diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfuri" "b/data_all_eng_slimpj/shuffled/split2/finalzzfuri" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfuri" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} For an algebraic variety $S$ over a commutative field $k$ of characteristic zero, let $A$ be a locally free ${\\mathcal O}_S$-module which is an associative ${\\mathcal O}_S$-algebra. In \\cite{G}, Getzler constructed a flat connection in the ${\\mathcal O}_S$-module $\\operatorname{HC}_{\\bullet}^{\\operatorname{per}}(A)$, the periodic cyclic homology of $A$ over the ring of scalars ${\\mathcal O}_S$. This connection is called the Gauss-Manin connection. In this paper we define this connection at the level of the periodic cyclic chain complex $\\operatorname{CC}_{\\bullet}^{\\operatorname{per}}(A)$.\n\nRecall that for an associative algebra over a commutative unital ring $K$ one can define the Hochschild chain complex $C_{\\bullet}(A)$, the negative cyclic complex ${\\operatorname{CC}}_{\\bullet}^- (A)$, and the periodic cyclic complex $\\operatorname{CC}_{\\bullet}^{\\operatorname{per}} (A)$, as well as the Hochschild cochain complex $C^{\\bullet}(A)$ (\\cite{L}, \\cite{T}, \\cite{Ge}). The latter is a differential graded Lie algebra, or a DGLA, if one shifts the degree by one: ${\\frak{g}}^{\\bullet}_A=C^{\\bullet +1}(A)$. Recall that $\\CCn(A) =(C_{\\bullet}(A)[[u]], b+uB)$ is a complex of $K[[u]]$-modules. Here $u$ is a formal variable of degree $-2$. We can view $\\CCn(A)$ as a {\\em cochain} complex if we reverse the grading. In particular, the {\\em cohomological} degree of $u$ is $2$. The complex $\\CCn(A)$ is known to be a DG module over the DGLA ${\\mathfrak{g}}^{\\bullet}_A$, the action of a cochain $D$ given by the standard operator $L_D$ (cf. \\cite{T} or \\ref{eq: L} below).\n\nConsider another formal variable, $\\epsilon$, of degree $1$. Now consider the DGLA \n\\begin{equation} \\label{eq:gA(u,e)}\n({\\mathfrak{g}}^{\\bullet}_A[u, \\epsilon], \\delta+u\\frac{\\partial}{\\partial \\epsilon})\n\\end{equation}\n\\begin{thm} \\label{thm:linfty structure}\nOn $\\CCn(A)$, there is a natural structure of an $L_{\\infty}$ module over $({\\mathfrak{g}}^{\\bullet}_A[u, \\epsilon], \\delta+u\\frac{\\partial}{\\partial \\epsilon})$. This structure is $K[[u]]$-linear and $(u)$-adically continuous. The induced structure of an $L_{\\infty}$ module over ${\\mathfrak{g}}^{\\bullet}_A$ is the standard one.\n\\end{thm}\nWe recall that an $L_{\\infty}$ module structure, or, which is the same, an $L_{\\infty}$ morphism ${\\mathfrak{g}}^{\\bullet}_A[u,\\epsilon] \\to \\operatorname{End}_{K[[u]]}(\\CCn(A))$, can be defined in two equivalent ways. One definition expresses it as a sequence of DGLA morphisms \n\\begin{equation}\\label{eq:L-infty as domik}\n{\\mathfrak{g}}^{\\bullet}_A[u,\\epsilon]\\leftarrow {\\mathcal L}\\to \\operatorname{End}_{K[[u]]}(\\CCn(A))\n\\end{equation}\nwhere the morphism on the left is a quasi-isomorphism. Alternatively, one can define this $L_{\\infty}$ morphism\nas a collection of $K[[u]]$-linear maps \n\\begin{equation} \\label{eq:linfty-morphism}\n\\phi _n: S^n( {\\mathfrak{g}}^{\\bullet}_A[u,\\epsilon] [1])\\to \\operatorname{End}_{K[[u]]}(\\CCn(A))[1]\n\\end{equation}\nsatisfying certain quadratic equations. Using that, one can define Getzler's Gauss-Manin connection at the level of chains as a morphism\n$$\n\\Omega ^{\\bullet}(S, \\CCp(A))\\to \\Omega ^{\\bullet}(S, \\CCp(A))\n$$\nof total degree one such that\n$$\\omega \\mapsto d\\omega +\\sum _{n=1}^{\\infty} \\frac{u^{-n}}{n!}\\phi _n(\\theta, \\ldots, \\theta)\n$$\nwhere $\\theta$ is the ${\\mathfrak{g}}^{\\bullet}_A[u,\\epsilon]$-valued one-form on $S$ given by \n$$\\theta (X)(s) = L_X m_s \\epsilon.$$\nHere $s\\in S$, $m_s$ is the multiplication on the fiber $A_s$ of $A$ at the point $s$, and $X$ is a tangent vector to $S$ at $s$.