop the dependence of the balls on the center and +write Bρ instead of Bρ(x0), and Qρ,ρsp instead of Qρ,ρsp(x0, T0). +By subtracting the weak formulation of the equations (2.10) for u and v with the same test function ϕ(x, t) ∈ +Lp(J; Xs,p +0 (Bρ, Bσ)) ∩ C1(J; L2(Bρ)) we get +− +� +J +� +Bρ +(u(x, t) − v(x, t)) ∂ +∂tϕ(x, t) dx dt ++ +� +J +� +Rn +� +Rn +� +Jp +� +u(x, t) − u(y, t) +� +− Jp +� +v(x, t) − v(y, t) +�� +(ϕ(x, t) − ϕ(y, t)) +|x − y|n+sp +dx dy dt += +� +Bρ +((u(x, T0 − ρsp) − v(x, T0 − ρsp))ϕ(x, T0 − ρsp) dx − +� +Bρ +((u(x, T0) − v(x, T0))ϕ(x, T0) dx ++ +� +J +� +Bρ +f(x, t)ϕ(x, t). +Now we take ϕ := u − v, which belongs to Lp(J; Xs,p +0 (Bρ; Bσ)) but it may not be in C1(J; L2(Bρ)). We justify +taking this as a test function in Appendix B. By Proposition 6.1 with F(t) = t, we get +� +J +�� +Rn×Rn +� +Jp +� +u(x, t) − v(x, t) +� +− Jp +� +u(y, t) − v(y, t) +���� +u(x, t) − u(y, t) +� +− +� +v(x, t) − v(y, t) +�� +|x − y|n+sp +dx dy dt += +� +J +� +Bρ +f(x, t)(u(x, t) − v(x, t)) dx dt +− 1 +2 +� +Bρ +((u(x, T0) − v(x, T0))2 − ((u(x, T0 − ρsp) − v(x, T0 − ρsp))2 dx += +� +J +� +Bρ +f(x, t)(u(x, t) − v(x, t)) dx dt − 1 +2 +� +Bρ +((u(x, T0) − v(x, T0))2 dx +≤ +� +J +� +Bρ +|f(x, t)(u(x, t) − v(x, t))| dx dt, +(3.8) +where in the third line we have used u(x, T0 − ρsp) = v(x, T0 − ρsp). The left hand side is essentially the W s, p +seminorm. By the pointwise inequality (2.1) +� +J +[u − v]p +W s,p(Rn) dt = +� +J +�� +Rn×Rn +|u(x, t) − v(x, t) − (u(y, t) − v(y, t))|p +|x − y|n+sp +dx dy dt +≤ C(p) +� +J +�� +Rn×Rn +� +Jp +� +u(x, t) − u(y, t) +� +− Jp +� +v(x, t) − v(y, t) +��� +u(x, t) − u(y, t) − +� +v(x, t) − v(y, t) +�� +|x − y|n+sp +dx dy dt. + +H ¨OLDER CONTINUITY +13 +Therefore, by (3.8) and H¨older’s inequality +� +J +[u − v]p +W s,p(Rn) dt ≤ C(p) +� +J +� +Bρ +|f(x, t)(u(x, t) − v(x, t))| dx dt +≤ C(p) +� +J +∥f( q, t)∥Lq(Bρ)∥(u − v)( q, t)∥Lq′(Bρ) dt +≤ C(p)∥f∥Lq,r(Qρ,ρsp)∥u − v∥Lq′,r′(Qρ,ρsp). +(3.9) +Now we consider three cases: sp < n, sp > n,and sp = n. +Case sp < n. By H¨older’s inequality (2.9) and Sobolev’s inequality (2.5) we have +∥u − v∥Lq′,r′ ≤ |Bρ| +1 +q′ − 1 +p⋆s +�� +J +∥u − v∥r′ +Lp⋆s (Bρ) dt +� 1 +r′ +≤ C(n, s, p)|Bρ| +1 +q′ − 1 +p⋆s +�� +J +[u − v]r′ +W s,p(Rn) dt +� 1 +r′ +≤ C(n, s, p)|Bρ| +1 +q′ − 1 +p⋆s |J| +1 +r′ − 1 +p +�� +J +[u − v]p +W s,p(Rn) dt +� 1 +p . +(3.10) +Combined with (3.9) this yields +�� +J +[u − v]p +W s,p(Rn) dt +� p−1 +p +≤ C|Bρ| +1 +q′ − 1 +p⋆s |J| +1 +r′ − 1 +p ∥f∥Lq,r(Qρ,ρsp) += C|Bρ| +1 +q′ − n−sp +np |J| +1 +r′ − 1 +p ∥f∥Lq,r(Qρ,ρsp ), +(3.11) +where C = C(n, s, p). By the Poincar´e inequality +− +� +J +− +� +Bρ +|u − v|p dx dt ≤ C|Bρ| +p′ +q′ −p′ (n−sp) +np ++ sp +n −1|J| +p′ +r′ − p′ +p −1∥f∥p′ +Lq,r(Qρ,ρsp). +Also from (3.11) and (3.10) we get +∥u − v∥Lq′,r′(Qρ,ρsp ) ≤ C(n, s, p)|Bρ| +p′ +q′ −p′ n−sp +np |J| +p′ +r′ − p′ +p ∥f∥ +1 +p−1 +Lq,r(Qρ,ρsp ). +Case sp > n. In this case, we use Morrey’s inequality (2.6) and H¨older’s inequality and obtain +∥u − v∥Lq′,r′(Qρ,ρsp ) ≤ C|Bρ| +1 +q′ +�� +J +∥u − v∥r′ +L∞(Bρ) dt +� 1 +r′ ≤ C|Bρ| +1 +q′ |J| +1 +r′ − 1 +p +�� +J +∥u − v∥p +L∞(Bρ) dt +� 1 +p +≤ C|Bρ| +1 +q′ + sp−n +np |J| +1 +r′ − 1 +p +�� +J +[u − v]p +W s,p(Rn) dt +� 1 +p . +(3.12) +Together with (3.9), this implies +�� +J +[u − v]p +W s,p(Rn) dt +� p−1 +p +≤ C|Bρ| +1 +q′ − n−sp +np |J| +1 +r′ − 1 +p ∥f∥Lq,r(Qρ,ρsp ). +(3.13) +By the Poincar´e inequality +− +� +J +− +� +Bρ +|u − v|p dx dt ≤ C|Bρ| +p′ +q′ −p′ (n−sp) +np ++ sp +n −1|J| +p′ +r′ − p′ +p −1∥f∥p′ +Lq,r(Qρ,ρsp). +Combining (3.12) and (3.13), we get +∥u − v∥Lq′,r′(Qρ,ρsp ) ≤ C(n, s, p)|Bρ| +p′ +q′ −p′ n−sp +np |J| +p′ +r′ − p′ +p ∥f∥ +1 +p−1 +Lq,r(Qρ,ρsp ). +Case sp = n. In this case, we use the critical case of Sobolev’s inequality (2.7) for l = q′ and obtain +∥u − v∥p +Lq′ (Bρ) ≤ C(n, s, p, q)|Bρ| +p +q′ [u − v]p +W s,p(Rn). + +14 +ALIREZA TAVAKOLI +Hence using H¨older’s inequality, we have for any r ≥ p′ +∥u − v∥Lq′,r′(Qρ,ρsp) = +�� +J +∥u − v∥r′ +Lq′ (Bρ) dt +� 1 +r′ +≤ C|Bρ| +1 +q′ +�� +J +[u − v]r′ +W s,p(Rn) dt +� 1 +r′ ≤ C|Bρ| +1 +q′ |J| +1 +r′ − 1 +p +�� +J +[u − v]p +W s,p(Rn) dt +� 1 +p . +The constant C = C(n, s, p, q) above does blow up as q goes to 1. In a similar way as in the prior cases, we get +for q > 1 and r ≥ p′ +− +� +J +− +� +Bρ +|u − v|p ≤ C(n, s, p, q)|Bρ| +p′ +q′ |J| +p′ +r′ − p′ +p −1∥f∥p′ +Lq,r(Qρ,ρsp) +and +∥u − v∥Lq′,r′(Qρ,ρsp) ≤ C(n, s, p, q)|Bρ| +p′ +q′ |J| +p′ +r′ − p′ +p ∥f∥ +1 +p−1 +Lq,r(Qρ,ρsp ). +Using that |Bρ| ∼ ρn and |I| ∼ ρsp we can conclude that +− +� +Qρ,ρs p(x0,T0) +|u − v|p dx dt ≤ Cρξ ∥f∥p′ +Lq,r(Qρ,ρs p), +and +∥u − v∥Lq′,r′(Qρ,ρsp ) ≤ Cρξ+n∥f∥ +1 +p−1 +Lq,r(Qρ,ρsp). +Here in the case of sp ̸= n, +ξ = np′ +q′ − p′ n − sp +p ++ sp − n + spp′ +r′ +− spp′ +p +− sp = p′( n +q′ − n +p′ − n − sp +p ++ sp +r′ − sp +p ) += p′( n +q′ − n + sp +r′ ) = p′(sp +r′ − n +q ) = spp′(1 − 1 +r − n +spq ) +and in the case sp = n, +ξ = p′( n +q′ + sp +r′ − sp +p − sp +p′ ) = spp′( 1 +q′ + 1 +r′ − 1 +p − 1 +p′ ) += spp′(1 − 1 +q + 1 − 1 +r − 1) += spp′(1 − 1 +r − 1 +q ). +□ +Next, we perform a Moser iteration to get an L∞ bound for the difference between the solution and its +(s, p)-caloric replacement. +Proposition 3.3. +Let u be a local weak solution of +∂tu + (−∆p)su = f, +in +Qσ,σsp(x0, T0), +with f ∈ Lq,r +loc(Qσ,σsp(x0, T0)) such that r ≥ p′, +1 +r + n +spq < 1 +and q ≥ (p⋆ +s)′ +in the case sp < n, +and +1 +r + 1 +q < 1 +and q > 1 +in the case sp ≥ n. +Let v be the (s, p)-caloric replacement of u in QR,Rsp(x0, T0), with R < σ. Then in the case of sp ̸= n, we have +∥(u − v)+∥L∞(QRRsp(x0,T0)) ≤ C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp(x0,T0)) +� +, +where ν = 1 − 1 +r − +n +spq , +ϑ = 1 + spν +n +if +sp < n, +and +ϑ = 2 − 1 +r − 1 +q +if +sp > n. + +H ¨OLDER CONTINUITY +15 +In the case of sp = n, given any l such that +p +r′ (1 − 1 +r − 1 +q )−1 < l < ∞ we get +∥(u − v)+∥L∞(QR,Rsp(x0,T0)) ≤ C(n, s, p, q, l)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp(x0,T0)) +� +, +where ϑ = 2 − 1 +r − 1 +q − +p +lr′ and ν = 1 − 1 +r − 1 +q . +Proof. Throughout the proof we write QR,Rsp instead of QR,Rs,p(x0, T0). We test the equations with powers of +u − v and perform a Moser iteration. Using Proposition 6.1 with +F(t) = (min {t+ , M} + δ)β − δβ, +and +δ = max {1, +� +Rspν∥f∥Lq,r(QR,Rsp) +� +1 +p−1 }, +(3.14) +we get +sup +t∈J +� +BR +F(u − v) dx + +� +J +�� +Rn×Rn +Jp(u(x, t) − u(y, t)) − Jp(v(x, t) − v(y, t)) +|x − y|n+sp +× (F(u(x, t) − v(x, t)) − F(u(y, t) − v(y, t))) dx dy dt +≤ +� +J +� +BR +|f(x, t)|F(u(x, t) − v(x, t)) +≤ ∥f∥Lq,r(BR×J)∥((u − v)+ +M + δ)β∥Lq′,r′ (BR×J). +(3.15) +In the last line, we have used H¨older’s inequality. Here F(t) = +� t +0 F(t) dt is +F(t) = + + + + + + + +0 +if +t ≤ 0, +1 +β+1(t + δ)β+1 − δβ+1 +β+1 − tδβ +if +0 ≤ t ≤ M, +1 +β+1(M + δ)β+1 − δβ+1 +β+1 − tδβ + (t − M)(M + δ)β +if +t ≥ M. +Notice that by Young’s inequality, for t ≥ 0 +(t + δ)β+1 +2(β + 1) + +β +β + 12δβ+1 ≥ t + δ +2 +1 +β+1 2 +β +β+1 δβ ≥ tδβ . +In particular for 0 ≤ t ≤ M +F(t) ≥ (t + δ)β+1 +2(β + 1) − 2β + 1 +β + 1 δβ+1 ≥ (t + δ)β+1 +2(β + 1) − 2δβ+1, +and for t ≥ M +1 +β + 1(M + δ)β+1 − δβ+1 +β + 1 − tδβ + (t − M)(M + δ)β = +1 +β + 1(M + δ)β+1 − δβ+1 +β + 1 − Mδβ ++ (t − M) +� +(M + δ)β − δβ� +≥ F(M) ≥ (M + δ)β+1 +2(β + 1) +− 2δβ+1. +Hence +F(t) ≥ (t+ +M + δ)β+1 +2(β + 1) +− 2δβ+1. +(3.16) +Using Lemma 2.2 for the second term in the left hand side of (3.15), and (3.16) in the first term we obtain +1 +2(β + 1) sup +t∈J +� +BR +((u − v)+ +M + δ)β+1 dx + +1 +3 · 2p−1 +βpp +(β + p − 1)p +� +J +[((u − v)+ +M + δ) +β+p−1 +p +]p +W s,p(Rn) dt +≤ sup +t∈J +� +BR +F(u − v) dx + 2δβ+1|BR| + +� +J +�� +Rn×Rn +Jp(u(x, t) − u(y, t)) − Jp(v(x, t) − v(y, t)) +|x − y|n+sp +× (F(u(x, t) − v(x, t)) − F(u(y, t) − v(y, t))) dx dy dt +≤ ∥f∥Lq,r(BR×J)∥((u − v)+ +M + δ)β∥Lq′,r′ (BR×J) + 2δβ+1|BR|. +(3.17) + +16 +ALIREZA TAVAKOLI +Let w(x, t) = ((u − v)+ +M + δ) +β +p . Since δ ≤ (u − v)+ +M + δ, we see that +δβ ≤ +∥w∥p +Lpq′,pr′ (QR,Rsp) +|BR|1− 1 +q |J|1− 1 +r +. +(3.18) +We consider three cases depending on whether sp > n, sp = n, or sp > n. +Case sp < n: Using Sobolev’s inequality in the second term in (3.17) and applying (3.18) we get +δ +2(β + 1)∥w∥p +Lp,∞(BR×J) + C(n, s, p)−1 +3 · 2p−1 +βpp +(β + p − 1)p +� +δp−1∥w∥p +Lp⋆s,p(BR×J) − δβ+p−1|BR| +n−sp +n |J| +� +≤ +δ +β + 1 sup +t∈J +� +BR +((u − v)+ +M + δ)β dx ++ C(n, s, p)−1 +3 · 2p−1 +βpp +(β + p − 1)p +� +J +∥((u − v)+ +M + δ) +β+p−1 +p +− δ +β+p−1 +p +∥p +Lp⋆s (BR) +≤ ∥f∥Lq,r(BR×J)∥w∥p +Lpq′,pr′(BR×J) + 2δ|BR| +∥w∥p +Lpq′,pr′ (BR×J) +|BR|1− 1 +q |J|1− 1 +r +. +(3.19) +Upon multiplying both sides by 3 · 2p−1 × C(n, s, p) (β+p−1)p +δβ +and taking J to have length Rsp, this implies +3 · 2p−2C(n, s, p)(β + p − 1)p +(β + 1)β +∥w∥p +Lp,∞(QR,Rsp) + δp−2pp∥w∥p +Lp⋆s,p(QR,Rsp) +≤ 3 · 2p−1 × C(n, s, p)(β + p − 1)p +δβ +∥w∥p +Lpq′,pr′ +� +∥f∥Lq,r(QR,Rsp) + +2(nωn) +1 +q δRn +Rn(1− 1 +q )+sp(1− 1 +r ) +� ++ ppδβ+p−2Rn−sp+sp. +(3.20) +By replacing the constant C(n, s, p) above in Sobolev’s inequality with max {1, C(n, s, p)}, we can assume +C(n, s, p) ≥ 1. Using this and that δ ≥ 1, and (β+p−1)p +β(β+1) +≥ 1 we obtain +∥w∥p +Lp,∞(QR,Rsp) + ∥w∥p +Lp⋆s,p(QR,Rsp) +≤ 3 · 2p−2C(n, s, p)(β + p − 1)p +(β + 1)β +∥w∥p +Lp,∞(QR,Rsp) + δp−2pp∥w∥p +Lp⋆s,p(QR,Rsp). +Using this together with (3.20), and (3.18) we get +∥w∥p +Lp,∞(QR,Rsp) + ∥w∥p +Lp⋆s,p(QR,Rsp) +≤ C (β + p − 1)p +β +∥w∥p +Lpq′,pr′(QR,Rsp) +�∥f∥Lq,r(QR,Rsp) +δ ++ δp−2 +Rn +Rn− n +q +sp− sp +r + +Rn +Rn− n +q +sp− sp +r +� +≤ C (β + p − 1)p +β +∥w∥p +Lpq′,pr′(QR,Rsp) +�∥f∥Lq,r(QR,Rsp) +δ ++ δp−2R−spν� +, +(3.21) +where C = C(n, s, p). Recalling our choice of δ, (3.14), in the case of δ > 1 +∥f∥Lq,r(QR,Rsp) +δ ++ δp−2R−spν = 2R +−spν +p−1 ∥f∥ +p−2 +p−1 +Lq,r(QR,Rsp) = 2δp−2R−spν, +(3.22) +and in the case of δ = 1 +∥f∥Lq,r(QR,Rsp) +δ ++ δp−2R−spν ≤ 2R−spν = 2δp−2R−spν. +(3.23) +Using the inequality (β+p−1)p +βppp +≤ 1, (3.22), and (3.23), in (3.21) we arrive at +∥w∥p +Lp,∞(QR,Rsp) + ∥w∥p +Lp⋆s,p(QR,Rsp) ≤ C(n, s, p)ppβp−1∥w∥p +Lpq′,pr′ (QR,Rsp) +�δp−2R−spν). +Now notice that since ν > 0, if we take ϑ = 1 + spν +n , the exponents (ϑr′)′, (ϑq′)′ satisfy the condition of Lemma +2.8. Indeed, +1 − +1 +(ϑr′)′ − +n +sp(ϑq′)′ = +1 +ϑr′ + +n +spϑq′ − n +sp = 1 +ϑ( 1 +r′ + +n +spq′ − ϑn +sp ) = 1 +ϑ(ν + n +sp − ϑn +sp ) = 0. + +H ¨OLDER CONTINUITY +17 +Using Lemma 2.8 for the exponents (ϑq′)′ and (ϑr′)′ we get +∥wϑ∥ +p +ϑ +Lpq′,pr′ (QR,Rsp) = ∥w∥p +Lϑpq′,ϑpr′(QR,Rsp) ≤ ∥w∥p +Lp,∞(QR,Rsp) + ∥w∥p +Lp⋆s,p(QR,Rsp) +≤ C(n, s, p)βp−1∥w∥p +Lpq′,pr′ (QR,Rsp) +� +R−spνδp−2� +(3.24) +Now we iterate this inequality with the following choice of exponents +β0 = 1, +βm+1 = ϑβm = ϑm+1. +With the notation +ϕm := ∥((u − v)+ +M + δ) +βm +p ∥ +p +βm +Lpq′,pr′(QR,Rsp) = ∥(u − v)+ +M + δ∥Lβmq′,βmr′(QR,Rsp), +(3.24) reads +ϕm+1 ≤ +� +C R−spνδp−2� +1 +ϑm ϑ +(p−1)m +ϑm +ϕn. +Iterating this yields +ϕm+1 ≤ +� +C R−spνδp−2��m +j=0 ϑ−j +ϑ(p−1) �m +j=0 jϑ−jϕ0. +(3.25) +Since ϑ > 1, we have the following convergent series +∞ +� +j=0 +ϑ−j = +ϑ +ϑ − 1 = n + spν +spν +, +and +∞ +� +j=0 +jϑ−j = +ϑ +(ϑ − 1)2 = n2 + nspν +s2p2ν2 +. +By (3.7) in Lemma 3.2, +ϕ0 =∥(u − v)+ +M + δ∥Lq′,r′ (QR,Rsp) ≤ C(n, s, p)Rspp′ν+n∥f∥ +1 +p−1 +Lq,r(QR,Rsp) + δR +n +q′ + sp +r′ +≤ C(n, s, p)Rn+spν(R +spν +p−1 ∥f∥ +1 +p−1 +Lq,r(QR,Rsp) + δ) ≤ 2C(n, s, p)Rn+spνδ. +(3.26) +Inserting (3.26) to (3.25) and sending n to infinity we obtain +∥(u − v)+ +M + δ∥L∞(QR,Rsp) ≤ Cϑ +(p−1)ϑ +(ϑ−1)2 R−n−spνδ(p−2) +ϑ +ϑ−1 Rspν+nδ += Cϑ +(p−1)ϑ +(ϑ−1)2 δ(p−1)+ p−2 +ϑ−1 +≤ Cϑ +(p−1)ϑ +(ϑ−1)2 max {1, +� +Rspν∥f∥Lq,r(QR,Rsp) +� +1 +p−1 } +p−1+ p−2 +ϑ−1 +≤ C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp) +� +. +Since the above estimate is independent of M, we get +∥(u − v)+∥L∞(QR,Rsp) ≤ C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp) +� +. +Which is the desired result. +Case sp > n. Here we use Morrey’s inequality (2.6) for the second term in (3.17). Instead of (3.19) we +obtain +δ +2(β + 1)∥w∥p +Lp,∞(BR×J) + (C(n, s, p)Rsp−n)−1 +3 · 2p−1 +βpp +(β + p − 1)p +� +δp−1∥w∥p +L∞,p(BR×J) − δβ+p−1|J| +� +≤ +δ +β + 1 sup +t∈J +� +BR +((u − v)+ +M + δ)β dx ++ C(n, s, p)−1 +3 · 2p−1 +βpp +(β + p − 1)p +� +J +∥((u − v)+ +M + δ) +β+p−1 +p +− δ +β+p−1 +p +∥p +L∞(BR) +≤ ∥f∥Lq,r(BR×J)∥w∥p +Lpq′,pr′(BR×J) + 2δ|BR| +∥w∥p +Lpq′,pr′ (BR×J) +|BR|1− 1 +q |J|1− 1 +r +. + +18 +ALIREZA TAVAKOLI +Following the same steps as in the case sp < n we arrive at +∥w∥p +Lp,∞(QR,Rsp) + Rn−sp∥w∥p +L∞,p(QR,Rsp) +≤ C(n, s, p)βp−1∥w∥p +Lpq′,pr′(QR,Rsp) +�∥f∥Lq,r(QR,Rsp +δ ++ R−spν + δp−2R−spν� +≤ C(n, s, p, l)βp−1∥w∥p +Lpq′,pr′ (QR,Rsp)(δp−2R−spν). +Now choose ϑ = 2 − 1 +r − 1 +q . Then ϑ > 1, since 1 +r + 1 +q < 1. Therefore, (ϑr′)′ and (ϑq′)′ satisfy +1 − +1 +(ϑr′)′ − +1 +(ϑq′)′ = 1 +ϑ( 1 +r′ + 1 +q′ − ϑ) = 0, +and we can apply Lemma 2.9 with the exponents (ϑq′)′ and (ϑr′)′. This gives +∥wϑ∥ +p +ϑ +Lpq′,pr′ (QR,Rsp) = ∥w∥p +Lpϑq′,pϑr′(QR,Rsp) ≤ R +sp−n +(ϑq′)′ (∥w∥p +L∞,p(QR,Rsp) + Rn−sp∥w∥p +Lp,∞(QR,Rsp)) +≤ C(n, s, p)βp−1R +sp−n +ϑr′ −spν∥w∥p +Lpq′,pr′ (QR,Rsp)δp−2. +(3.27) +We apply (3.27) with the exponents +β0 = 1, +βm+1 = ϑβm = ϑn+1. +Let +ϕm := ∥((u − v)+ +M + δ) +βm +p ∥ +p +βm +Lpq′,pr′(QR,Rsp) = ∥(u − v)+ +M + δ∥Lβnq′,βmr′ (QR,Rsp). +Then (3.27) reads +ϕm+1 ≤ +� +C R +sp−n +ϑr′ −spνδp−2� +1 +ϑm θ +(p−1)m +ϑm +ϕm. +By iterating the above inequality, we get +ϕm+1 ≤ +� +C R +sp−n +ϑr′ −spνδp−2��m +j=0 ϑ−j +ϑ(p−1) �m +j=0 jϑ−jϕ0. +(3.28) +Since ϑ > 1, we have the following convergent series +∞ +� +j=0 +ϑ−j = +ϑ +ϑ − 1 = 1 + +1 +1 − 1 +r − 1 +q +and +∞ +� +j=0 +jϑ−j = +ϑ +(ϑ − 1)2 . +By (3.7) in Lemma 3.2 we have +ϕ0 = ∥(u − v)+ +M + δ)∥Lq′,r′ (QR,Rsp) ≤ C(n, s, p)Rspp′ν+n∥f∥ +1 +p−1 +Lq,r(QR,Rsp) + δR +n +q′ + sp +r′ ≤ C(n, s, p)Rn+spν(2δ). +Inserting this into (3.28), and sending n to infinity we get +∥(u − v)+ +M + δ∥L∞(QR,Rsp) ≤ C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 R +ϑ +ϑ−1 ( sp−n +ϑr′ −spν)δ(p−2) +ϑ +ϑ−1 Rspν+nδ += C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 R +sp−n +(ϑ−1)r′ − spν +ϑ−1 +nδ1+(p−2) +ϑ +ϑ−1 += C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 R +n +ϑ−1 (ϑ−1− 1 +r′ )+ +sp +ϑ−1 ( 1 +r′ −(1− 1 +r − +n +spq )δp−1+ p−2 +ϑ−1 += C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 R +n +ϑ−1 ( −1 +q )+ +sp +ϑ−1 ( +n +spq )δp−1+ p−2 +ϑ−1 +≤ C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp) +� +. +Hence, we arrive at the desired estimate +∥(u − v)+∥L∞(QR,Rsp) ≤ C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp) +� +. + +H ¨OLDER CONTINUITY +19 +Case sp=n. Here we use the critical case of Sobolev-Morrey inequality, (2.7) with +max { p +r′ (1 − 1 +r − 1 +q )−1, q′} < l < ∞. +(3.29) +This applied for the second term in (3.17) implies +δ +2(β + 1)∥w∥p +Lp,∞(BR×J) + (C(n, s, p, l)R +np +l )−1 +3 · 2p−1 +βpp +(β + p − 1)p +� +δp−1∥w∥p +Ll,p(BR×J) − δβ+p−1|BR| +p +l |J| +� +≤ +δ +β + 1 sup +t∈J +� +BR +((u − v)+ +M + δ)β dx ++ C(n, s, p, l)−1 +3 · 2p−1 +βpp +(β + p − 1)p +� +J +∥((u − v)+ +M + δ) +β+p−1 +p +− δ +β+p−1 +p +∥p +Ll(BR) +≤ ∥f∥Lq,r(BR×J)∥w∥p +Lpq′,pr′(BR×J) + 2δ|BR| +∥w∥p +Lpq′,pr′ (BR×J) +|BR|1− 1 +q |J|1− 1 +r +. +Following the same step as in the previous two cases, we arrive at +∥w∥p +Lp,∞(QR,Rsp) + R +−np +l ∥w∥p +Ll,p(QR,Rsp) +≤ C(n, s, p, l)βp−1∥w∥p +Lpq′,pr′ (QR,Rsp) +�∥f∥Lq,r(QR,Rsp +δ ++ R−spν + δp−2R−spν� +≤ C(n, s, p, l)βp−1∥w∥p +Lpq′,pr′ (QR,Rsp)(δp−2R−spν). +Now we choose ϑ = 2 − 1 +r − 1 +q − +p +lr′ . Notice that due to the choice of l, (3.29), we have ϑ > 1. Then the +exponents (ϑr′)′ and (ϑq′)′ satisfy +1 − +1 +(ϑr′)′ − +1 +(ϑq′)′ = +p +lϑr′ . +Therefore, we can apply Lemma 2.10 with the exponents (ϑr′)′ and (ϑq′)′ to get +∥wϑ∥ +p +ϑ +Lpq′,pr′(QR,Rsp) = ∥w∥p +Lpϑq′,pϑr′(QR,Rsp) ≤ Rsp(1− +1 +(ϑr′)′ − +1 +(ϑq′)′ )(∥w∥p +Lp,∞(QR,Rsp) + R +−np +l ∥w∥p +L∞,p(QR,Rsp)) += R +np +lϑr′ (∥w∥p +Lp,∞(QR,Rsp) + R +−np +l ∥w∥p +L∞,p(QR,Rsp)) +≤ C(n, s, p, l)βp−1R +np +lϑr′ −spν∥w∥p +Lpq′,pr′ (QR,Rsp)δp−2. +(3.30) +We apply (3.30) with the exponents +β0 = 1, +βm+1 = ϑβm = ϑm+1. +Let +ϕn := ∥((u − v)+ +M + δ) +βm +p ∥ +p +βm +Lpq′,pr′(QR,Rsp) = ∥(u − v)+ +M + δ∥Lβmq′,βmr′ (QR,Rsp). +Then (3.30) reads +ϕm+1 ≤ +� +C R +np +lϑr′ −spνδp−2� +1 +ϑm θ +(p−1)m +ϑm +ϕm. +By iterating the above inequality, we get +ϕm+1 ≤ +� +C R +np +lϑr′ −spνδp−2��m +j=0 ϑ−j +ϑ(p−1) �m +j=0 jϑ−jϕ0. +(3.31) +Since ϑ > 1, we have the following convergent series +∞ +� +j=0 +ϑ−j = +ϑ +ϑ − 1 +and +∞ +� +j=0 +jϑ−j = +ϑ +(ϑ − 1)2 . + +20 +ALIREZA TAVAKOLI +By (3.7) in Lemma 3.2 we obtain +ϕ0 = ∥(u − v)+ +M∥Lq′,r′ (QR,Rsp) ≤ C(n, s, p, q)Rspp′ν+n∥f∥ +1 +p−1 +Lq,r(QR,Rsp) + δR +n +q′ + sp +r′ ≤ C(n, s, p, q)Rn+spνδ. +Inserting this into (3.30), and sending n to infinity we get +∥(u − v)+ +M + δ∥L∞(QR,Rsp) ≤ C(n, s, p, q, l)ϑ +(p−1)ϑ +(ϑ−1)2 R +ϑ +ϑ−1 ( np +lϑr′ −spν)δ(p−2) +ϑ +ϑ−1 Rn+spνδ += C(n, s, p, q, l)ϑ +(p−1)ϑ +(ϑ−1)2 R +np +(ϑ−1)lr′ − spν +ϑ−1 +nδ1+(p−2) +ϑ +ϑ−1 += C(n, s, p, q, l)ϑ +(p−1)ϑ +(ϑ−1)2 R +n +(ϑ−1) (ϑ−1+ p +lr′ )− spν +ϑ−1 δp−1+ (p−2) +ϑ−1 += C(n, s, p, q, l)ϑ +(p−1)ϑ +(ϑ−1)2 R +nν +(ϑ−1) − spν +ϑ−1 δp−1+ (p−2) +ϑ−1 +≤ C(n, s, p, q, l)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp) +� +. +Hence we arrive at the desired estimate +∥(u − v)+∥L∞(QR,Rsp) ≤ δ + C(n, s, p, q, l)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp) +� +. +□ +Notice that −u is a solution to the same type of problem, and we can apply the above proposition to −u. +Since −v is the (s, p)-caloric replacement of −u we get the same bound on ∥(−u + v)+∥L∞(QR,Rsp), as a result +we get a bound on the ∥u − v∥L∞(QR,Rsp). +Corollary 3.4. +Let u be a solution of ∂tu + (−∆p)su = f in Qσ,σsp(x0, T0) with f ∈ Lq,r +loc(Qσ,σsp(x0, T0)) and +let v be the (s, p)-caloric replacement of u in QR,Rsp. Assume further that r ≥ p′, +1 +r + n +spq < 1 +and q ≥ (p⋆ +s)′ +in the case sp < n, +and +1 +r + 1 +q < 1 +and q > 1 +in the case sp ≥ n. +If sp ̸= n then +∥u − v∥L∞(QR,Rsp(x0,T0)) ≤ C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp(x0,T0)) +� +, +where ν = 1 − 1 +r − +n +spq and +ϑ = 1 + spν +n +if +sp < n, +and +ϑ = 2 − 1 +r − 1 +q +if +sp > n. +If sp = n, then for any l such that +p +r′ (1 − 1 +r − 1 +q )−1 < l < ∞, we have +∥u − v∥L∞(QR,Rsp(x0,T0)) ≤ C(n, s, p, q, l)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp(x0,T0)) +� +, +where ϑ = 2 − 1 +r − 1 +q − +p +lr′ and ν = 1 − 1 +r − 1 +q . +Now we combine the local boundedness results for the equations with zero right hand side (see [S19] and +also [DZZ21]) with Proposition 3.3 to prove local boundedness for the equation with nonzero right hand side. +Proof of Theorem 1.1. For u, a local weak solution of +∂tu + (−∆p)su = f(x, t), +in Q2R,(2R)sp(x0, T0), +we consider v to be the (s, p)-caloric replacement in QR,Rsp(x0, T0), + + + + + + + +vt + (−∆p)sv = 0 +in QR,Rsp(x0, T0), +v = u +in (Rn \ BR(x0)) × [T0 − Rsp, T0], +v(x, T0 − Rsp) = u(x, T0 − Rsp) +in BR(x0). + +H ¨OLDER CONTINUITY +21 +By [DZZ21, Theorem 1.1] +∥v∥L∞(Q R +2 ,( R +2 )sp (x0,T0)) ≤ C +� +1 + +� +− +� +QR,Rsp(x0,T0) +|v|p dx dt +� 1 +p � ++ +sup +T0−Rsp 1 +in the case sp ≥ n. +Consider a bounded, local weak solution u ∈ Lp(I; W s,p +loc (BR1(z))) ∩ C(I; L2 +loc(BR1(z))) ∩ L∞(I; Lp−1 +sp (Rn)) ∩ +L∞(QR1,Rsp +1 (z, T1)) of the equation +∂tu + (−∆p)su = f +in QR1,Rsp +1 (z, T1). + +22 +ALIREZA TAVAKOLI +Then u is locally H¨older continuous in time and space. In particular, there exists a ζ > 0, such that for σ < 1, +(x1, t1), (x2, t2) ∈ QσR1,(σR1)sp(z, T1), there holds +|u(x1, t1) − u(x2, t2)| ≤ C(|x1 − x2|ζ + |t1 − t2| +ζ +sp ), +with C depending on +n, s, p, R1, σ, +sup +T1−Rsp +1 +0. +For any point (x0, T0) ∈ QσR1,(σR1)sp(z, T1) consider the (s, p)-caloric replacement of u in the cylinder +QR,Rsp(x0, T0) with R ≤ min {1, d}. The choice of d implies that QR,Rsp(x0, T0) ⊂ QR1,Rsp +1 (z, t). First, we +observe that: +− +� +Qρ,ρsp (x0,T0) +|u − ¯u(x0,T0),ρ|p dx dt ≤ C(p) − +� +Qρ,ρsp(x0,T0) +|u − v|p dx dt ++ C(p) − +� +Qρ,ρsp (x0,T0) +|¯u(x0,T0),ρ − ¯v(x0,T0),ρ|p dx dt + C(p) − +� +Qρ,ρsp (x0,T0) +|v − ¯v(x0,T0),ρ|p dx dt +≤ 2C(p) − +� +Qρ,ρsp(x0,T0) +|u − v|p dx dt + C(p) − +� +Qρ,ρsp(x0,T0) +|v − ¯v(x0,T0),ρ|p dx dt. +(3.34) +For ρ ≤ R +2 , v is H¨older continuous in Qρ,ρsp(x0, T0) by Theorem 5.1, and by the Mean Value Theorem there is +a point (˜x0, ˜t0) ∈ Qρ,ρsp such that ¯vx0,t0 = v(˜x0, ˜t0). With the notation +M := 1 + ∥v∥L∞(QR,(R)sp ) + +sup +T0−Rsp 0 by our assumptions on q and r. Inserting (3.36) and (3.35) +in (3.34) we arrive at +− +� +Qρ,ρsp(x0,T0) +|u − ¯u(x0,T0),ρ|p dx dt ≤ C(n, s, p) +�R +ρ +�n+spRξ ∥f∥p′ +Lq,r(QR,Rsp(x0,T0)) ++ C(n, s, p) +� ρ +R +�δp� +1 + ∥v∥p +L∞(QR,Rsp(x0,T0)) + +sup +T0−Rsp≤t≤T0 +Tailp−1,sp(u( q, t); x0, R)p�p−1 +≤ C(n, s, p) +�R +ρ +�n+spRξ ∥f∥p′ +Lq,r(QR,Rsp(x0,T0)) + C(n, s, p) +� ρ +R +�δp� +∥u∥p +L∞(QR,Rsp(x0,T0)) ++ ∥u − v∥p +L∞(QR,Rsp(x0,T0)) + +sup +T0−Rsp≤t≤T0 +Tailp−1,sp(u( q, t); x0, R)p�p−1 +. +Using Corollary 3.4 we get: +− +� +Qρ,ρsp (x0,T0) +|u − ¯u(x0,T0),r|p dx dt ≤ C(n, s, p)(R +ρ )n+spRξ∥f∥p′ +Lq,r(QR,Rsp(x0,T0)) ++ C(n, s, p)( ρ +R)δp� +∥u∥p +L∞(QR,Rsp(x0,T0)) + +sup +T0−Rsp≤t≤T0 +Tailp−1,sp(u( q, t); x0, R)p ++ C(n, s, p) +� +ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥f∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp(x0,T0)) +��p�p−1 +, +with ϑ and ν defined in Corollary 3.4, here the estimate is only written in the case sp ̸= n for simplicity. Since +QR,Rsp(x0, T0) ⊂ QR1,Rsp +1 (z, T1) the above expression is less than +≤ C(n, s, p)(R +ρ )n+spRξ∥f∥p′ +Lq,r(QR1,Rsp +1 (z,T1)) ++ C(n, s, p)( ρ +R)δp� +1 + ∥u∥p +L∞(QR1,Rsp +1 (z,T1)) + +sup +T0−Rsp 1 +in the case sp ≥ n. +Let u be a local weak solution to the equation +ut + (−∆p)su = f +in Q2R,(2R)sp, +with +∥u∥L∞(QR,Rsp) + +sup +−Rsp ε > 0. +(3.38) +Using (3.11) from Lemma 3.2, we have +lim +n→∞ +� T0 +T0−Rsp[un − ϕn]p +W s,p(Rn) dt ≤ C(n, s, p, q, r, R) lim +n→∞ ∥fn∥p′ +Lq,r(QR,Rs p) = 0. +(3.39) +By assumption, un is uniformly bounded in L∞(QR,Rsp). Now we show that ϕn is also uniformly bounded in +L∞(QR,Rsp). +∥ϕn∥L∞(QR,Rs p(x0,T0)) ≤ ∥un∥L∞(QR,(σR)s p)(x0,T0) + ∥un − ϕn∥L∞(QR,Rs p)(x0,T0) +≤ M + ∥un − ϕn∥L∞(QR,Rs p). +(3.40) +By Corollary 3.4 +∥un − ϕn∥L∞(QR,Rs p(x0,T0)) ≤ C(n, s, p)ϑ +(p−1)ϑ +(ϑ−1)2 � +1 + Rspν+ +(p−2)spν +(p−1)(ϑ−1) ∥fn∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(QR,Rsp) +� +. +(3.41) +Since ∥fn∥p′ +Lq,r(QR,Rs p) is uniformy bounded, (3.41) and (3.40) gives us a uniform bound on ∥ϕn∥L∞(QR,Rs p(x0,T0)). +Now we are in a position to use Theorem 3.5 for both of the sequences un and ϕn, which gives us a uniform +bound on the H¨older seminorms of un and ϕn in QσR,(σR)sp. Therefore, by Arzela-Ascoli’s Theorem un − ϕn +has a uniformly convergent subsequence in QσR,(σR)sp. By (3.39) the limit is 0, contradicting (3.38). +□ + +26 +ALIREZA TAVAKOLI +4. Improved H¨older regularity for nonhomogeneous equation +Proposition 4.1. +Let f ∈ Lq,r(Q1,2) with q, r satisfying r ≥ p′, +1 +r + n +spq < 1 +and q ≥ (p⋆ +s)′ +in the case sp < n, +and +1 +r + 1 +q < 1 +and q > 1 +in the case sp ≥ n. +Assume u is a weak solution of ut + (−∆p)su = f in Q1,2 that satisfies +∥u∥L∞(Q1,2) ≤ 1 , +sup +−2≤t≤0 +Tailp−1,sp(u; 0, 1) ≤ 1. +Then there exists ω such that if +∥f∥Lq,r(Q1,2) ≤ ω(n, s, p, q, r, α), +u is locally H¨older continuous in Q 1 +2 , +1 +2sp with exponents α in space and +α +sp−(p−2)α in time, as long as +α < min +� +Θ, +r(spq − n) − spq +q(r(p − 1) − (p − 2)) +� +. +(4.1) +Recall that Θ = min +� +sp +p−1, 1 +� +. +More precisely for (x1, t1), (x2, t2) ∈ Q 1 +2 , +1 +2sp we have +|u(x2, t2) − u(x1, t1)| ≤ C(n, s, p, q, r, α) +� +|x2 − x1|α + |t2 − t1| +α +sp−(p−2)α +� +. +Proof. Step 1: Decay at the origin. +For this part, we prove a decay at the origin for u under the assumptions +∥u∥L∞(Q1,1) ≤ 1 , +sup +−1≤t≤0 +Tailp−1,sp(u; 0, 1) ≤ 1, +and ∥f∥Lq,r(Q1,1) ≤ ω. +(4.2) +We introduce the parabolic cylinder +Gr := Br(0) × (−rβ, 0], +with β = sp − (p − 2)α. We show that for any exponent α satisfying (4.1), the following holds for r < 1 +∥u(x, t) − u(0, 0)∥L∞(Gr) ≤ Crα. +It is enough to prove the inequality for a sequence of r = λk, (k)∞ +0 , for some λ < 1. Without loss of generality, +we assume u(0, 0) = 0. Consider the rescaled functions +vk(x, t) := u(λkx, λkβt) +λαk +, +with λ small enough to be determined later. We will prove the following by induction, +∥vk(x, t)∥L∞(G1) ≤ 1 +and +sup +−1≤t≤0 +� +Rn\B1 +|vk(x, t)| +|x|n+s p dx ≤ 1. +(4.3) +For k = 0, (4.3) follows from our assumptions (4.2). +Observe that + + + +∂vk(x,t) +∂t += λβk−αkut(λkx, λβkt) +(−∆p)svk(x, t) = λk[sp−(p−1)α](−∆p)su(λkx, λβkt) +With β = sp − (p − 2)α, vk(x, t) solves +∂vk +∂t + (−∆p)svk = λk[sp−(p−1)α]f(λkx, λβkt) =: fk(x, t) +in Q 1 +λk , +1 +λβk . + +H ¨OLDER CONTINUITY +27 +Moreover, +∥fk∥r +Lq,r(G1) = +� 0 +−1 +�� +B1 +|fk(x, r)|q dx +� r +q dt += +� 0 +−1 +�� +Bλk +λkq[sp−(p−1)α]−kn|f(x, λβkt)|q dx +� r +q dt += +� 0 +−1 +λrk[sp−(p−1)α]− krn +q +�� +Bλk +|f(x, λβkt)|q dx +� r +q dt += λrk[sp−(p−1)α]− krn +q +−βk∥f∥Lq,r(Gλk). +Since λ < 1, and the exponent of λ is positive by (4.1), we get ∥fk∥Lq,r(G1) ≤ ω. +Assume that (4.3) holds for k. Now we prove that it holds for k + 1. Consider the (s, p)-caloric replacement +of vk(x, t) in Q1,1, say ϕk(x, t). Then +|vk(x, t)| ≤ |vk(x, t) − ϕk(x, t)| + |ϕk(x, t) − ϕk(0, t)| + |ϕk(0, t) − vk(0, t)|. +By Theorem 5.1, ϕk is locally H¨older continuous in Q1,1 and for (x, t) ∈ Q 1 +2 , +1 +2sp +|ϕk(x, t) − ϕk(0, 0)| ≤ C1|x|Θ−ε + C2|t|Γ− ε +β . +Here we take ε = Θ−α +2 . Since ∥fk∥Lq,r(Q1,1) ≤ ω, Lemma 3.6 implies +|vk(x, t)| ≤ 2δ(ω) + C1|x|Θ−ε + C2|t|Γ− ε +β , +in +Q 1 +4 , +1 +4sp . +(4.4) +In Theorem 5.1 the H¨older constants are bounded by +(C2) +1 +p−1 ≤ C1 ≤ C +� +1 + ∥ϕk∥L∞(Q1,1) + +sup +− +1 +2sp ≤t≤0 +Tailp−1,sp(ϕk; 0, 1) +� +≤ C +� +1 + ∥ϕk∥L∞(Q1,1) + +sup +−1≤t≤0 +Tailp−1,sp(vk; 0, 1) +� +≤ C +� +1 + ∥vk − ϕk∥L∞(Q1,1) + ∥vk∥L∞(Q1,1) + +sup +−1≤t≤0 +Tailp−1,sp(vk; 0, 1) +� +. +Therefore, using (4.3), for vk we have +(C2) +1 +p−1 ≤ C1 ≤ C(n, s, p, α)(3 + ∥vk − ϕk∥L∞(Q1,1)). +By Corollary 3.4 +C1 ≤ C(3 + C(n, s, p, q, r)(1 + ∥fk∥ +1+ +1 +ϑ−1 +p−2 +p−1 +Lq,r(Q1,1) )) ≤ C(3 + C(n, s, p, q, r)(1 + ω1+ +1 +ϑ−1 +p−2 +p−1 )). +This is a bound independent of k. We can take ω to be less than 1 and take C1 = C(n, s, p)(3 + 2C(n, s, p, q, r), +with the C(n, s, p, q, r) coming from Corollary 3.4, so that the constants C1, C2 are independent of ω as well. +Now we proceed and prove (4.3) for k + 1. First, we state our choice of λ +λ ≤ min +�1 +4, 1 +4 +sp +β , +1 +(2C1 + 2C2) +2 +Θ−α , +� +1 + ωn(4sp − 1) +sp ++ (1 + C1 + C2)p−1 +(p − 1)(Θ − α)/2 +� +2 +(p−1)(Θ−α) � +. +(4.5) +Since λ < 1 +4, and λβ < +1 +4s p , Qλ,λβ ⊂ Q 1 +4 , +1 +4s p . Therefore, from (4.4) we obtain +∥vk(x, t)∥L∞(Gλ) ≤ δ(ω) + C1λΘ−ε + C2λβ(Γ− ε +β ). +Notice that βΓ ≥ Θ, by the above choice of β. Thus, +∥vk(x, t)∥L∞(Gλ) ≤ δ(ω) + (C1 + C2)λΘ−ε. +(4.6) +Recall that ε = Θ−α +2 +and by the assumption (4.5) +(C1 + C2)λΘ−ε < 1 +2λα. +Now we choose ω so that +2δ(ω) ≤ 1 +2λΘ ≤ 1 +2λα. + +28 +ALIREZA TAVAKOLI +This is possible since δ(ω) converges to zero as ω → 0. Then, (4.6) implies +∥vk(x, t)∥L∞(Gλ) ≤ λα, +which translates to +∥vk+1(x, t)∥L∞(G1) = +���vk(λx, λβt) +λα +��� +L∞(G1) ≤ 1, +(4.7) +which is the first part of (4.3). For the second part, we want to show +sup +−1 0. Notice that in the case of sp ≥ n, we are assuming 1 − 1 +r − 1 +q > 0 which is +a stronger assumption. Now we verify the assumption on the tail. +sup +−1≤t≤0 +� +Rn\B1 +|˜u|p−1 +|x|n+sp dx = 2−sp +Lp−1 +sup +t0− L2−p +2sp ≤t≤t0 +� +Rn\B 1 +2 (x0) +|u(y)|p−1 +|y − x0|n+sp dy +≤ +1 +Lp−1 +sup +−2≤t≤0 +Tailp−1,sp(u( q, t); x0, 1 +2)p−1 +≤ +1 +Lp−1 (1 +2)sp( +1 +1 − |x − x0|)n+sp +sup +−2≤t≤0 +Tailp−1,sp(u( q, t); 0, 1)p−1 ++ +2n +Lp−1 +sup +−2≤t≤0 +∥u( q, t)∥p−1 +Lp−1(B1(0)) +≤ +2n +Lp−1 +� +1 + |B1|∥u∥L∞(Q1,2) +� +≤ 2n(1 + |B1|) +Lp−1 +≤ 1. +Now we can apply Step 1 to ˜u and we get the decay +∥˜u − ˜u(0, 0)∥L∞(Gr) ≤ Crα, +for 0 < r < 1 +or in other words +|˜u(x, t) − ˜u(0, 0)| ≤ C(|x|α + |t| +α +β ), +for (x, t) ∈ Q1,1. +In terms of u, this means +|u(x, t) − u(x0, t0)| ≤ CL(2α|x − x0|α + (2spLp−2) +α +β |t − t0| +α +β ), +for (x, t) ∈ Q 1 +2 , +1 +2spLp−2 (x0, t0). +(4.8) +Now take two points (x1, t1) , (x2, t2) ∈ Q 1 +2 , +1 +2sp and split the line joining them into 1 + [Lp−2] pieces, say +(yi, τi)1+[Lp−2] +i=0 +with (x1, t1) = (y0, τ0), (x2, t2) = (y1+[Lp−2], τ1+[Lp−2]) , |yi+1 − yi| = +|x2−x1| +1+[Lp−2] < 1 +2 and |τi+1 − +τi| = +|t2−t1| +1+[Lp−2] < +1 +2spLp−2 so that (yi+1, τi+1) ∈ Q 1 +2 , +1 +2spLp−2 (yi, τi). By (4.8) applied in each of Q 1 +2 , +1 +2spLp−2 (yi, τi) + +30 +ALIREZA TAVAKOLI +obtain +|u(x2, t2) − u(x1, t1)| ≤ +[Lp−2] +� +i=0 +|u(yi+1, τi+1) − u(yi, τi)| +≤ CL +[Lp−2] +� +i=0 +2α|yi+1 − yi|α + (2spLp−2) +α +β |τi+1 − τi| +α +β +≤ C(1 + L)p−1� +(2 |x2 − x1| +1 + [Lp−2])α + (2spLp−2 +|t2 − t1| +1 + ⌊Lp−2⌋) +α +β +� +≤ C(n, s, p, q, r, α)(|x2 − x1|α + |t2 − t1| +α +β ). +□ +Now we prove the H¨older regularity at any scale. +Proof of Theorem 1.2. We will consider the rescaled functions +˜uι(x, t) = 1 +µu(Rx + x0, µ2−pRspt + ι + T0) +with +µ =1 + ∥u∥L∞(QR,2Rsp(x0,T0)) + +sup +T0−2Rsp≤t≤T0 +Tailp−1,sp(u( q, t); x0, R) ++ +� +Rsp− n +q − sp +r ∥f∥Lq,r(QR,2Rsp(x0,T0)) +ω +� +1 +p−1+ p−2 +r +, +where ω = ω(n, s, p, q, r, α) is the same as in the proof of Proposition 4.1 and ι ∈ [−(R/2)sp(1 − µ2−p), 0]. The +interval [−(R/2)sp(1−µ2−p), 0] is chosen so that the cylinders Q R +2 , µ2−pRsp +2sp +(x0, T0+ι) cover all of Q R +2 ,( R +2 )sp(x0, T0) +by varying ι over. Note that for these choices of ι we have QR,2µ2−pRsp(x0, T0 + ι) ⊂ QR,2Rsp(x0, T0). Then ˜u +is a solution of +∂t˜uι + (−∆s +p)˜uι = Rsp f(Rx + x0, µ2−pRspt + ι + T0) +µp−1 +, +in +Q1,2. +We now verify that ˜uι satisfies the conditions of Proposition 4.1. The Lq,r norm of the right hand side is +���Rsp f(Rx, µ2−pRspt + ι) +µp−1 +��� +Lq,r(Q1,2) = +µ +p−2 +r +2 +1 +r µ(p−1) Rsp− n +q − sp +r ∥f∥Lq,r(QR,2µ2−pRsp(x0,T0+ι)) +≤ Rsp− n +q − sp +r ∥f∥Lq,r(QR,2Rsp(x0,T0)) +2 +1 +r µp−1− p−2 +r +≤ ω +2 +1 +r < ω. +The L∞ norm of ˜uι satisfies +∥˜uι∥L∞(Q1,2(0,0)) = 1 +µ∥u∥L∞(QR,2µ2−pRsp(x0,T0+ι)) ≤ 1 +µ∥u∥L∞(QR,2Rsp) ≤ 1. +Similarly +sup +−2≤t≤0 +Tailp−1,sp(˜u( q, t); 0, 1) ≤ 1 +µ +sup +T0+ι−2µ2−pRsp≤t≤T0+ι +Tailp−1,sp(u( q, t); x0, R) +≤ 1 +µ +sup +T0−2Rsp≤t≤T0 +Tailp−1,sp(u( q, t); x0, R) ≤ 1. +Hence, using Proposition 4.1 for ˜uι, we get +|˜uι(˜x2, ˜t2) − ˜uι(˜x1, ˜t1)| ≤ C(|˜x2 − ˜x1|α + |˜t2 − ˜t1| +α +sp−(p−2)α ) +for (˜x1, ˜t1), (˜x2, ˜t2) ∈ Q 1 +2 , +1 +2s p (0, 0), +with C = C(n, s, p, q, r, α). This translates to +|u(x2, τ2) − u(x1, τ1)| ≤ µC +��|x2 − x1| +R +�α ++ +� |τ2 − τ1| +Rs pµ2−p +� +α +sp−(p−2)α � +, +(4.9) + +H ¨OLDER CONTINUITY +31 +for (x1, τ1), (x2, τ2) ∈ Q R +2 , Rs pµ2−p +2s p +(x0, T0 + ι). Now we vary ι to obtain an estimate in the whole Q R +2 ,( R +2 )s p. +Specifically we split the interval [t1, t2] into 1 + ⌊µp−2⌋ pieces, say [τi+1, τi], with τi − τi+1 = +|t2−t1| +1+⌊µp−2⌋, τ0 = t2, +and τ⌊1+µp−2⌋ = t1. Using (4.9) we obtain +|u(x2, t2) − u(x1, t1)| ≤ |u(x2, t1) − u(x1, t1)| + |u(x2, t2) − u(x2, t1)| +≤ µC +�|x2 − x1| +R +�α ++ +⌊µp−2⌋ +� +i=0 +|u(x2, τi) − u(x2, τi+1)| +≤ µC +��|x2 − x1| +R +�α ++ +⌊µp−2⌋ +� +i=0 +�|τi − τi+1| +Rs pµ2−p +� +α +sp−(p−2)α � += µC +��|x2 − x1| +R +�α ++ +⌊µp−2⌋ +� +i=0 +� +|t2 − t1| +Rs pµ2−p(1 + ⌊µp−2⌋) +� +α +sp−(p−2)α � +≤ µC +��|x2 − x1| +R +�α ++ +⌊µp−2⌋ +� +i=0 +�|t2 − t1| +Rs p +� +α +sp−(p−2)α � +≤ µC +��|x2 − x1| +R +�α ++ 2µp−2�|t2 − t1| +Rs p +� +α +sp−(p−2)α � +, +which concludes the desired result. +□ +5. Appendix A +In this section, we spell out the necessary modifications to prove the following theorem 5.1 which is a modified +version of [BLS21, Theorem 1.2]. +Theorem 5.1. +Let Ω ⊂ Rn be a bounded and open set, I = (t0, t1], p ≥ 2 and 0 < s < 1. Suppose u is a local +weak solution of +ut + (−∆p)su = 0 +in Ω × I, +such that +u ∈ L∞ +loc(I; L∞ +loc(Ω)) ∩ L∞ +loc(I; Lp−1 +sp (Rn)). +(5.1) +Define the exponents +Θ(s, p) := + + + + + + + + + +s p +p − 1, +if s < p − 1 +p +, +1, +if s ≥ p − 1 +p +, +and +Γ(s, p) := + + + + + + + + + +1, +if s < p − 1 +p +, +1 +s p − (p − 2), +if s ≥ p − 1 +p +. +(5.2) +Then +u ∈ Cδ +x,loc(Ω × I) ∩ Cγ +t,loc(Ω × I), +for every 0 < δ < Θ(s, p) and 0 < γ < Γ(s, p). +More precisely, for every 0 < δ < Θ(s, p), 0 < γ < Γ(s, p), R > 0, x0 ∈ Ω and T0 such that +QR,Rs p(x0, T0) ⋐ Ω × (t0, t1], +there exists a constant C = C(n, s, p, δ, γ, σ) > 0 such that +|u(x1, τ1)−u(x2, τ2)| ≤ C +� +∥u∥L∞(QR,Rsp(x0,T0)) + +sup +t∈[T0−Rsp,T0] +Tailp−1,sp(u; x0, R) + 1 +� �|x1 − x2| +R +�δ ++ C +� +∥u∥L∞(QR,Rsp(x0,T0) + +sup +t∈[T0−Rsp,T0] +Tailp−1,sp(u; x0, R) + 1 +�p−1 +�|τ1 − τ2| +Rs p +�γ +. +(5.3) +for any (x1, τ1), (x2, τ2) ∈ QσR,(σR)s p(x0, T0). + +32 +ALIREZA TAVAKOLI +First we reproduce a modified version of [BLS21, Proposition 4.1], where instead of a global L∞ bound we +assume ∥u∥L∞(B1×[−1,0]) + supt∈[−1,0] Tailp−1,sp(u; 0, 1)) ≤ 1 . +Proposition 5.2. +Assume p ≥ 2 and 0 < s < 1. Let u be a local weak solution of ut + (−∆p)su = 0 in +B2 × (−2, 0]. We assume that +∥u∥L∞(B1×[−1,0]) + +sup +t∈[−1,0] +Tailp−1,sp(u( q, t); 0, 1) ≤ 1, +and that, for some q ≥ p and 0 < h0 < 1/10, we have +� T1 +T0 +sup +0<|h| 0 and C ր +∞ as h0 ց 0 or µ ց 0. +Proof. In the proof of [BLS21, Proposition 4.1], the L∞(Rn × [0, 1]) boundedness is only used in Step 3, in the +estimation of the nonlocal terms I2 and I3, which are defined by +I2(t) := +� +B R+r +2 +×(Rn\BR) +� +Jp(uh(x) − uh(y)) − Jp(u(x) − u(y)) +� +|h|1+ϑ β +× Jβ+1(uh(x) − u(x)) η(x)p dµ, +and +I3(t) := − +�� +(Rn\BR)×B R+r +2 +� +Jp(uh(x) − uh(y)) − Jp(u(x) − u(y)) +� +|h|1+ϑ β +× Jβ+1(uh(y) − u(y)) η(y)p dµ. +We also recall the definition of ˜I2 and ˜I3 +˜Ii := +� T1 +T0 +Ii(t) τ(t) dt, +i = 2, 3, +where τ is smooth function 0 ≤ τ ≤ 1 such that +τ ≡ 1 +on [T0 + µ, +∞), +τ ≡ 0 +on (−∞, T0], +τ ′ ≤ C +µ . +The general argument is the same but instead of using the L∞ norm of u(y) we can keep the inequality as it is +and write +��(Jp(uh(x) − uh(y)) − Jp(u(x) − u(y)))Jβ+1(δhu(x)) +�� ≤ C(1 + |uh(y)|p−1 + |u(y)|)|δhu(x)|β, +where x ∈ BR−2h0 and 4h0 < R < 1 − 5h0. Therefore, |x − y| ≥ (1 − R−2h0 +R +)|y| ≥ C(h0)|y| and we get +� +Rn\BR +1 + |u(y)|p−1 + |uh(y)|p−1 +|x − y|n+sp +dy ≤ C(n, s, p, h0) + (C(h0))n+sp +� +Rn\BR +|u(y)|p−1 +|y|n+sp +dy ++ (C(h0))n+sp +� +Rn\BR +|uh(y)|p−1 +|y|n+sp +dy. +Now +� +Rn\BR +|u(y)|p−1 +|y|n+sp +dy ≤ +� +Rn\B1 +|u(y)|p−1 +|y|n+sp +dy + R−n−sp +� +B1 +|u|p−1 dy +≤ 1 + nωnR−n−sp ≤ 1 + nωn(4h0)−n−sp, + +H ¨OLDER CONTINUITY +33 +and for uh +� +Rn\BR +|u(y + h)|p−1 +|y|n+sp +dy ≤ +� +Rn\BR(h) +|u(y)|p−1 +|y − h|n+sp dy ≤ (3 +2)n+sp +� +Rn\BR(h) +|u(y)|p−1 +|y|n+sp +dy +≤ (3 +2)n+sp�� +Rn\B1 +|u(y)|p−1 +|y|n+sp +dy + R−n−sp +� +B1 +|u(y)|p−1 dy +� +≤ +�3 +2 +�n+sp(1 + nωnR−n−sp) ≤ +�3 +2 +�n+sp(1 + nωn(4h0)−n−sp). +Here we have used BR(h) ⊂ B1, and |y−h| +|y| += | y +|y| − h +|y|| ≥ | y +|y|| − | h +|y|| ≥ 1 − | +h0 +R−h0 | ≥ 2 +3 . Using this we get +� +Rn\BR +1 + |u(y)|p−1 + |uh(y)|p−1 +|x − y|n+sp +dy ≤ C(n, s, p, h0), +and we can conclude +|˜I2| + |˜I3| ≤ C(n, s, p, h0) +� T1 +T0 +� +B R+r +2 +|δhu|β +|h|1+ϑ β τ dx dt ≤ C(h0, n, s, p, q, β) +� T1 +T0 +� +1 + +� +BR +��� +δhu +|h| +1+ϑ β +β +��� +βq +q−p+2 � +τ dt. +Which is the same as equation (4.6) in [BLS21]. +□ +We can estimate the W s,p semi-norm of a solution as follows. The