\n\nA few words about the proof of the main theorem. We define the $L_{\\infty}$ morphism by explicit formulas (Theorem \\ref{thm:action of twisted bar} and Lemma \\ref{quis of twisted and untwisted}), but the proof that they do satisfy $L_{\\infty}$ axioms is somewhat roundabout. Recall that the Hochschild cochain complex $C^{\\bullet}(A)$, with the cup product, is a differential graded algebra (DGA). One can consider the negative cyclic complex ${\\operatorname{CC}}_{\\bullet}^-(C^{\\bullet}(A))$ of this DGA. In \\cite{TT} and \\cite{T}, an $A_{\\infty}$ structure on this complex is constructed. The negative cyclic complex $\\CCn(A)$ is an $A_{\\infty}$ module over this $A_{\\infty}$ algebra. From this, we deduce that $\\CCn(A)$ is a DG module over some DGA which is related to the universal enveloping algebra $U(g_A [u, \\epsilon])$ by a simple chain of quasi-isomorphisms.\n\nA statement close to Theorem \\ref{thm:linfty structure} was proven in \\cite{DT}. Our proof substantially simplifies the proof given there. Note that a much stronger statement can be proven. Namely, $C^{\\bullet}(A)$ is in fact a $G_{\\infty}$ algebra in the sense of Getzler-Jones \\cite{GJ1} whose underlying $L_{\\infty}$ algebra is ${\\mathfrak{g}}^{\\bullet}_A$, (\\cite{Ta}, \\cite{H}); moreover, the pair ($C^{\\bullet}(A)$, $C_{\\bullet}(A)$) is a {\\it homotopy calculus}, or a $\\operatorname{Calc}_{\\infty}$ algebra (\\cite{KS}, \\cite{TT}, \\cite{T}). The underlying $L_{\\infty}$ module structure on $\\CCn(A)$ is the standard one. From this, Theorem \\ref{thm:linfty structure} follows immediately. (The interpretation of the $A_{\\infty}$ algebra ${\\operatorname{CC}}_{\\bullet}^-(C^{\\bullet}(A))$ in terms of the $\\operatorname{Calc}_{\\infty}$ structure is given in \\cite{TT}). However, theorems from \\cite{KS}, \\cite{TT}, \\cite{T} are extremely inexplicit and the constructions are not canonical, i.e. dependent on a choice of a Drinfeld associator. Our construction here is much more canonical and explicit, though still not perfect in that regard. It does not depend on an associator; it provides an explicit structure of a DG module over an auxiliary DG algebra (denoted in this paper by $B^{\\operatorname{tw}}({\\mathfrak{g}}^{\\bullet}_A[u, \\epsilon])$). Unfortunately, this auxiliary DGA is regated to our DGLA somewhat inexplicitly.\n\nTheorem \\ref{thm:linfty structure} implies the existence on $\\CCn(A)$ of a structure of an $A_{\\infty}$ module over $U({\\mathfrak{g}}^{\\bullet}_A [u, \\epsilon])$; the induced $A_{\\infty}$ module structure over $U({\\mathfrak{g}}^{\\bullet}_A)$ is defined by the standard operators $L_D$. An explicit linear map \n$$U({\\mathfrak{g}}^{\\bullet}_A [u, \\epsilon])\\otimes _{U({\\mathfrak{g}}^{\\bullet}_A)}\\CCn(A) \\to \\CCn(A)$$\nwas defined in \\cite{NT}. It is likely to coincide with the first term of the above $A_{\\infty}$ module structure. \n\nThis paper is, to a large extent, an effort to clarify and streamline our work \\cite{DT} with Yu.~L.~Daletsky. I greatly benefited from conversations with P.~Bressler, K.~Costello, V.~Dolgushev, E.~Getzler, M.~Kontsevich, Y.~Soibelman, and D.~Tamarkin.\n\nThis work was partially supported by an NSF grant.\n\n\\subsection{The Hochschild cochain complex} \\label{hocochain}\n\nLet $A$ be a graded algebra with unit over a commutative unital ring\n$K$ of characteristic zero. A Hochschild $d$-cochain is a linear map $A^{\\otimes d}\\to A$. Put,\nfor $d\\geq 0$,\n$$\n C^d(A) = C^d (A,A) =\\operatorname{Hom}_K({\\overline{A}}^{\\otimes d},A)\n$$\nwhere ${\\overline{A}}=A\/K\\cdot 1$. Put\n$$\n |D|\\;=({\\em {degree\\; of\\; the\\; linear\\; map\\; }}D)+d \n$$\n Put for cochains $D$ and $E$ from $C^{\\bullet}(A,A)$\n$$\n (D\\smile E)(a_1,\\dots,a_{d+e})=(-1)^{| E|\\sum_{i \\leq d}(|a_i| + 1)}\n D(a_1,\\dots,a_d) E(a_{d+1},\\dots,a_{d+e});\n$$\n$$\n (D\\circ E)(a_1,\\dots,a_{d+e-1})=\\sum_{j \\geq 0}\n (-1)^{(|E|+1)\\sum_{i=1}^{j}(|a_i|+1)}\nD(a_1,\\dots,a_j, \n E(a_{j+1},\\dots,a_{j+e}),\\dots); \n$$\n$$\n [D, \\; E]= D\\circ E - (-1)^{(|D|+1)(|E|+1)}E\\circ D\n$$\nThese operations define the graded associative algebra\n$(C^{\\bullet}(A,A)\\;,\\smile)$ and the graded Lie algebra\n($C^{\\bullet + 1}(A,A)$, $[\\;,\\;]$) (cf. \\cite{CE}; \\cite{G}).\nLet\n$$\n m(a_1,a_2)=(-1)^{\\deg a_1}\\;a_1 a_2;\n$$\nthis is a 2-cochain of $A$ (not in $C^2$). Put\n$$\n \\delta D=[m,D]; \n$$\n$$\n (\\delta D)(a_1,\\dots,a_{d+1})=(-1)^{|a_1||D|+|D|+1} \n a_1 D(a_2,\\dots,a_{d+1})+ \n$$\n$$\n +\\sum\n_{j=1}^{d}(-1)^{|D|+1+\\sum_{i=1}^{j}(|a_i|+1)}\n D(a_1,\\dots,a_ja_{j+1},\\dots,a_{d+1}) \n +(-1)^{|D|\\sum_{i=1}^{d }(|a_i|+1)}D(a_1,\\dots,a_d)a_{d+1}\n$$\n \nOne has\n$$\n \\delta^2=0;\\quad\\delta(D\\smile E)=\\delta D\\smile E+(-1)^{|deg D|}\n D\\smile\\delta E \n$$\n$$\n \\delta[D,E]=[\\delta D,E]+(-1)^{|D|+1}\\;[D,\\delta E]\n$$\n($\\delta^2=0$ follows from $[m,m]=0$).\n\nThus $ C^{\\bullet}(A,A)$ becomes a complex; we will denote it also by $C^{\\bullet}(A)$. The cohomology of this complex \nis $H^{\\bullet}(A,A)$ or the Hochschild cohomology. We denote it also by \n$H^{\\bullet}(A) $. The $\\smile$ product induces the\nYoneda product on $H^{\\bullet}(A,A)=Ext_{A\\otimes A^0}^{\\bullet}(A,A)$. The operation\n$[\\;,\\;]$ is the Gerstenhaber bracket \\cite{Ge}. \n\nIf $(A, \\;\\; \\partial)$ is a differential graded algebra then one can define \nthe differential $\\partial$ acting on $A$ by \n$$\n\\partial D \\;\\; = \\; [\\partial , D]\n$$\n\n\\begin{thm} \\cite{Ge} The cup product and the Gerstenhaber bracket induce a Gerstenhaber algebra structure on $H^{\\bullet}(A)$.\n\\end{thm}\nFor cochains $D$ and $D_i$ define a new Hochschild cochain by the following formula of Gerstenhaber (\\cite{Ge}) and Getzler (\\cite{G}):\n\n$$D_0\\{D_1, \\ldots , D_m\\}(a_1, \\ldots, a_n) = \n\\sum (-1)^{ \\sum_{k\\leq {i_p}}(|a_k| + 1)(| D_p|+1)} D_0(a_1, \\ldots ,$$\n$$a_{i_1} , D_1 (a_{i_1 + 1}, \\ldots ),\\ldots ,\nD_m (a_{i_m + 1}, \\ldots ) , \\ldots)$$\n\\begin{proposition}\\label{prop:brace structure}\n One has\n$$\n(D\\{E_1, \\ldots , E_k \\})\\{F_1, \\ldots, F_l \\}=\\sum (-1)^{\\sum _{q \\leq i_p}(|E_p|+1)(|F_q|+1)} \\times \n$$\n$$\n\\times D\\{F_1, \\ldots , E_1 \\{F_{i_1 +1}, \\ldots , \\} , \\ldots , E_k \\{F_{i_k +1}, \\ldots , \\}, \\ldots, \\}\n$$\n\\end{proposition}\n\nThe above proposition can be restated as follows. \nFor a cochain $D$ let $D^{(k)}$ be the following $k$-cochain of the DGA $C^{\\bullet}(A)$:\n$$\nD^{(k)}(D_1, \\ldots, D_k) = D\\{D_1, \\ldots, D_k\\}\n$$\n\\begin{proposition} \\label{etoee}\n The map \n$$\nD \\mapsto \\sum_{k \\geq 0} D^{(k)}\n$$\nis a morphism of differential graded algebras\n$$ C^{\\bullet}(A) \\rightarrow C^{\\bullet}( C^{\\bullet}(A))$$\n\\end{proposition}\n\\subsection{Hochschild chains} \\label{ss:hochschild-1}\nLet $A$ be an associative unital dg algebra over a ground ring $K$. The differential on $A$ is denoted by $\\delta$. Recall that by definition\n$$\\overline{A} = A \/ K\\cdot 1$$\nSet\n$$C_p (A,A) = C_p(A) = A \\otimes \\overline{A} ^{\\otimes p}$$\nDefine the differentials $\\delta: C_{\\bullet}(A) \\to C_{\\bullet}(A)$, $b: C_{\\bullet}(A) \\to C_{\\bullet - 1}(A)$, $B: C_{\\bullet}(A) \\to C_{\\bullet + 1}(A)$ as follows.\n\\[\n\\delta (a_0\\otimes\\cdots\\otimes a_p ) = \n\\sum_{i=1}^p {(-1)^{\\sum_{k1.$\nHere $D, \\; D_i, \\; E_j \\in {\\mathfrak{g}}^{\\bullet}_A$ and $Y=E_1 \\ldots E_n \\in S({\\mathfrak{g}}^{\\bullet}_A)^+.$\n\\end{thm}\nWe will start the proof in section \\ref{a-infty} below by recalling the ${A_{\\infty}}$ structure from \\cite{T}, \\cite{TT}. Then, in section \\ref{s:proof of twisted theorem}, we will re-write the definitions in term of this ${A_{\\infty}}$ structure. The proof will follow from the definition of an ${A_{\\infty}}$ module.\n\\section{The ${A_{\\infty}}$ algebra $C_{\\bullet}(C^{\\bullet}(A))$} \\label{a-infty}\n\nIn this section we will construct an ${A_{\\infty}}$ algebra structure on the negative cyclic complex of the DGA of Hochschild cochains of any algebra $A$. The negative cyclic complex of $A$ itself will be a right ${A_{\\infty}}$ module over the above ${A_{\\infty}}$ algebra. Our construction is a direct generalization of the construction of Getzler and Jones \\cite{GJ} who constructed an ${A_{\\infty}}$ structure on the negative cyclic complex of any commutative algebra $C$. We adapt their definition to the case when $C$ is {\\em a brace algebra}, in particular the Hochschild cochain complex. \n\nNote that all our constructions can be carried out for a unital ${A_{\\infty}}$ algebra $A$. The Hochschild and cyclic complexes of ${A_{\\infty}}$ algebras are introduced in \\cite{GJ}; as shown in \\cite{G}, the Hochschild cochain complex becomes an ${A_{\\infty}}$ algebra; all the formulas in this section are good for the more general case. In fact they are easier to write using the ${A_{\\infty}}$ language, even if $A$ is a usual algebra.