proof follows the argument in [BLS21, +Lemma 7.1]. +Lemma 5.3. +Let p ≥ 2 and 0 < s < 1. Let u be a local weak solution of +∂tu + (−∆p)su = 0, +in B2R × (−2 Rs p, 0], +such that u ∈ L∞(B2R × [−Rs p, 0]). Then +� +R−n +� 0 +− 7 +8 Rs p[u]p +W s,p(BR(x0)) dt +� 1 +p +≤ C +� +∥u∥L∞(B2R×[−Rs p,0]) + +sup +t∈[−Rsp,0] +Tailp−1,sp(u; 0, 2R) + 1 +� +, +for some C = C(n, s, p) > 0. +Proof. Without loss of generality, we may suppose that x0 = 0. Let +k = ∥u∥L∞(B2R×[−Rs p,0]) + +sup +t∈[−Rsp,0] +Tailp−1,sp(u( q, t); 0, 2R) + 1 +and +�u = u + k. +Then �u is a local weak solution in B2 × (−2 Rs p, 0] and �u ≥ 1 in B2R × [−Rs p, 0]. We choose ϕ and ψ exactly +as [BLS21, Lemma 7.1], that is +η ∈ C∞ +0 (2R), +η ≡ 1 in BR, +|∇η| ≤ C +R +and +η ≡ 0 in Rn \ B 3 +2 R; +and +ψ ∈ C∞(R), +ψ(t) = 0 for t ≤ −Rs p, +ψ ≡ 1 in +� +−7 +8 Rs p, 0 +� +and +|ψ′| ≤ +C +Rs p . + +34 +ALIREZA TAVAKOLI +Then for ϕ(x, t) = η(x)ϕ(t) we get +� 0 +− 7 +8 Rs p[˜u( q, t)]p +W s,p(BR) dt ≤ +� 0 +−Rs p +� +�u( q, t) ϕ( q, t) +�p +W s,p(B2R) dt +≤ C +� 0 +−Rs p +�� +B2R×B2R +max +� +�u(x, t), �u(y, t) +�p +|ϕ(x, t) − ϕ(y, t)|p dµ dt ++ C +� +sup +x∈supp η +� +Rn\B2R +dy +|x − y|n+s p +��� 0 +−Rs p +� +B2R +�u(x, t)p ϕ(x, t)p dx dt +� ++ C +� +sup +t∈[−Rs p,0] +sup +x∈supp η +� +Rn\B2R +(u(y, t)+)p−1 +|x − y|n+s p +dy +� � 0 +−Rs p +� +B2R +�u(x, t) ϕ(x, t)p dx dt ++ 1 +2 +� 0 +−Rs p +� +B2R +�u(x, t)2 +�∂ϕp +∂t +�+ +dx dt + +� +B2R +�u(x, 0) dx +≤ C Rn (kp + k2 + k) ≤ C Rn kp. +The only difference in the proof is in estimating the term +sup +x∈supp η +� +Rn\B2R +(u(y, t)+)p−1 +|x − y|n+s p dy. +Noticing that for x ∈ supp η ⊂ B 3 +2 R we have |x−y| +|y| +≥ 1 − |x| +|y| ≥ 1 − 3/2R +2R += 1 +4, we get +� +Rn\B2R +(u(y, t)+)p−1 +|x − y|n+s p dy ≤ 4n+s pR−s p Tailp−1 +p−1,sp(u; 0, 2R) ≤ C R−s pkp−1. +□ +We can now prove the following modified version of [BLS21, Theorem 4.2]. +Theorem 5.4 (Spatial almost Cs regularity). +Let Ω ⊂ Rn be a bounded and open set, I = (t0, t1], p ≥ 2 +and 0 < s < 1. Suppose u is a local weak solution of +ut + (−∆p)su = 0 +in Ω × I, +such that u ∈ L∞ +loc(I; L∞(Ω)) ∩ L∞ +loc(I; Lp−1 +sp (Rn)). Then u ∈ Cδ +x,loc(Ω × I) for every 0 < δ < s. +More precisely, for every 0 < δ < s, R > 0 and every (x0, T0) such that +Q2R,2Rs p(x0, T0) ⋐ Ω × (t0, t1], +there exists a constant C = C(n, s, p, δ) > 0 such that +sup +t∈[T0− Rs p +2 +,T0] +[u(·, t)]Cδ(BR/2(x0)) ≤ C +Rδ +� +1 + ∥u∥L∞(B2R(x0)×[T0−Rs p,T0]) + +sup +t∈[T0−Rsp,T0] +Tailp−1,sp(u; x0, 2R) +� +(5.5) +Proof. The proof is essentially the same as the proof of [BLS21, Theorem 4.2]. We assume for simplicity that +x0 = 0 and T0 = 0, and set +MR = ∥u∥L∞(B2R×[−Rs p,0]) + +sup +t∈[−Rsp,0] +Tailp−1,sp(u; 0, R) + +� +R−n +� 0 +− 7 +8 Rs p[u]p +W s,p(BR) dt +� 1 +p ++ 1. +Notice that by Lemma 5.3 we have +MR ≤ C +� +∥u∥L∞(B2R×[−Rs p,0]) + +sup +t∈[−Rsp,0] +Tailp−1,sp(u; 0, 2R) +� ++ 1. + +H ¨OLDER CONTINUITY +35 +Let α ∈ [−Rs p(1 − M2−p +R +), 0] and set +uR,α(x, t) := +1 +MR +u +� +R x, +1 +Mp−2 +R +Rs p t + α +� +, +for x ∈ B2, t ∈ (−2, 0]. +Then uR,α(x, t) is a local weak solution of +ut + (−∆p)su = 0, +in B2 × (−2, 0], +that satisfies +∥uR,α∥L∞(B2×[−1,0]) + +sup +t∈[−1,0] +Tailp−1,sp(u; 0, 1) ≤ 1, +� 0 +− 7 +8 +[uR,α]p +W s,p(B1) dt ≤ 1. +This function satisfies the assumption of Proposition 5.2, and we can do the same argument as in [BLS21] to +obtain +sup +t∈[−1/2,0] +[uR,α(·, t)]Cδ(B1/2) ≤ C(n, s, p, δ), +for a C independent of α and by scaling back we get +sup +α− 1 +2 M2−p +R +Rs p≤t≤0 +[u(·, t)]Cδ(BR/2) ≤ C +Rδ MR. +By varying α ∈ [−Rs p(1 − M2−p +R +), 0] we get the desired result. +□ +We now address the improved regularity and start with the following modified version of [BLS21, Proposition +5.1]. +Proposition 5.5. +Assume p ≥ 2 and 0 < s < 1. Let u be a local weak solution of ut + (−∆p)su = 0 in +B2 × (−2, 0], such that +∥u∥L∞(B2×[−1,0]) + +sup +t∈[−1,0] +Tailp−1,sp(u; 0, 2) ≤ 1, +Assume further that for some 0 < h0 < 1/10 and ϑ < 1, β ≥ 2 such that (1 + ϑ β)/β < 1, we have +� T1 +T0 +sup +0<|h|≤h0 +����� +δ2 +hu +|h| +1+ϑ β +β +����� +β +Lβ(BR+4 h0 ) +dt < +∞, +for a radius 4 h0 < R ≤ 1 − 5 h0 and two time instants −1 < T0 < T1 ≤ 0. Then +� T1 +T0+µ +sup +0<|h| 0, x0 ∈ Ω and T0 such that +B2R(x0) × [T0 − 2 Rs p, T0] ⋐ Ω × (t0, t1], +there exists a constant C = C(n, s, p, δ) > 0 such that +sup +t∈[T0− Rs p +2 +,T0] +[u(·, t)]Cδ(BR/2(x0)) ≤ C +Rδ +� +∥u∥L∞(B2R×[T0−Rs p,T0]) + +sup +t∈[T0−Rsp,T0] +Tailp−1,sp(u; x0, 2R) + 1 +� +. +(5.7) +Now we modify the argument regarding the regularity in time (see [BLS21, Proposition 6.2]). +Proposition 5.7. +Suppose that u is a local weak solution of +∂tu + (−∆p)su = 0, +in B2 × (−2, 0], +such that +∥u∥L∞(B2×[−1,0]) + +sup +t∈[−1,0] +Tailp−1,sp(u; 0, 2) ≤ 1, +and +sup +t∈[−1/2,0] +[u(·, t)]Cδ(B1/2) ≤ Kδ, +for any s < δ < Θ(s, p), +(5.8) +where Θ(s, p) is the exponent defined in (5.2). Then there is a constant C = C(n, s, p, Kδ, δ) > 0 such that +|u(x, t) − u(x, τ)| ≤ C |t − τ|γ, +for every (x, t), (x, τ) ∈ Q 1 +4 , 1 +4 , +where +γ = +1 +s p +δ − (p − 2) +. +In particular, u ∈ Cγ +t (Q 1 +4 , 1 +4 ) for any γ < Γ(s, p), where Γ(s, p) is the exponent defined in (5.2). +Proof. The only part that needs to be modified is the estimation of the nonlocal term J2 +J2 := +� T1 +T0 +�� +(Rn\Br(x0))×Br/2(x0) +Jp(u(x, τ) − u(y, τ)) η(x) dµ(x, y) dτ, +here T0, T1 ∈ (t0 − θ, t0) with T0 < T1. We recall that 0 < θ < 1 +8, x0 ∈ B 1 +4 , and r < 1 +8. Thus x ∈ B r +2 (x0) implies +x ∈ B 5 +16 . +For y ∈ B 1 +2 (0), assumption (5.8) implies +|u(x, τ) − u(y, τ)| ≤ Kδ|x − y|δ. +For y ∈ B2(0) \ B 1 +2 (0) the L∞ bound on u implies +|u(x, τ) − u(y, τ)| ≤ 2 ≤ C(δ)|x − y|δ. + +H ¨OLDER CONTINUITY +37 +Also notice that for x ∈ Br/2(x0) and y ∈ Rn \ Br(x0),we have |x − y| ≥ 1 +2|y − x0|. Using these we obtain +J2 ≤ 2 (T1 − T0) ∥η∥L∞(Br/2(x0)) +sup +t∈[− 1 +2 ,0] +�� +(Rn\Br(x0))×Br/2(x0) +|u(x, t) − u(y, t)|p−1 +|x − y|n+s p +dy dx +≤ 2 (T1 − T0) ∥η∥L∞(Br/2(x0)) +� +sup +t∈[− 1 +2 ,0] +�� +(Rn\B2)×Br/2(x0) +|u(x, t) − u(y, t)|p−1 +|x − y|n+s p +dy dx ++ C(δ, Kδ) +�� +(B2\Br(x0))×Br/2(x0)) +|x − y|δ(p−1)−n−sp dy dx +� +≤ Cθ +� +sup +t∈[− 1 +2 ,0] +�� +(Rn\B2)×Br/2(x0) +1 + |u(y, t)|p−1 +|x0 − y|n+sp +dy dx ++ +�� +(B2(0)\Br(x0)×Br/2(x0)) +|x0 − y|δ(p−1)−n−sp dy dx +� +≤ Cθ +� +Br/2(x0) +� +sup +t∈[− 1 +2 ,0] +� +Rn\B2 +1 + |u(y, t)|p−1 +|y|n+sp +dy + +� +B2\Br(x0) +|x0 − y|δ(p−1)−n−sp dy +� +dx +≤ C θ rn� +2−sp + 1 + rδ(p−1)−sp� +≤ C θ rn−sp+δ(p−1). +(since δ(p − 1) − sp is not positive) +□ +Finally, we are ready to prove a modified version of [BLS21, Theorem 1.1], which is Theorem 5.1. +Proof of Theorem 5.1. Consider a cylinder Q2ρ,2ρsp(˜x, τ) ⋐ Ω×I, first, we prove the following type of bound on +the H¨older seminorm in Qρ/4,ρs p/4(˜x, τ), and later with the aid of a covering argument we conclude the claim +of the theorem. +Claim: For any (x1, τ1), (x2, τ2) ∈ Qρ/4,ρs p/4(˜x, τ) we have +|u(x1, τ1)−u(x2, τ2)| ≤ C +� +∥u∥L∞(B2ρ×[T0−ρs p,T0]) + +sup +t∈[T0−ρsp,T0] +Tailp−1,sp(u; x0, 2ρ) + 1 +� �|x1 − x2| +ρ +�δ ++ C +� +∥u∥L∞(B2ρ×[T0−Rs p,T0]) + +sup +t∈[T0−ρsp,T0] +Tailp−1,sp(u; x0, 2ρ) + 1 +�p−1 +�|τ1 − τ2| +ρs p +�γ +. +(5.9) +The regularity in space variable has been proven in Theorem 5.6. To prove the part on time regularity we set +Mρ(˜x, τ) := 1 + ∥u∥L∞(Q2ρ,ρsp(˜x,τ)) + +sup +τ−ρs≤t≤τ +Tailp−1,sp(u; ˜x, 2ρ) +and consider the rescaled functions +˜uρ,ι(x, t) := +1 +Mρ(˜x, τ)u(ρx + ˜x, Mρ(˜x, τ)2−pρspt + τ + ι), +for ι ∈ (− ρsp +4 (1 − M2−p +ρ +), 0). Then ˜uρ,ι(x, t) is a solution of +∂t˜uρ,ι + (−∆p)˜uρ,ι = 0, +in +Q2,2. +Moreover, ˜uρ,ι(x, t) satisfies the conditions of Proposition 5.7. Indeed by construction +∥˜uρ,ι∥L∞(B2×[−1,0]) + +sup +t∈[−1,0] +Tailp−1,sp(˜uρ,ι; 0, 2) ≤ 1 +and the estimate (5.8) follows from (5.7) in Theorem 5.6. From Proposition 5.7 we obtain +sup +x∈B 1 +4 +[˜uρ,ι(x, q)]Cγ[− 1 +4 ,0] ≤ C, +with C = C(n, s, p, γ) for every 0 < γ < Γ(s, p). By scaling back this translates to +|u(x, t1) − u(x, t2)| ≤ Mρ(˜x, τ)C +� |t1 − t2| +ρspM2−p +ρ +�γ +for +(x, t1), (x, t2) ∈ Q ρ +4 , ρsp +4 M2−p +ρ +(˜x, τ + ι). +(5.10) + +38 +ALIREZA TAVAKOLI +By varying ι with an argument similar to the proof of Theorem 1.2 we arrive at the claim (5.9). We have to +point out that the H¨older constant does change, unlike what is suggested in the proof of [BLS21, Theorem 1.1]. +Here is a detailed computation +We split the time interval [t1, t2] into 1+⌊Mρ(˜x, τ)p−2⌋ pieces, say [τi+1, τi], with τi −τi+1 = +|t2−t1| +1+⌊Mρ(˜x,τ)p−2⌋, +τ0 = t2, and τ⌊1+µp−2⌋ = t1. Then using (5.10) and the triangle inequality we get +|u(x, t2) − u(x, t1)| ≤ |u(x, t2) − u(x, t1)| +≤ +⌊Mp−2 +ρ +⌋ +� +i=0 +|u(x2, τi) − u(x2, τi+1)| +≤ CMρ +⌊Mp−2 +ρ +⌋ +� +i=0 +� |τi − τi+1| +Rs pM2−p +ρ +�γ += CMρ(˜x, τ) +⌊Mp−2 +ρ +⌋ +� +i=0 +� +|t2 − t1| +Rs pM2−p +ρ +(1 + ⌊Mp−2 +ρ +⌋) +�γ +≤ CMρ +⌊Mp−2 +ρ +⌋ +� +i=0 +�|t2 − t1| +Rs p +�γ +≤ CMρ +� +Mp−2 +ρ +�|t2 − t1| +Rs p +�γ� +≤ CMp−1 +ρ +�|t2 − t1| +Rs p +�γ +. +Now use (5.9) in cylinders of the form +Q r +4 , rsp +4 (y, t), +for (y, t) ∈ QσR,(σR)sp, +where the radius r = +R +C(n,s,p,σ) is so small, such that +Q2r,2rs p(y, t) ⊂ QR,Rsp. +Consider a sequence of points (˜xi, ˜τi) on the segment joining (x1, τ1) and (x2, τ2) such that +(˜xi, ˜τi) ∈ Q r +4 , rsp +4 (xi−1, τi−1). +Using (5.9) together with the triangle inequality, we obtain +|u(x1, τ1)−u(x2, τ2)| ≤ C +� +∥u∥L∞(QR,Rsp(x0,T0)) + +sup +t∈[T0−Rsp,T0] +Tailp−1,sp(u; x0, R) + 1 +� �|x1 − x2| +R +�δ ++ C +� +∥u∥L∞(QR,Rsp(x0,T0) + +sup +t∈[T0−Rsp,T0] +Tailp−1,sp(u; x0, R) + 1 +�p−1 +�|τ1 − τ2| +Rs p +�γ +, +with C = C(n, s, p, δ, γ, σ), which is the desired result. +□ +6. Appendix B +Here we will justify the insertion of u − v and |u − v|p−2(u − v) as test functions. +Proposition 6.1. +Let B = BR(x0) be a ball of radius r, B2 = BσR(x0) with σ > 1, and I = (τ0, τ1] be an +interval. Let f ∈ L(p⋆ +s)′,p′(B × I) and assume that u ∈ Lp(I, W s,p(B2)) ∩ Lp−1(I; Lp−1 +sp (Rn)) ∩ C(I; L2(B)) be a +local weak solution of +ut + (−∆p)su = f, +in B2 × I +with +Tailp−1,sp(u( q, t); x0, R) ∈ Lp(I). + +H ¨OLDER CONTINUITY +39 +(in particular, this will be the case under the stronger assumption supt∈I Tailp,sp(u( q, t); x0, R) < ∞ that we use +in this article.) For any time interval [T0, T1] ⋐ I and let v ∈ Lp([T0, T1], W s,p(B2))∩Lp−1([T0, T1]; Lp−1 +sp (Rn))∩ +C([T0, T1]; L2(B)) be a weak solution to + + + + + + + +vt + (−∆p)sv = 0 +in B × [T0, T1] +v = u +in (Rn \ B) × [T0, T1] +v(x, T0) = u(x, T0) +in B +In addition, assume that F is a globally Lipschitz function with F(0) = 0, which is either bounded or F(a) = a. +Then we have: +� T1 +T0 +�� +Rn×Rn +� +Jp(u(x, t) − u(y, t)) − Jp(v(x, t) − v(y, t)) +� +× +� +F(u(x, t) − v(x, t)) − F(u(y, t) − v(y, t)) +� +dµ dt ++ +� +B +F(u(x, t) − v(x, t)) dx +��T1 +T0 += +� T1 +T0 +� +B +F(u − v)f dx dt, +where F(a) := +� a +0 f(t) dt is the primitive function of F. +Proof. The proof is essentially the same as [BLS21, Lemma 3], except that here we don’t use a cut off function +and don’t have the global boundedness of u in the ball. For simplicity we assume x0 = 0, R = 1 and σ = 2. +For a function ϕ ∈ C((T0, T1); L2(B)) ∩ Lp((T0, T1); Xs,p +0 (B, B2)), we use the following regularization of +functions +ϕε(x, t) := 1 +ε +� t+ ε +2 +t− ε +2 +ζ(t − ℓ +ε +)ϕ(x, ℓ) dℓ = +� +1 +2 +− 1 +2 +ζ(−σ)ϕ(x, t + εσ) dσ, +where ζ(σ) is a smooth function with compact support in (− 1 +2, 1 +2) satisfying +|ζ| ≤ 1, +and +|ζ′| ≤ 8. +This regulization process gives us a test function ϕε ∈ C1((T0+ε, T1−ε); L2(B))∩Lp((T0+ε, T1−ε); Xs,p +0 (B, B2)). +Let t0 = T0 + ε0 and t1 = T1 − ε0 and we test the equation with ϕε as above, for ε < ε0 +2 . First, we will show +the claim for the smaller interval [t0, t1] ⊂ [T0, T1], and then through a limiting argument, prove the