\n\nRecall \\cite{LS}, \\cite{St} that an $A_{\\infty}$ algebra is a graded vector space ${\\cal{C}}$ together with a Hochschild cochain $m$ of total degree $1$, \n$$ m = \\sum _{n=1}^{\\infty} m_n$$\nwhere $m_n \\in C^n({\\cal{C}})$ and \n$$[m,m] = 0$$\n\nConsider the Hochschild cochain complex of a graded algebra $A$ as a differential graded associative algebra $(C^{\\bullet}(A), \\; \\smile, \\; \\delta)$. Consider the Hochschild {\\it chain} complex of this differential graded algebra. The total differential in this complex is $b+ \\delta$; the degree of a chain is given by \n$$|D_0 \\otimes \\ldots \\otimes D_n | = |D_0| + \\sum _{i=1}^{n}(|D_i|+1)$$\nwhere $D_i$ are Hochschild cochains.\n\nThe complex $C_{\\bullet}(C^{\\bullet}(A))$ contains the Hochschild cochain complex $C^{\\bullet}(A)$ as a subcomplex (of zero-chains) and has the Hochschild chain complex $C_{\\bullet}(A)$ as a quotient complex:\n$$C^{\\bullet}(A) \\stackrel {i}{\\longrightarrow} C_{\\bullet}(C^{\\bullet}(A))\\stackrel {\\pi}{\\longrightarrow}C_{\\bullet}(A)$$\n(this sequence is not by any means exact). The projection on the right splits if $A$ is commutative. If not, $C_{\\bullet}(A)$ is naturally a graded subspace but not a subcomplex.\n\n\\begin{thm} \\label{thm: a-infty}\nThere is an $A_{\\infty}$ structure ${\\bf m}$ on $C_{\\bullet}(C^{\\bullet}(A))[[u]]$ such that:\n\\begin{itemize}\n\\item All ${\\bf m}_n$ are $k[[u]]$-linear, $(u)$-adically continuous\n\\item ${\\bf m}_1 = b+\\delta +uB$\n\nFor $x,\\;y \\in C_{\\bullet}(A)$:\n\\item $(-1)^{|x|}{\\bf m}_2 (x,y) = (\\operatorname{sh} + u \\operatorname{sh}')(x,y)$\n\nFor $D,\\;E \\in C^{\\bullet}(A)$:\n\\item $(-1)^{|D|}{\\bf m}_2(D,E) = D \\smile E$\n\\item ${\\bf m}_2(1\\otimes D, \\; 1 \\otimes E) + (-1)^{|D||E|}{\\bf m}_2(1\\otimes E, \\; 1 \\otimes D) = (-1)^{|D|}1 \\otimes [D,\\; E]$\n\\item ${\\bf m}_2( D, \\; 1 \\otimes E) + (-1)^{(|D|+1)|E|}{\\bf m}_2(1\\otimes E, \\;D) = (-1)^{|D|+1}[D,\\; E]$\n\\end{itemize}\n\\end{thm}\n\nHere is an explicit description of the above $A_{\\infty}$ structure. We define for $n \\geq 2$ \n$${\\bf m}_n = {\\bf m}_n^{(1)} + u{\\bf m}_n^{(2)}$$\nwhere, for \n$$a^{(k)} = D_0^{(k)} \\otimes \\ldots \\otimes D_{N_k}^{(k)}, $$\n\n\n$${\\bf m}_n^{(1)}(a^{(1)}, \\ldots, a^{(n)}) = \\sum \\pm \n{ m}_k \\{{\\underline{\\ldots}} , D^{(0)}_0\\{{\\underline{\\ldots}}\\} , {\\underline{\\ldots}} , D^{(n)}_0 \\{{\\underline{\\ldots}}\\}{\\underline{\\ldots}}\\}\\otimes {\\underline{\\ldots}}\n$$\nThe space designated by $\\;\\underline \\;$ is filled with $D_i^{(j)},\\;i>0,$ in such a way that:\n\\begin{itemize}\n\\item the cyclic order of each group $D_0 ^{(k)}, \\ldots, D_{N_k}^{(k)}$ is preserved;\n\\item any cochain $D^{(i)}_j$ may contain some of its neighbors on the right inside the braces, provided that all of these neighbors are of the form $D^{(p)}_q$ with $p < i$.\nThe sign convention: any permutation contributes to the sign; the parity of $D^{(i)}_j$ is always $|D^{(i)}_j|+1 $.\n\\end{itemize}\n\n\n$${\\bf m}_n^{(2)}(a^{(1)}, \\ldots, a^{(n)}) = \\sum \\pm 1 \\otimes \n{\\underline{\\ldots}} \\otimes D^{(0)}_0\\{{\\underline{\\ldots}}\\} \\otimes {\\underline{\\ldots}} \\otimes D^{(n)}_0\\{{\\underline{\\ldots}}\\} \\otimes {\\underline{\\ldots}}\n$$\nThe space designated by $\\;\\underline \\;$ is filled with $D_i^{(j)},\\;i>0,$ in such a way that:\n\\begin{itemize}\n\\item the cyclic order of each group $D_0 ^{(k)}, \\ldots, D_{N_k}^{(k)}$ is preserved;\n\\item any cochain $D^{(i)}_j$ may contain some of its neighbors on the right inside the braces, provided that all of these neighbors are of the form $D^{(p)}_q$ with $p < i$.\nThe sign convention: any permutation contributes to the sign; the parity of $D^{(i)}_j$ is always $|D^{(i)}_j|+1 $.\n\\end{itemize}\n\n\\begin{remark} \\label{rmk:hj} Let $A$ be a commutative algebra. Then $C_{\\bullet}(A)[[u]]$ is not only a subcomplex but an $A_{\\infty}$ subalgebra of $C_{\\bullet}(C^{\\bullet}(A))[[u]]$. The $A_{\\infty}$ structure on $C_{\\bullet}(A)[[u]]$ is the one from \\cite{GJ}. \n\\end{remark}\n{\\bf Proof of the Theorem.} First let us prove that ${\\bf m}^{(1)}$ is an ${A_{\\infty}}$ structure on $C_{\\bullet}(C^{\\bullet}(A))$. Decompose it into the sum $\\delta + {\\widetilde {\\bf m}^{(1)}}$ where $\\delta$ is the differential induced by the differential on $C^{\\bullet}(A)$. We want to prove that $[\\delta, {\\widetilde {\\bf m}^{(1)}}]+\\frac{1}{2}[{\\widetilde {\\bf m}^{(1)}},{\\widetilde {\\bf m}^{(1)}}]=0$. We first compute $\\frac{1}{2}[{\\widetilde {\\bf m}^{(1)}},{\\widetilde {\\bf m}^{(1)}}]$. It consists of the following terms:\n\n(1) $ m\\{\\ldots D_0^{(1)} \\ldots m\\{ \\ldots D_0^{(i+1)} \\ldots D_0^{(j)}\\ldots\\} \\ldots D_0^{(n)} \\ldots\\}\\otimes \\ldots$\n\nwhere the only elements allowed inside the inner $m\\{\\ldots\\}$ are $D_p^{(q)}$ with $i+1\\leq q\\leq j;$\n\n(2) $ m\\{\\ldots D_0^{(1)} \\ldots m\\{ \\ldots \\} \\ldots D_0^{(n)} \\ldots\\}\\otimes \\ldots$\n\nwhere the only elements allowed inside the inner $m\\{\\ldots\\}$ are $D_p^{(q)}$ for one and only $q$ (these are the contributions of the term ${\\widetilde {\\bf m}^{(1)}}(a^{(1)}, \\ldots, ba^{(q)}, \\ldots, a^{(n)}$);\n\n(3) $ m\\{\\ldots D_0^{(1)} \\ldots D_0^{(n)} \\ldots\\} \\otimes \\ldots \\otimes m\\{ \\ldots \\} \\otimes \\ldots$\n\nwith the only requirement that the second $m\\{\\ldots\\}$ should contain elements $D_p^{(q)}$ and $D_{p'}^{(q')}$ with $q\\neq q'$.(The terms in which the second $m\\{\\ldots\\}$ contains $D_p^{(q)}$ where all $q$'s are the same cancel out: they enter twice, as contributions from $b{\\widetilde {\\bf m}^{(1)}}(a^{(1)}, \\ldots, a^{(q)}, \\ldots, a^{(n)}$ and from ${\\widetilde {\\bf m}^{(1)}}(a^{(1)}, \\ldots, ba^{(1)}, \\ldots, a^{(n)}$). \n\nThe collections of terms (1) and (2) differ from \n\n(0) $ \\frac{1}{2}[m,m]\\{\\ldots D_0^{(1)} \\ldots \\ldots D_0^{(n)} \\ldots\\}\\otimes \\ldots$\n\nby the sum of all the following terms:\n\n$(1')$ terms as in (1), but with a requirement that in the inside $m\\{\\ldots \\}$ an element $D_p^{(q)}$ must me present such that $q\\leq i$ or $q>j;$\n\n$(2')$ terms as in (1), but with a requirement that the inside $m\\{\\ldots \\}$ must contain elements $D_p^{(q)}$ and $D_{p'}^{(q')}$ with $q\\neq q'$.\n\nAssume for a moment that $D_p^{(q)}$ are elements of a commutative algebra (or, more generally, of a $C_{\\infty}$ algebra, i.e. a homotopy commutative algebra). Then there is no $\\delta$ and ${\\widetilde {\\bf m}^{(1)}}={{\\bf m}^{(1)}}.$ But the terms $(1')$ and $(2')$ all cancel out, as well as (3). Indeed, they all involve $m\\{\\ldots\\}$ with some shuffles inside, and $m$ is zero on all shuffles. (the last statement is obvious for a commutative algebra, and is exactly the definition of a $C_{\\infty}$ algebra).\n\nNow, we are in a more complex situation where $D_p^{(q)}$ are Hochschild cochains (or, more generally, elements of a {\\em brace algebra}). Recall that all the formulas above assume that cochains $D_p^{(q)}$ may contain their neighbors on the right inside the braces. We claim that \n\n(A) the terms ($1'$), $(2')$ and (3), together with (0), cancel out with the terms constituting $[\\delta, {\\widetilde {\\bf m}^{(1)}}]$. \n\nTo see this, recall from \\cite{KS1} the following description of brace operations. To any rooted planar tree with marked vertices one can associate an operation on Hochschild cochains. The operation\n$$D\\{\\ldots E_1 \\{\\ldots \\{Z_{1,1}, \\ldots, Z_{1,k_1}\\}, \\ldots\\}\\ldots E_n \\{\\ldots \\{Z_{n,1}, \\ldots, Z_{n,k_n}\\}\\ldots\\} \\ldots \\}$$\ncorresponds to a tree where $D$ is at the root, $E_i$ are connected to $D$ by edges, and so on, with $Z_{ij}$ being external vertices. The edge connecting $D$ to $E_i$ is to the left from the edge connecting $D$ to $E_j$ for $i1.