result for +the whole interval. As in equation (3.5) in [BLS21], we get +� t1 +t0 +�� +Rn×Rn +� +Jp(u(x, t) − u(y, t))(ϕε(x, t) − ϕε(y, t)) dµ dt ++ +� +B +� t1− ε +2 +t0+ ε +2 +∂tuε(x, t)ϕ(x, t) dt dx + Σu(ε) += +� +B +� +u(x, t0)ϕ(x, t0) − uε(x, t0 + ε +2)ϕ(x, t0 + ε +2) +� +dx +− +� +B +� +u(x, t1)ϕ(x, t1) − uε(x, t1 − ε +2)ϕ(x, t1 − ε +2) +� +dx + +� t1 +t0 +� +B +ϕεf dx dt, +and we obtain a similar identity for v without +� t1 +t0 +� +B ϕεf dx dt in the right hand side. Here Σu is defined by +Σu(ε) = − +� +B +� t0+ ε +2 +t0− ε +2 +� +1 +ε +� ℓ+ ε +2 +t0 +u(x, t) ζ +�ℓ − t +ε +� +dt +� +∂ℓϕ(x, ℓ) dℓ dx +− +� +B +� t1+ ε +2 +t1− ε +2 +� +1 +ε +� t1 +ℓ− ε +2 +u(x, t) ζ +�ℓ − t +ε +� +dt +� +∂ℓϕ(x, ℓ) dℓ dx. + +40 +ALIREZA TAVAKOLI +Observe that by using an integration by parts, the term Σu(ε) can be rewritten as +Σu(ε) = − +� +B +� +1 +ε +� T0+ε +T0 +u(x, t) ζ +�T0 − t +ε ++ 1 +2 +� +dt +� +ϕ +� +x, T0 + ε +2 +� +dx ++ +� +B +� T0+ ε +2 +T0− ε +2 +� +1 +ε2 +� ℓ+ ε +2 +T0 +u(x, t) ζ′ +�ℓ − t +ε +� +dt +� +ϕ(x, ℓ) dℓ dx ++ +� +B +� +1 +ε +� T1 +T1−ε +u(x, t) ζ +�T1 − t +ε +− 1 +2 +� +dt +� +ϕ +� +x, T1 − ε +2 +� +dx +− +� +B +� T1+ ε +2 +T1− ε +2 +� +1 +ε2 +� T1 +ℓ− ε +2 +u(x, t) ζ′ +�ℓ − t +ε +� +dt +� +ϕ(x, ℓ) dℓ dx, +(6.1) +where we also used that ζ has compact support in (−1/2, 1/2). By subtracting the identities for u and v, we +obtain +� t1 +t0 +�� +Rn×Rn +� +Jp(u(x, t) − u(y, t)) − Jp(v(x, t) − v(y, t)) +� +(ϕε(x, t) − ϕε(y, t)) dµ dt ++ +� +B +� t1− ε +2 +t0+ ε +2 +∂t(u − v)ε(x, t)ϕ(x, t) dt dx + Σu(ε) − Σv(ε) += +� +B +� +(u − v)(x, t0)ϕ(x, t0) − (u − v)ε(x, t0 + ε +2)ϕ(x, t0 + ε +2) +� +dx +− +� +B +� +(u − v)(x, t1)ϕ(x, t1) − (u − v)ε(x, t1 − ε)ϕ(x, t1 − ε +2) +� +dx + +� t1 +t0 +� +B +ϕε(x, t)f(x, t) dx dt. +Now we take ϕ to be F(uε − vε). Observe that +∂t(u − v)εF(uε − vε) = ∂tF(uε − vε). +After an integration by parts we get +� t1 +t0 +�� +Rn×Rn +� +Jp(u(x, t) − u(y, t)) − Jp(v(x, t) − v(y, t)) +� +× ([F(uε − vε)(x, t)]ε − [F(uε − vε)(y, t)]ε) dµ dt ++ +� +B +F(uε − vε) dx +�t1− ε +2 +t0+ ε +2 ++ Σu(ε) − Σv(ε) += +� +B +� +(u − v)(x, t0)F(uε − vε)(x, t0) − (u − v)ε(x, t0 + ε +2)F(uε − vε)(x, t0 + ε +2) +� +dx +− +� +B +� +(u − v)(x, t1)F(uε − vε)(x, t1) − (u − v)ε(x, t1 − ε +2)F(uε − vε)(x, t1 − ε +2) +� +dx ++ +� t1 +t0 +� +B +(F(uε − vε))ε(x, t)f(x, t) dx dt := I1 − I2 + I3. +(6.2) +We now wish to pass to the limit in I1, I2, and I3. Let w = u − v, we now treat I1. The fact that F is globally +Lipschitz together with F(0) = 0 implies |F(t)| ≤ C|t|. Therefore, +|w(x, t0)F(wε)(x, t0) − wε(x, t0 + ε +2)F(wε(x, t0 + ε +2))| +≤ |(w(x, t0) − wε(x, t0 + ε +2))F(wε(x, t0))| + |wε(x, t0 + ε +2)(F(wε(x, t0)) − F(wε(x, t0 + ε +2)))| +≤ C +� +|(w(x, t0) − wε(x, t0 + ε +2))wε(x, t0)| + |wε(x, t0 + ε +2)(wε(x, t0) − wε(x, t0 + ε +2))| +� +, + +H ¨OLDER CONTINUITY +41 +where C is the Lipschitz constant of F. After integrating and using H¨older’s inequality, we obtain +I1 = +� +B +� +w(x, t0)F(wε)(x, t0) − wε(x, t0 + ε +2)F(wε)(x, t0 + ε) +� +dx +≤ C +� +∥w( q, t0) − wε( q, t0 + ε +2)∥L2(B)∥wε( q, t0)∥L2(B) ++ ∥wε( q, t0 + ε +2)∥L2(B)∥wε( q, t0) − wε( q, t0 + ε +2)∥L2(B) +� +. +Since wε ∈ C((T0 + ε0, T1 − ε0); L2(B)), uniformly, we have +Observe that +� +B +|w(x, t0) − wε(x, t0 + ε +2)|2 dx = +� +B +��� +� +1 +2 +− 1 +2 +ζ(−σ)[w(x, t0) − w(x, t0 + ε +2 + εσ)] dσ +��� +2 +dx +≤ +� +B +� +1 +2 +− 1 +2 +��ζ(−σ)[w(x, t0) − w(x, t0 + ε +2 + εσ)] +��2 dσ dx = +� +1 +2 +− 1 +2 +� +B +��ζ(−σ)[w(x, t0) − w(x, t0 + ε +2 + εσ)] +��2 dx dσ +≤ +� +1 +2 +− 1 +2 +� +B +��w(x, t0) − w(x, t0 + ε +2 + εσ) +��2 dx dσ ≤ sup +0≤t≤ε +� +B +��w(x, t0) − w(x, t0 + t) +��2 dx +(6.3) +which tends to zero since w is in C([T0, T1], L2(B)). In a similar way oe can argue that +lim +ε→0 ∥wε( q, t0) − wε( q, t0 + ε +2)∥L2(B) = 0. +(6.4) +Using the traingle inequality we get +∥wε( q, t0) − wε( q, t0 + ε +2)∥L2(B) ≤ ∥wε( q, t0) − w( q, t0)∥L2(B) + ∥w( q, t0) − wε( q, t0 + ε +2)∥L2(B). +using a computation similar to (6.3) we obtain +∥wε( q, t0) − w( q, t0)∥L2(B) ≤ +sup +− ε +2 ≤t≤ ε +2 +∥w( q, t0 + t) − w( q, t0)∥L2(B) +and +∥w( q, t0) − wε( q, t0 + ε +2)∥L2(B) ≤ sup +0≤t≤ε +∥w( q, t0 + t) − w( q, t0)∥L2(B). +These two expressons converge to zero, since w ∈ C([T0, T1], L2(B)) and (t0 − ε, t1 + ε) ⋐ (T0, T1). This shows +that I1 converges to zero. By similar reasoning, I2 also tends to zero. For the term I3, we have +��� +� t1 +t0 +� +B +� +(F(wε))εf − F(w)f +� +dx dt +��� = +��� +� t1 +t0 +� +B +� +F(wε))ε − F(wε) +� +f + +� +F(wε) − F(w) +� +f dx dt +��� +≤ +� t1 +t0 +� +B +|(F(wε))ε − F(wε)||f(x, t)| + C|wε − w||f(x, t)| dx dt. +The sequence wε is bounded in Lp⋆ +s,p(B × (t0, t1)), therefore, it has a weakly convergent subsequence. Using the +pointwise convergence of wε to w, we get the weak convergence of wε − w to zero. By the assumptions on q, r +together with H¨older’s inequality (2.9), f(x, t) belongs to the dual space L(p⋆ +s)′,p′(B × (t0, t1)). Therefore, +lim +ε→0 +� t1 +t0 +� +B +|wε − w||f(x, t)| dx dt = 0. + +42 +ALIREZA TAVAKOLI +On the other hand, +� t1 +t0 +� +B +|(F(wε))ε − F(wε)||f(x, t)| dx dt += +� t1 +t0 +� +B +���� +� +1 +2 +− 1 +2 +ζ(−σ)(F(wε(x, t + εσ))) − F(wε(x, t)) dσ +����|f(x, t)| dx dt +≤ +� t1 +t0 +� +B +� +1 +2 +− 1 +2 +ζ(−σ)|(F(wε(x, t + εσ)) − F(wε(x, t))||f(x, t)| dσ dx dt +≤ C +� t1 +t0 +� +B +� +1 +2 +− 1 +2 +ζ(−σ)|(wε(x, t + εσ) − wε(x, t)||f(x, t)| dσ dx dt +≤ C +� +1 +2 +− 1 +2 +� t1 +t0 +� +B +|(wε(x, t + εσ) − wε(x, t)||f(x, t)| dx dt dσ +≤ C +� +1 +2 +− 1 +2 +�� ∥wε(x, t + εσ) − wε(x, t)∥Lp⋆s (B) +�� +Lp(t0,t1)∥f∥L(p⋆s)′,p′(B×(t0,t1)) dσ. +Recalling that the shift operator, +T (a)(g) := ∥g(t + a)∥Lp((t0,t1)) +for a function g ∈ Lp(t0 − ε0, t1 + ε0) is continuous for −ε0 ≤ a ≤ ε0. Hence, we get +lim +ε→0 ∥wε(x, t)∥Lp⋆s,p(B×(t0,t1)) = lim +ε→0 ∥wε(x, t + εσ)∥Lp⋆s,p(B×(t0,t1)) = ∥w∥Lp⋆s,p(B×(t0,t1)). +Upon passing to a subsequence wε(x, t + εσ) and wε(x, t) converge weakly in Lp⋆ +s,p(B × (t0, t1)), since they +converge to w(x, t) pointwise, we get the weak convergence +wε(x, t) ⇀ w(x, t) +and +wε(x, t + εσ) ⇀ w(x, t) +in +Lp⋆ +s,p(B × (t0, t1)). +Combined with the convergence of the norms, this implies the strong convergence in the norm, in particular, +we have +�� ∥wε(x, t + εσ) − wε(x, t)∥Lp⋆s (B) +�� +Lp(t0,t1) → 0. +Now we turn our attention to the terms on the left hand side of (6.2). The terms Σu(ε) and Σv(ε) converge +to zero. To show this we start with the following computation, borrowed from [BLS21, Lemma 3.3.]. Using a +suitable change of variables in (6.1) and recalling ϕ = F(wε), we can also write +Σu(ε) = − +� +B +�� +1 +2 +− 1 +2 +u(x, t0 − ερ + ε +2) ζ(ρ) dρ +� +F(wε) +� +x, t0 + ε +2 +� +dx ++ +� +B +� +1 +2 +− 1 +2 +�� ρ +− 1 +2 +u(x, ε ρ + t0 − ε σ) ζ′ (σ) dσ +� +F(wε)(x, ε ρ + t0) dρ dx ++ +� +B +�� +1 +2 +− 1 +2 +u(x, t1 − ερ − ε +2) ζ(ρ) dρ +� +F(wε) +� +x, t1 − ε +2 +� +dx +− +� +B +� +1 +2 +− 1 +2 +�� ρ +1 +2 +u(x, ε ρ + T1 − ε σ) ζ′ (σ) dσ +� +F(wε)(x, ε ρ + T1) dρ dx +:= Σ1 +u(ε) + Σ2 +u(ε) + Σ3 +u(ε) + Σ4 +u(ε). +(6.5) +In a similar way to the argument for convergence of I1, we can see that +lim +ε→0 Σ1 +u(ε) = − +� +B +u(x, t0)F(w)(x, t0) dx. + +H ¨OLDER CONTINUITY +43 +We spell out the details of the arguments for convergence of Σ2 +u(ε). +���Σ2 +u(ε) − +� +B +u(x, t0)F(w)(x, t0) dx +��� += +���� +� +B +� +1 +2 +− 1 +2 +�� ρ +− 1 +2 +(u(x, ε ρ + t0 − ε σ) − u(x, t0)) ζ′ (σ) dσ +� +F(wε)(x, ε ρ + t0) dρ dx ++ +� +B +� +1 +2 +− 1 +2 +�� ρ +− 1 +2 +u(x, t0)ζ′(σ) dσ +� � +F(wε)(x, ερ + t0) − F(w)(x, t0) +� +dρ dx +���� +≤ +� +1 +2 +− 1 +2 +� ρ +− 1 +2 +�� +B +���u(x, ερ + t0 − εσ) − u(x, t0))F(wε)(x, ερ + t0)ζ′(σ) +��� dx +� +dσ dρ ++ +� +1 +2 +− 1 +2 +� +B +|ζ(ρ)| +���u(x, t0) +� +F(wε)(x, ερ + t0) − F(w)(x, t0) +���� dx dρ +≤ 8 +� +1 +2 +− 1 +2 +� ρ +− 1 +2 +∥u( q, ερ + t0 − εσ) − u( q, t0)∥L2(B)∥F(wε)( q, ερ + t0)∥L2(B) dσ dρ ++ +� +1 +2 +− 1 +2 +∥u( q, t0)∥L2(B)∥F(wε)( q, ερ + t0) − F(w)( q, t0)∥L2(B) dρ +≤ 8 +sup +t0≤t≤t0+ε +∥u( q, t0 + t) − u( q, t0)∥L2(B) +sup +t0− ε +2 ≤t≤t0+ ε +2 +∥F(wε)( q, t0 + t)∥L2(B) ++ C∥u( q, t0)∥L2(B) +sup +t0− ε +2 ≤t≤t0+ ε +2 +∥wε( q, t0 + t) − w( q, t0)∥L2(B), +where C is the Lipschitz constant of F. +We have used |ζ| ≤ 1 and |ζ′| ≤ 8 in the computation. +Since +u ∈ C([T0, T1]; L2(B)), we get +lim +ε→0 +sup +t0≤t≤t0+ε +∥u( q, t0 + t) − u( q, t0)∥L2(B) = 0. +Using a computation similar to (6.3) we obtain +sup +t0− ε +2 ≤t≤t0+ ε +2 +∥wε( q, t0 + t) − w( q, t0)∥L2(B) ≤ +sup +t0−ε≤t≤t0+ε +∥w( q, t0 + t) − w( q, t0)∥L2(B). +This converges to zero since w ∈ C([T0, T1]; L2(B)), and (t0 − ε, t1 + ε) ⋐ (T0, T1) due to the choice of ε. In +conclusion +lim +ε→0 Σ1 +u(ε) + Σ2 +u(ε) = 0. +In a similar fashion, we can argue that +lim +ε→0 Σ3 +u(ε) + Σ4 +u(ε) = 0. +Hence, limε→0 Σu(ε) = 0. The treatment of Σv(ε) is similar. +The term +� +B +F(uε − vε) dx +�t1− ε +2 +t0+ ε +2 += +� +B +F(wε)(x, t1 − ε +2) dx − +� +B +F(wε)(x, t0 + ε +2) dx, +converges to +� +B +F(w)(x, t1) dx − +� +B +F(w)(x, t0) dx. +To show this, we consider two cases. +Case A: F is bounded. In this case F is globally Lipschitz, that is |F(a) − F(b)| ≤ C|a − b|, therefore, +��� +� +B +F(wε(x, t0 + ε +2)) − F(w(x, t0)) dx +��� ≤ +� +B +C|wε(x, t0 + ε +2) − w(x, t0)| dx +≤ C|B| +1 +2 ∥wε( q, t0 + ε +2) − w( q, t0)∥L2(B) dx, +which converges to zero as was explained before, see (6.4). + +44 +ALIREZA TAVAKOLI +Case B: In this case, we have F(a) = a2. Therefore, +��� +� +B +F(wε(x, t0 + ε +2)) − F(w(x, t0)) dx +��� ≤ +� +B +|wε(x, t0 + ε +2))2 − w(x, t0)2| dx +≤ +� +B +|wε(x, t0 + ε +2)) − w(x, t0)||wε(x, t0 + ε +2)) − w(x, t0)| dx +≤ ∥wε( q, t0 + ε +2)) − w( q, t0)∥L2(B)∥wε( q, t0 + ε +2)) + w( q, t0)∥L2(B) +and since w ∈ C([T0, T1]; L2(B)), with an argument similar to the treatment of I1, as we let ε go to zero this +term converges to zero. +Now we discuss the convergence of the nonlocal term. Our treatment is similar to the argument in [BLS21, +Appendix B]. The aim is to show that the following converges to zero. +� t1 +t0 +�� +Rn×Rn(Jp(u(x) − u(y)) − Jp(v(x) − v(y))) × +� +(F(wε(x, t)))ε − F(w(x, t) +− +� +(F(wε(y, t)))ε − F(w(y, t)) +�� +dµ dt. +We split it into the two parts +� t1 +t0 +�� +B2×B2 +(Jp(u(x) − u(y)) − Jp(v(x) − v(y))) × +� +(F(wε(x, t)))ε − F(w(x, t) +− +� +(F(wε(y, t)))ε − F(w(y, t)) +�� +dµ dt ++ 2 +� t1 +t0 +�� +B× (Rn\B2) +(Jp(u(x) − u(y)) − Jp(v(x) − v(y))) × +� +(F(wε(x, t)))ε − F(w(x, t) +� +dµ dt +:= Θ1(ε) + 2Θ2(ε). +Here we have used the boundary condition u = v(w = 0) for y ∈ Rn \ B . Since |F(a) − F(b)| ≤ C|a − b| we +have +� t1 +t0 +∥(F(wε))ε∥p +W s,p(B2) dt ≤ C +� t1 +t0 +∥wε∥p +W s,p(B2) dt. +After passing to a subsequence this sequence converges weakly in Lp((t0, t1); W s,p(B2)) to F(w(x, t)) or in +another words +(F(wε(x, t))ε − (F(wε(y, t)))ε +|x − y| +n +p +s +converges weakly in Lp� +(t0, t1); Lp(B2 × B2) +� +and since +Jp(u(x) − u(y)) − Jp(v(x) − v(y)) +|x − y| +n +p′ +(p−1)s +belongs to Lp′� +(t0, t1); Lp′(B2 × B2) +� +we get the desired convergence for Θ1(ε). Now for Θ2(ε) consider +G(x, t) := +� +Rn\B2 +Jp(u(x) − u(y)) − Jp(v(x) − v(y)) +|x − y|n+sp +dy. +Then for almost every x ∈ B +|G(x, t)| ≤ C(n, s, p) +� +Rn\B2 +|u(x, t)|p−1 + |u(y, t)|p−1 + |v(x, t)|p−1 + |v(y, t)|p−1 +|y|n+sp +dy +≤ C +� +2Tailp−1,sp(u( q, t); 0, 2)p−1 + |u(x, t)|p−1 + |v(x, t)|p−1� +(6.6) +The terms |u(x, t)|p−1 and |v(x, t)|p−1 belongs to Lp′� +(t0, t1); Lp′(B) +� +since u, v ∈ Lp((t0, t1); Lp(B)). The tail +term its independent of x and belongs to Lp′(t0, t1) by the assumption +� T1 +T0 +� +Tailp−1,sp(u( q, t); 0, 2)) +�p′ +dt ≤ ∞. + +H ¨OLDER CONTINUITY +45 +Thus, G(x, t) ∈ Lp′([T0, T1]; Lp′(B2)) and as before after extracting a subsequence: +F(wε(x, t))ε ⇀ F(w(x, t) +in Lp([t0, t1]; Lp(B)). +This shows that +Θ2(ε) = +� t1 +t0 +� +B +G(x, t) +� +F(wε(x, t))ε − F(w(x, t) +� +converges to zero. +Finally, we let ε0 go to zero to get the desired result for [T0, T1]. We need to show that the following converge +to zero as ε0 tends to 0. +J1 := +� +B +F(w(x, T0)) − F(w(x, T0 + ε0)) dx +, +J2 := +� +B +F(w(x, T1)) − F(w(x, T1 − ε0)) dx, +J3 := +� T0+ε0 +T0 +� +B +F(w(x, t))f(x, t) dx dt +, +J4 := +� T1 +T1−ε0 +� +B +F(w(x, t))f(x, t) dx dt, +and +N1 := +� T0+ε0 +T0 +�� +Rn×Rn +�Jp(u(x, t) − u(u, t)) − Jp(v(x, t) − v(y, t)) +|x − y|n+sp +� +× +� +F(w(x, t)) − F(w(y, t)) +� +dx dy dt, +and +N2 := +� T1 +T1−ε0 +�� +Rn×Rn +�Jp(u(x, t) − u(u, t)) − Jp(v(x, t) − v(y, t)) +|x − y|n+sp +� +× +� +F(w(x, t)) − F(w(y, t)) +� +dx dy dt. +The arguments will be reminiscent of the ideas in the previous part. +We start with J2, in the case of a bounded F, F is globally Lipschitz and we have +��J2 +�� ≤ +� +B +��F(w(x, T1)) − F(w(x, T1 − ε0)) +�� dx ≤ C +� +B +|w(x, T1) − w(x, T1 − ε0)| dx +≤ C|B| +1 +2 ∥w( q, T1) − w( q, T1 − ε0)∥L2(B). +This converges to 0 since w ∈ C([T0, T1]; L2(B)), in the case of F(a) = a, we have +��J2 +�� ≤ +� +B +��F(w(x, T1)) − F(w(x, T1 − ε0)) +�� dx ≤ +� +B +��w(x, T1)2 − w(x, T1 − ε0)2�� dx +≤ +� +B +��w(x, T1) − w(x, T1 − ε0) +����w(x, T1) + w(x, T1 − ε0) +�� dx +≤ ∥w( q, T1) − w( q, T1 − ε0)∥L2(B)∥w( q, T1) + w( q, T1 − ε0)∥L2(B). +Again since w ∈ C([T0, T1]; L2(B)), this term converges to 0. +J1 can be treated in a similar way. For the term J4, using |F(a)| ≤ C|a| we get +��J4 +�� ≤ C +� T1 +T1−ε0 +� +B +|w(x, t)||f(x, t)| dx dt. +Since w ∈ Lp⋆ +s,p(B × [T0, T1]) and f ∈ L(p⋆ +s)′,p′(B × [T0, T1]), using H¨older’s inequality (2.9), one can see that +w(x, t)f(x, t) ∈ L1(B × [T0, T1]). +Now using the absolute continuity of the integral for integrable functions we can conclude that J4 converges to +0. The reasoning for convergence of J3 is similar. +Now we turn our attention to the nonlocal terms. +N2 = +� T1 +T1−ε0 +�� +B2×B2 +�Jp(u(x, t) − u(y, t)) − Jp(v(x, t) − v(y, t)) +|x − y|n+sp +� +× +� +F(w(x, t)) − F(w(y, t)) +� +dx dy dt ++ 2 +� T1 +T1−ε0 +�� +B×(Rn\B2) +Jp(u(x, t) − u(y, t)) − Jp(v(x, t) − v(y, t)) +|x − y|n+sp +F(w(x, t)) dx dy dt +:= Θ1 + 2Θ2 + +46 +ALIREZA TAVAKOLI +First, we treat Θ1. Notice that since u, v ∈ Lp([T0, T1]; W s,p(B2)) we have +Jp(u(x, t) − u(y, t)) − Jp(v(x, t) − v(y, t)) +|x − y| +n +p′ +(p−1)s +∈ Lp′([T0, T1]; Lp′(B2 × B2)]), +and using Lipschitz continuity of F and the fact that w ∈ Lp([T0, T1]; W s,p(B2)) we have +F(w(x, t)) − F(w(y, t)) +|x − y| +n +p +s +∈ Lp([T0, T1]; Lp(B2 × B2)). +This implies that the integrand involved in Θ1 belongs to L1([T0, T1]; L1(B2×B2)). And similar to the treatment +of J4, since the volume of the integration region is shrinking to 0, Θ1 converges to 0. To deal with Θ2, notice +that +F(w(x, t)) ∈ Lp([T0, T1]; Lp(B)) +and define +G(x, t) := +� +Rn\B2 +Jp(u(x, t) − u(y, t)) − Jp(v(x, t) − v(y, t)) +|x − y|n+sp +dy. +We can estimate this integration in terms of the Tail, that is +��G(x, t) +�� ≤ C(n, s, p) +� +Tailp−1,sp(u( q, t); 0, 2) + |u(x, t)|p−1 + |v(x, t)|p−1� +, +see for example (6.6). Therefore, G(x, t) ∈ Lp′([T0, T1]; Lp′(B)). Hence using H¨older’s inequality +G(x, t)F(x, t) ∈ L1([T0, T1]; L1(B)). +This concludes the result. N1 can be treated in an exactly similar manner. +□ +References +[AABP18] B. Abdellaoui, A. Attar, R. Bentifour, I. Peral, On fractional p-Laplacian parabolic problem with general data, Ann. +Mat. Pura Appl., 197 (2018), 329-356. +[APT22] +K. Adimurthi, H. Prasad, V. Tewary, Local H¨older regularity for nonlocal parabolic p-Laplace equations, preprint, +http://arxiv.org/abs/2205.09695v1 +[AMRT10] F. Andreu-Vaillo; J. Maz´on; J.D. Rossi;J.J. Toledo-Melero, Nonlocal diffusion problems Mathematical Surveys and +Monographs, 165. American Mathematical Society, Providence, RI; Real Sociedad Matematica Espan˜nla, Madrid, +2010. +[AS67] +D.G. Aronson, J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25, +81-122 (1967) +[BLS18] +L. Brasco, E. Lindgren, A. 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Warma, Local Lipschitz continuity of the inverse of the fractional p-Laplacian, H¨older type continuity and continuous +dependence of solutions to associated parabolic equations on bounded domains, Nonlinear Anal., 135 (2016), 129-157. +Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden +Email address: alirezat@kth.se + diff --git a/9tE2T4oBgHgl3EQfmAcG/content/tmp_files/load_file.txt b/9tE2T4oBgHgl3EQfmAcG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ffcfc912c989efbbc20caab5d590cc017425cc11 --- /dev/null +++ b/9tE2T4oBgHgl3EQfmAcG/content/tmp_files/load_file.txt @@ -0,0 +1,1440 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf,len=1439 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content='03993v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content='AP] 10 Jan 2023 A PERTURBATIVE APPROACH TO H¨OLDER CONTINUITY OF SOLUTIONS TO A NONLOCAL p-PARABOLIC EQUATION ALIREZA TAVAKOLI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' We study local boundedness and H¨older continuity of a parabolic equation involving the fractional p-Laplacian of order s, with 0 < s < 1, 2 ≤ p < ∞, with a general right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' We focus on obtaining precise H¨older continuity estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' The proof is based on a perturbative argument using the already known H¨older continuity estimate for solutions to the equation with zero right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Introduction In this paper, we study the local boundedness and H¨older regularity of solutions to the inhomogeneous equation ut + (−∆p)su = f(x, t), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content='1) where f ∈ Lr loc(I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Lq loc(Ω)) with q ≥ 1, r ≥ 1, p ≥ 2 and s ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Here, (−∆p)s is the fractional p-Laplacian, arising as the first variation of the Sobolev-Slobodecki˘ı seminorm (−∆p)su(x) := 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' � Rn |u(x) − u(y)|p−2(u(x) − u(y)) |x − y|n+s p dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Nonlocal equations involving operators of the type above, with a singular kernel, were first considered in [IN10] to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' In this study, continuing the work in [BLS21], we perform a perturbative argument to obtain H¨older continuity estimates, with explicit exponents for the equations with a right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Our approach closely follows the arguments in [TU14] and [BLS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' In such perturbative arguments it is often possible to establish H¨older regularity results for bounded solutions using only L∞ estimates for the equations with zero right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' This is not the case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Due to the presence of a suprimim in time in the tail (see section 3), we are led to proving a L∞ bound for equations with right hand sides, this is Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' The proof is inspired by the work [AS67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Below, we state the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' For the definition of the tail and relevant function spaces, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' We use the following notation of parabolic cylinders QR,r(x, T ) := BR(x0) × (T − r, T ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' The exponent p⋆ s = np n−sp is the critical exponent for the Sobolev embedding therem, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' We denote by p′, the H¨older conjugate of p, that is p′ = p p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded and open set, I = (t0, t1], p ≥ 2, 0 < s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Consider q and r such that r ≥ p′, 1 r + n spq < 1 and q ≥ (p⋆ s)′ in the case sp < n, and 1 r + 1 q < 1 and q > 1 in the case sp ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' 35K55, 35K65, 35R11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Fractional p-Laplacian, Local H¨older regularity, Nonlocal diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' 1 2 ALIREZA TAVAKOLI Suppose u is a local weak solution of ut + (−∆p)su = f in Ω × I, such that u ∈ L∞ loc(I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Lp−1 sp (Rn)) and f ∈ Lr loc(I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' Lq loc(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' then u is locally bounded in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfmAcG/content/2301.03993v1.pdf'} +page_content=' More specifically, if Q2R,(2Rsp)(x0,T0) ⊂ Ω × I, u bounded in QR/2,(R/2)sp(x0, T0) and in the case sp ̸= n, the estimate reads ∥u∥L∞(Q R 2 ,( R 2 )sp) ≤ 2 sup T0−Rsp 1.25 V , induced charges in system C, are greater +than system B until both systems reach saturation in ∆V = 3.5 V . +On the other hand, charge storage in SCs is described by the critical quantities of the +Differential Capacitance, Cd(V ), and the energy density. The Cd(V ) is the derivative of +the electrodes surface charge with respect to the applied potential difference between the +electrodes:33 +Cd(V ) = dq +dV . +(8) +According to the Eq.8, the Cd(V ) is the derivative of the curves in Fig.4. Fig.5, which is +obtained by the numeric derivation of smooth curves of q(V ), shows the Cd(V ) plots for +three systems. +Based on Kornyshev theory for most ILs, Cd(V ) displays a bell-shape, with a maximum at +11 + + 0 + 5 + 10 + 15 + 20 + 25 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +Charge Density (µC/cm2) +Voltage (V) +System A +System B +System C +Figure 4: Mean induced charge density on the electrodes surface. The points in the figure +display the simulation data and the solid lines in the plots represent the smoothed data. +the Potential of Zero Charge (PZC) or a camel-shape with two peaks. In both cases, Cd(V ) +decreases at large potential difference. The reason is, the ions are only allowed to pack up to a +given maximum density in the double layer. Therefore, by increasing the potential difference, +the ions concentration near the electrodes’ surface reaches their maximum value and the +effective diffuse layer thickness actually grows larger, leading to a decrease in Cd(V ).34,34 +Fig.5 is a plot of the Cd(V ) for three systems, which display camel-shaped figures. There +is limited change in the Cd(V ) curve for system B, indicating the surface charge density +changes less rapidly in system B with increasing potential. In contrast, there is a peak at +∆V = 1.5 V in the Cd(V ) plot of system C which points out that more counter-ions can be +condensed on the electrodes’ surface of system C in comparison with system B. The Cd(V ) +peak for system A is the highest among systems B and C. The result of Korneyshev theory +is observed in Fig.4 and Fig.5. In systems A, B and C, the saturation voltage is 2, 3 and 3.5 +12 + + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +Differential Capacitance (µF/cm2) +Voltage (V) +System A +System B +System C +Figure 5: Differential Capacitance for three systems. +V . Due to this, in the Cd(V ) plot, the DC drops at voltages above these values. +As mentioned before, energy density is another useful parameter that shows the efficiency +of SCs. SC with a higher energy density is more efficient in electrical devices. The stored +energy density in SCs is obtained:35 +E(V ) = +� V +0 +V Cd(V ) dV +(9) +Based on the above equation, the plot of energy density for the three systems is calculated +and plotted in Fig.6. According to this plot, for ∆V < 2 V the energy density of system +A is greater than systems B and C. For ∆V > 2 V systems B and C have higher energy +density which indicates that the GPEs-based SC has more efficiency than IL-based SCs in +higher potential difference. +13 + + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +Energy Density (µW/cm2) +Voltage (V) +System A +System B +System C +Figure 6: The stored energy density in three systems. +Structural investigation: Ion density profile +Further insight into the ion structure near the electrodes’ surface is gained in this section. +Due to the constant density in all three systems, system A has a smaller bulk region between +its electrodes than systems B and C. It is expected that the ion density profiles will oscillate +near the charged surface in IL-based SCs,34 as shown in Fig.7a for ∆V = 2 V . Fig.7b and +Fig.7c depict ion density profiles for systems B and C at ∆V = 2 V after reaching equilibrium. +Similar to ILs, the ion density profile of these systems exhibits layers and oscillations near +electrode surfaces. While the conductive electrodes repel co-ions and attract counter-ions, +co-ions can still be observed inside the pores. The highest peak in Fig.7a can be seen at the +entrance and end of the pores, which contributes to the high induced charge on the surface +of the electrodes. Fig.7c, the ion density profile of system C, displays a higher peak at the +entrance of pores in analogy with the profile of system B. This leads to more induced charge +14 + + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 +-15 +-10 +-5 + 0 + 5 + 10 + 15 + 20 +2 V +0 V +Ion Density Profile (nm-3) +X (nm) +(a) + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 +-15 -10 -5 + 0 + 5 + 10 15 20 25 30 +2 V +0 V +Ion Density Profile (nm-3) +X (nm) +(b) + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 +-15 -10 -5 + 0 + 5 + 10 15 20 25 30 +2 V +0 V +Ion Density Profile (nm-3) +X (nm) +(c) +Figure 7: Ion density profile of three systems along x axis for (a) System A (b) System B +and (c) System C. Green and red graphs display positive and negative ions. +on the electrodes’ surface and eventually higher energy density. +The linear polymers can be placed near the electrode surfaces as well as in bulk space in +system B, in contrast to system C, which places the polymer network only in bulk space. As +a result, the accessible space for ions near electrode surfaces is reduced. Accordingly, system +B exhibits a lower ion density at the pore entrance than systems A and C. +Dynamical investigation: Mean Squared Displacement and +Diffusion coefficient +Gels with cross-linked polymers demonstrated higher mobility.5 Since the mobility of ions +is proportional to the diffusion coefficient (D) we need to calculate the diffusion coefficient. +The ionic diffusion coefficient is derived from the Mean-Squared Displacement (MSD) curve +using the 3D diffusion relation: +MSD ≡< ∆r(t)2 >= 1 +N +N +� +i=1 +< ri(t)2 − ri(0)2 >, +(10) +where N is the number of particles. Diffusion coefficient can be calculate using the following +equation; +D = lim +t→∞ +< ∆r(t)2 > +6t +. +(11) +15 + +According to the above equation, also known as Einstein’s relation,36 the self-diffusion of +particles is calculated from the slope of the MSD curve over time. +We calculated MSD for clusters of linear polymers in system B and cross-linked polymers +in system C. According to Fig.8, MSD for linear polymers and cross-linked polymers display +superdiffusive motion. On the other hand, by taking the slope of the MSD into account, the +diffusion coefficient for cross-linked polymers is greater than linear polymers and therefore +the mobility of cross-linked polymers is higher. This result can also be seen in Fig.9. In this +figure, two snapshots of both polymers are shown for equal time intervals ∆t = 10 timestep. +Fig.9a shows lower mobility (for linear polymers) than Fig.9b (for cross-linked polymers). + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 0 + 2 + 4 + 6 + 8 + 10 + 12 +~ t1.91 +~ t1.99 +Mean Squared Displacement (× 106 nm2) +Time (ns) +Linear polymers in system B +Network polymer in system C + 1×10-5 + 1×10-4 + 1×10-3 + 1×10-2 + 1×10-1 + 1×100 + 1×101 + 1×10-1 + 1×100 + 1×101 +Figure 8: Mean Squared Displacement of polymers in systems B and C in absence of electric +field. Inner is the same result but in logarithmic scale, indicating on power-law behavior of +MSD. +In addition, we obtained MSD for cations and anions for all systems. Due to volume +effects and the high mobility of cross-linked polymers, these polymers provide a limited path +for charged particles in system C in comparison with linear polymers in system B. Therefore +16 + +(a) +(b) +Figure 9: Two different snapshots of polymers for (a) linear polymers in system B and (b) +cross-linked polymers in system C. +as shown in Fig.10, the charged particles’ diffusion in system C is lower than in systems A +and B. Since a lower diffusion coefficient is associated with a larger viscosity, the viscosity +of system C is greater than systems A and B. Consequently, increasing the viscosity of the +electrolyte improves its mechanical stability. Accordingly, system C has higher mechanical +stability than systems A and B. +Furthermore, it is important to note that cations and anions differ slightly in their results +due to their mass differences. Compared to a cation, an anion has a lower weight, so its +MSD plot shows a higher value. +17 + + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 + 2 + 4 + 6 + 8 + 10 + 12 +D ~ 0.21538 × 10-4 (m2/s) +D ~ 0.1375 × 10-4 (m2/s) +D ~ 0.07313 × 10-4 (m2/s) +Mean Squared Displacement (× 105 nm2) +Time (ns) +anion in system A +cation in system A +anion in system B +cation in system B +anion in system C +cation in system C + 1×10-4 + 1×10-3 + 1×10-2 + 1×10-1 + 1×100 + 1×101 + 1×10-1 + 1×100 + 1×101 +Figure 10: Mean Squared Displacement of ions in all three systems at ∆V = 2 V . Inner plot +is the same as outer but in logarithmic scale. +Conclusion +In summary, simulations demonstrate a difference between the performance of two types +of electrolytes, i.e. liquid and solid electrolytes in supercapacitors. Solid electrolytes are +used as an alternative to reduce the problems associated with liquid electrolytes. Therefore, +simulation and comparison of these two categories can give us a clear insight into improving +the efficiency of supercapacitors. +Liquid electrolyte-based supercapacitors have a smaller operating voltage window there- +fore, the amount of energy stored is less. Polymer electrolytes can be considered as a cluster +of linear polymers or as cross-linked polymers in which linear polymers are connected and +form a network. In linear polymers, since polymers can be present near the walls of the elec- +trodes as well as inside the pores, the available space for ions in the vicinity of the electrodes +is reduced. 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The +Journal of Chemical Physics 2020, 153, 100903. +24 + diff --git a/BNAzT4oBgHgl3EQfGPsJ/content/tmp_files/load_file.txt b/BNAzT4oBgHgl3EQfGPsJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a1aac5f213ba75ea290671a0acf35d815144d05 --- /dev/null +++ b/BNAzT4oBgHgl3EQfGPsJ/content/tmp_files/load_file.txt @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf,len=673 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='01023v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='comp-ph] 3 Jan 2023 Performance investigation of supercapacitors with PEO-based gel polymer & ionic liquid electrolytes: Molecular Dynamics Simulation Nasrin Eyvazi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='† Davood Abbaszadeh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='† Morad Biagooi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='‡ and SeyedEhsan Nedaaee Oskoee∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='¶ †Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Institute for Advanced Studies in Basic Sciences (IASBS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Zanjan 45137-66731,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Iran ‡Intelligent Data Aim Ltd (IDA Ltd),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Science and Technology Park of Institute for Advanced studies in Basic Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Zanjan 45137-65697,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Iran ¶Research Center for Basic Sciences & Modern Technologies (RBST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Institute for Advanced Studies in Basic Sciences (IASBS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Zanjan 45137-66731,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Iran E-mail: nedaaee@iasbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='ir Phone: (+98) 241-415-2217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Fax: (+98) 241-415-2104 Abstract Due to the importance of using supercapacitors in electronic storage devices, im- proving their efficiency is one of the topics that has attracted the attention of many researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Choosing the proper electrolyte for supercapacitors is one of the most sig- nificant factors affecting the performance of supercapacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' In this paper, two classes of electrolytes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' liquid electrolyte (ionic liquid electrolyte) and solid electrolyte (polymer electrolyte) are compared by molecular dynamics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' We consider the polymer electrolyte in linear and network configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The results show that 1 although ionic liquid-based supercapacitors have a larger differential capacitance, since they have a smaller operation voltage, the amount of energy stored is less than poly- mer electrolyte-based supercapacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Also, our investigations indicate that polymer electrolyte-based supercapacitors have more mechanical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Therefore, they can be considered a very suitable alternative to liquid electrolyte-based supercapacitors that do not have known liquid electrolyte problems and display better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Introduction Supercapacitors (SCs), also known as Electric Double Layer Capacitors (EDLCs), have re- cently attracted much attention in the field of electrical energy storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The SCs fill the gap between batteries and conventional capacitors in terms of energy and power density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' They consist of two porous electrodes immersed in an electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Due to the potential differ- ence between the electrodes, the charged electrodes repel the co-ions in the electrolyte while attracting their counter-ions, resulting in charge separation and charge storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The SCs have higher energy density in comparison with conventional capacitors due to their porous electrodes with large surface areas and small charge separation distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='1 Also, compared to batteries, SCs have the advantages of higher power density induced by a fast charging/discharging rate (in seconds), a long cycle life (4,100,000 