$\nHere $D, \\; D_i, \\; E_j \\in {\\mathfrak{g}}^{\\bullet}_A$ and $Y=E_1 \\ldots E_n \\in S({\\mathfrak{g}}^{\\bullet}_A)^+.$\n\n\\begin{lemma} \\label{lemma:Y bar as A infty morphism}\nFor $Y\\in S({\\mathfrak{g}}^{\\bullet}_A)^+$ and $D\\in {\\mathfrak{g}}^{\\bullet}_A$, \n$${\\overline{({\\operatorname{ad}}_D Y)}}=\\sum D\\{{\\overline{Y}}_1^+,{\\overline{Y}}_2^+, \\ldots, {\\overline{Y}}_n^+ \\}-(-1)^{(|D|+1)(|Y|+1)}{\\overline Y}\\{D\\}.$$\nIn particular, for $Y\\in S({\\mathfrak{g}}^{\\bullet}_A)^+$,\n$${\\overline{(\\delta Y)}}=\\delta {\\overline{Y}}+\\sum m_2\\{{\\overline{Y}}_1^+, {\\overline{Y}}_2^+\\}= \\delta {\\overline{Y}}+\\sum (-1)^{|{Y_1^+}|+1}({\\overline{Y}}_1^+\\smile {\\overline{Y}}_2^+)$$\n\\end{lemma}\nIndeed, let $Y=E_1\\ldots E_n.$ For $n=2$, the lemma follows from Proposition \\ref{prop:brace structure}; it general, it is obtained from the same proposition by induction on $n$.\n \nTo prove the theorem, we have to show that \n\\begin{equation}\\label{eq:right module over twisted bar}\n-(b+uB)(x\\cdot c)+((b+uB)x)\\cdot c)+(-1)^{|x|}x\\cdot\\partial c+(-1)^{|x|}x\\cdot\\delta c+\n\\end{equation}\n\\begin{equation*}\n+\\sum(-1)^{|x|+|c_1^+|+1}x\\cdot c_1^+\\cdot c_2^++(-1)^{|x|}u\\sum\\frac{1}{n!}x\\cdot \\partial_{\\epsilon}c_1\\cdot \\ldots \\cdot \\partial_{\\epsilon}c_n=0\n\\end{equation*}\nLet us start by applying the ${A_{\\infty}}$ identity \n$$\\sum \\sum _{0\\leq i \\leq n} \\pm \\mu _1(\\mu _{n+3}(x, {\\overline{Y}}_1^+, \\ldots ,{\\overline{Y}}_i^+ , 1\\otimes D, {\\overline{Y}}_{i+1}^+ , \\ldots, {\\overline{Y}}_n^+, T)))+\\ldots=0\n$$\nwhere $T=1\\otimes F_1\\otimes \\ldots\\otimes F_m$ is a cycle with respect to $b$ (and, automatically, to $B$). \nBy virtue of Proposition \\ref{prop:key property 1}, all the terms containing $1\\otimes D$ in the middle vanish. The only surviving terms produce the identity\n\\begin{eqnarray*}\\label{eq:identity 1}\n\\sum \\pm \\mu _{n-i+1}(\\mu _{i+2}(x, {\\overline{Y}}_1^+, \\ldots ,{\\overline{Y}}_i^+ , 1\\otimes D), {\\overline{Y}}_{i+1}^+ , \\ldots, {\\overline{Y}}_n^+, T))\\\\\n+\\sum \\pm \\mu_{n+1} (x, {\\overline{Y}}_1^+, \\ldots ,{\\overline{Y}}_i^+\\{D\\} ,\\ldots, {\\overline{Y}}_n^+, T)+\\\\\n+\\sum \\pm \\mu_{n+2-j} (x, {\\overline{Y}}_1^+, \\ldots ,D\\{{\\overline{Y}}_{i+1}^+, \\ldots ,{\\overline{Y}}_{i+j}^+ \\}, \\ldots, {\\overline{Y}}_n^+, T)=0\n\\end{eqnarray*}\nWhen $T=1\\otimes F$, $F\\in {\\mathfrak{g}}^{\\bullet}_A$, we obtain, using the first part of Lemma \\ref{lemma:Y bar as A infty morphism}, the identity \\eqref{eq:right module over twisted bar} for $c=\\epsilon E_1\\wedge \\ldots \\wedge \\epsilon E_n\\wedge D\\wedge F$. An identical computation without a $T$ at the end yields \\eqref{eq:right module over twisted bar} for $c=\\epsilon E_1\\wedge \\ldots \\wedge \\epsilon E_n\\wedge D$. Now apply the ${A_{\\infty}}$ identity \n$$\\sum \\pm \\mu _1(\\mu _{n+1}(x, {\\overline{Y}}_1^+, \\ldots , {\\overline{Y}}_n^+))+\\ldots=0\n$$\nWe obtain\n$$\\sum \\pm \\mu _1(\\mu _{n+1}(x, {\\overline{Y}}_1^+, \\ldots , {\\overline{Y}}_n^+))+\n\\sum \\pm \\mu _{n+1}(\\mu _1( x), {\\overline{Y}}_1^+, \\ldots , {\\overline{Y}}_n^+)+$$\n$$\\sum \\pm \\mu _{n+1}( x, {\\overline{Y}}_1^+, \\ldots ,m_1({\\overline{Y}}_i^+),\\ldots, {\\overline{Y}}_n^+)+\n\\sum \\pm \\mu _{n+1}( x, {\\overline{Y}}_1^+, \\ldots ,m_2\\{{\\overline{Y}}_i^+,{\\overline{Y}}_{i+1}^+\\},\\ldots, {\\overline{Y}}_n^+)+$$\n$$\n\\sum \\pm \\mu_{n-i+1} (\\mu _{i+1}( x, {\\overline{Y}}_1^+, \\ldots ,{\\overline{Y}}_i^+), {\\overline{Y}}_{i+1}^+,\\ldots, {\\overline{Y}}_n^+)+$$\n$$\\sum \\pm \\mu _{i+2}( x, {\\overline{Y}}_1^+, \\ldots ,{\\overline{Y}}_i^+, 1\\otimes {\\overline{Y}}_{i+1}^+\\otimes \\ldots \\otimes {\\overline{Y}}_n^+)=0.