cycles), and high power density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='1 Despite their higher power density, they cannot store the same amount of energy as batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='1 Extensive efforts and research have been devoted to increasing the energy density of SCs to 20-30 Wh/L to solve the problems and satisfy the performance demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='2–4 According to relation E = 1 2CV 2, the energy density (E) of SCs is proportional to the capacitance (C) and the square of the voltage (V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Therefore, increasing either the capacitance or the voltage of a cell can be an effective way to achieve high energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='2 The efficiency of SCs depends mainly on both electrolyte and electrode structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The pore size and surface area of electrodes, ionic conductivity, and electrolyte operating voltage window play a significant role in developing high-performance and flexible SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='5 Especially 2 in the case of liquid electrolytes, they have some disadvantages for use in flexible SCs like being toxic and corrosive,2 requiring high-cost packaging to fabricate, and are associated with leakage problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='2,6 In general, the critical features of an ideal electrolyte include: (1) a wide voltage and temperature window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' (2) a high ionic conductivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' (3) a high chemical and mechanical stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' (4) well-matched with the electrolyte materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' (5) low volatility and flammability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' (6) safety;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' and (7) simple processing with low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='6–8 To overcome the limitations of liquid electrolytes, polymer electrolytes (PEs) were in- troduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' In 1970, Armand first used PE in Lithium Ion Batteries (LIBs) and proposed LIBs with improved efficiency and energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='9 PEs consist of a macromolecule matrix dissolved in a low viscosity and high dielectric constant organic solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='9 PEs have many advantages such as avoiding liquid leakage and corrosion problems, good ionic conductivity, high chemical and mechanical stability, high energy density, safety, solvent-free condition, being light in weight, low cost, and simple manufacturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='6–9 Due to the advantages of PEs, they are ideal candidates for use in SCs as electrolytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' PEs for SCs can be classified into three categories: (1) solid polymer electrolytes (SPEs), (2) gel polymer electrolytes (GPEs), and (3) polyelectrolytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The SPE is composed of a polymer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=', PEO) and a salt (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=', LiCl), without any solvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The ions in the SPE are transported through the polymer2,5 and the polymer works as a host matrix for ion move- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='2,10 In contrast, the GPE consists of a polymer host (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=', PVA) and a liquid electrolyte or a conducting salt dissolved in a solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='2,11 The polymer in GPE is swollen by the solvent and acts as a dynamic moving matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The conductivity of ions occurs through the solvent instead of the polymer phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='2,11 In GPEs, the liquid electrolyte generally provides free ions that participate in conductivity enhancement and also acts as a conductive medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' In addition, the polymer provides perfect mechanical stability by increasing the viscosity of the electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='5 Recently, researchers have shown that using Ionic Liquids (ILs) can im- prove ionic conductivity and cell voltage, resulting to improvement in the electrochemical performance of GPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='5 By its softening effect on the polymer chains, IL can increase the 3 electrolyte’s ionic conductivity and facilitate ion transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='5 GPE based on ILs and linear polymers usually exhibit poor mechanical properties, including both strength and flexibil- ity, because of their few polymer chain entanglements separated by small molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='12 To generate polymer networks, cross-linking strategies have been proposed in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The behaviors and performance of polymer gels are largely determined by the structure of the polymer network that makes up the gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' This is due to the interaction between the net- work and the solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='5 Gels generally have high mobility because the polymer networks are dissolved by a large amount of entrapped solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='5 In the polyelectrolyte, ionic conductivity is created by charged polymer chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='12 As it turns out, each type of these solid-state electrolytes has its advantages and disadvan- tages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Typically, GPEs have the highest ionic conductivity among the three types of solid- state electrolytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='5,12 Due to the liquid phase in the GPE, its ionic conductivity is signif- icantly higher than dry SPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Therefore, GPE-based SCs currently dominate the products of solid electrolyte-based SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Several polymer matrices have been explored for prepar- ing GPEs in the role of host polymer including: poly (vinyl alcohol) (PVA), poly (acrylic acid) (PAA), potassium polyacrylate (PAAK), poly (ethyl oxide) (PEO), poly (methyl- methacrylate) (PMMA), poly (ether ether ketone) (PEEK), and poly (vinylidene fluoride- co-hexafluoro-propylene) (PVDF-HFP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='2 Beyond all of the experimental achievements,6,13–16 modeling the EDLCs under differ- ent physical conditions would provide a lot of insight into the system’s physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Modeling the microstructure will reveal how the related dynamics for charge carriers happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' So far, numerous types of research have been done on liquid electrolyte-based SCs to explore the performance of different kinds of liquid electrolyte-based SCs and the effect of electrode struc- ture and its pore size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='17–20 Our previous work,21 models the third classification of polymer electrolytes (polyelectrolytes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' In this work, considering the advantages of GPEs compared to other polymer electrolytes and their useful applications, we investigated the behavior of GPE-based SCs using the molecular dynamics simulation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Here, we compared IL- 4 based and GPE-based SCs in order to find out the effects of adding host polymers to the IL electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' In the first section, we describe the method of simulation under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Based on the results we get from our model, we discuss the pros and cons of using GPEs and IL electrolytes in SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Method Systems’ model and force field parameters Molecular Dynamics (MD) simulations have been widely used to describe the behavior of SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' Using MD simulations, we compared liquid electrolyte-based SCs with PE-based SCs under various conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The simulations were carried out by the CAVIAR software package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='22 We have investigated the performance of three different systems with similar electrodes but different electrolytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The electrolytes were confined between two electrodes with single slit- pore geometry, placed at a distance of 15 nm from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' This model called a slit-pore model was introduced by Breitsprecher et al18,23 for simulating porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The slit-pore length was set to 15 nm, in order to be compared with the bulk region and the width of the pore is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='5 nm (larger than the size of two ionic particles) so that at least two ionic particles can pass through the pore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='24 To compare the electrolyte role we have simulated the above mentioned pore structure with different electrolyte systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' liquid electrolyte (system A), linear polymer electrolyte (system B), and polymer network electrolyte (system C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' By comparing the results, we can discuss the effects and the function of different electrolytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' System A: Liquid electrolyte-based supercapacitors Our first model contains IL electrolytes confined between those described electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' We used Coarse-Grained (CG) models of ILs, where ions are soft spheres with diameters dcation = danion = 1 nm and valency q = ± 1, in units of the elementary charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' The mass of 5 cations and anions are mcation = 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='17 g mol and manion = 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content='81 g mol corresponding to EMIM+ and BF4-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' All the particles interact with the non-bonding Lennard-Jones (LJ) potential: VLJ(rij) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 4εij[( σij rij )12 − ( σij rij )6] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' r ≤ rcut 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' r ≥ rcut (1) Where rij is the relative distance between each pair of particles, ε and σ are the length and energy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' In calculating the LJ potential, rcut = 6√ 2 σ is the cutoff length to ensure short-range repulsive force at any distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfGPsJ/content/2301.01023v1.pdf'} +page_content=' In addition, charged particles also interact through the electrostatic non-bonding potential: VC(rij) = � i � i