$$\nThe first two sums in the above formula correspond to the first two terms in \\eqref{eq:right module over twisted bar}; the second two sums, by virtue of the second part of Lemma \\ref{lemma:Y bar as A infty morphism}, corresponds to the third term of \\eqref{eq:right module over twisted bar};the fourth term of \\eqref{eq:right module over twisted bar} is in our case equal to zero. The fifth sum corresponds to the fifth term of \\eqref{eq:right module over twisted bar}. Now, consider the last sum in the above formula. Use Proposition \\ref{prop:key property 2}, and apply the computation right after \\eqref{eq:right module over twisted bar} in the case when $T=\\sum 1\\otimes {\\overline{Y}}^+_{i+2}\\otimes \\ldots \\otimes {\\overline{Y}}^+_{n}$ and $D=1\\otimes {\\overline{Y}}^+_{i+1}$. Then proceed by induction on $i$. We see that the sixth sum in the formula corresponds to the sixth term of \\eqref{eq:right module over twisted bar}.\n\\subsection{End of the proof}\\label{ss:end of the proof}\nIt remains to pass from $B^{\\operatorname {tw}}({\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u])$ to $U({\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u])$.\n\\begin{lemma}\\label{quis of twisted and untwisted} \nThe formulas \n$$D\\to D;$$\n$$\\epsilon E_1 \\wedge \\ldots \\wedge \\epsilon E_n \\mapsto \\frac{1}{n!}\\sum_{\\sigma \\in S_n}\\frac{1}{n!}(\\epsilon E_{\\sigma_1}) E_{\\sigma_2}\\ldots E_{\\sigma_n};$$\n$$D_1\\wedge \\ldots D_k \\wedge \\epsilon E_1 \\wedge \\ldots \\wedge \\epsilon E_n\\mapsto 0$$\nfor $k>1$ or $k=1, \\, n\\geq 1$\ndefine a quasi-isomorphism of DGAs\n$$B^{\\operatorname {tw}}({\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u]) \\to U({\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u]).$$\n\\end{lemma}\n{\\bf{Proof.}} The fact that the above map is a morphism of DGAs follows from an easy direct computation. \nTo show that this is a quasi-isomorphism, consider the increasing filtration by powers of $\\epsilon$. At the level of graduate quotients, $B^{\\operatorname {tw}}({\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u])$ becomes the standard free resolution of $(U({\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u]), \\delta)$, and the morphism is the standard map from the resolution to the algebra, therefore a quasi-isomorphism. The statement now follows from the comparison argument for spectral sequences.\n\nTo summarize, we have constructed explicitly a DGA $B^{\\operatorname {tw}}({\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u])$ and the morphisms of DGAs\n$$U({\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u]) \\leftarrow B^{\\operatorname {tw}}({\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u]) \\to \\operatorname{End} _{K[[u]]}(\\CCn(A))$$\nwhere the morphism on the left is a quasi-isomorphism. This yields an $A_{\\infty}$ morphism \n$$U({\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u])\\to \\operatorname{End} _{K[[u]]}(\\CCn(A))$$\nand therefore an $L_{\\infty}$ morphism \n$${\\mathfrak{g}}^{\\bullet}_A[\\epsilon, u] \\to \\operatorname{End} _{K[[u]]}(\\CCn(A)).$$\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n The dynamical and physical properties of the extreme trans-Neptunian objects or ETNOs (semimajor axis, $a$, greater than 150 au, and \n perihelion distance, $q$, greater than 30 au, Trujillo \\& Sheppard 2014) are intriguing in many ways. Their study can help probe the \n orbital distribution of putative planets going around the Sun between the orbit of Pluto and the Oort Cloud as well as understand the \n formation and evolution of the Solar system as a whole. The first ETNO was found in 2000, (148209)~2000~CR$_{105}$, and its discovery \n was soon recognised as a turning point in the study of the outer Solar system (e.g. Gladman et al. 2002; Morbidelli \\& Levison 2004). \n The current tally stands at 21 ETNOs. \n\n Trujillo \\& Sheppard (2014) were first in suggesting that the dynamical properties of the ETNOs could be better explained if a yet to \n be discovered planet of several Earth masses is orbiting the Sun at hundreds of au. This interpretation was further supported by de la \n Fuente Marcos \\& de la Fuente Marcos (2014) with a Monte Carlo-based study confirming that the observed patterns in ETNO orbital \n parameter space cannot result from selection effects and suggesting that one or more trans-Plutonian planets may exist. A plausible \n multi-planet dynamical scenario was explored by de la Fuente Marcos, de la Fuente Marcos \\& Aarseth (2015). Based on observational \n data, and analytical and numerical work, Batygin \\& Brown (2016) presented their Planet Nine hypothesis that was further developed by \n Brown \\& Batygin (2016), but questioned by Shankman et al. (2016). The orbits of seven ETNOs ---Sedna, 148209, (474640) 2004~VN$_{112}$, \n 2007~TG$_{422}$, 2010~GB$_{174}$, 2012~VP$_{113}$ and 2013~RF$_{98}$--- were used by Brown \\& Batygin (2016) to predict the existence \n of the so-called Planet Nine, most probably a trans-Plutonian super-Earth in the sub-Neptunian mass range. Out of the 21 known ETNOs, \n only Sedna has been observed spectroscopically (see Fornasier et al. 2009). \n \n Among the known ETNOs, the pair 474640--2013~RF$_{98}$ clearly stands out (de la Fuente Marcos \\& de la Fuente Marcos 2016). The \n directions of their perihelia (those of the vector from the Sun to the respective perihelion point) are very close (angular separation \n of 9\\fdg8), their orbital poles are even closer (4\\fdg1), and consistently the directions of their velocities at perihelion\/aphelion \n are also very near each other (9\\fdg5), although improved values are given in Section~3.3; in addition, they have similar aphelion \n distances, $Q$ (589 au vs. 577 au). Assuming that the angular orbital elements of the ETNOs follow uniform distributions (i.e. they are \n unperturbed asteroids moving in Keplerian orbits around the Sun), the probability of finding by chance two objects with such a small \n angular separation between their directions of perihelia and, what is more important, also between their orbital poles is less than \n 0.0001, which suggests a common dynamical origin. However, a probable common dynamical origin does not imply a common physical origin. \n In an attempt to unravel their physical nature, visible spectroscopy of the two targets was obtained on 2016 September using the OSIRIS \n camera-spectrograph at the 10.4~m Gran Telescopio Canarias (GTC) telescope, located in La Palma (Canary Islands, Spain). Here, we \n present and discuss the results of these observations. This Letter is organized as follows. Section~2 reviews the state of the art for \n this pair of ETNOs. The new observations ---including spectroscopy, photometry and astrometry--- and their results are presented in \n Section~3. The possible origin of this pair is explored in Section~4, making use of the new observational results and $N$-body \n simulations. Conclusions are summarized in Section~5.\n\n \\section{The pair 474640--2013~RF$_\\mathbf{98}$: state of the art}\n Asteroid (474640)~2004~VN$_{112}$ was discovered on 2004 November 6 by the ESSENCE supernova survey observing with the 4~m Blanco \n Telescope from Cerro Tololo International Observatory (CTIO) at an apparent magnitude $R$ of 22.7 (Becker et al. 2007). Its absolute \n magnitude, $H$ = 6.4 (assuming a slope parameter, $G$ = 0.15), suggests a diameter in the range 130--300~km for an assumed albedo in \n the range 0.25--0.05. The orbital solution (2016 August) for this object is based on 31 observations spanning a data-arc of 5113 d or \n 14 yr, from 2000 September 26 to 2014 September 26, its residual rms amounts to 0\\farcs19.\\footnote{Orbit available from JPL's \n Small-Body Database and \\textsc{Horizons} On-Line Ephemeris System: $a=318\\pm1$ au, $e=0.8513\\pm0.0005$, $i=25\\fdg5748\\pm0\\fdg0004$, \n $\\Omega=65\\fdg9990\\pm0\\fdg0007$ and $\\omega=327\\fdg121\\pm0\\fdg010$, referred to the epoch 2457600.5 JD TDB.} Such an object would have \n been visible to ESSENCE for only about 2 per cent of its orbit, suggesting that the size of the population of minor bodies moving in \n orbits similar to that of 474640 could be very significant (Becker et al. 2008). Sheppard (2010) gives a normalised spectral gradient \n of 11$\\pm$4~\\%\/0.1~$\\mu$m for this object based on Sloan $g'$, $r'$, $i'$ optical photometry acquired in 2008. Some additional \n photometry was obtained with the Hubble Wide Field Camera 3 (Fraser \\& Brown 2012). Its optical colours are relatively neutral and this \n was interpreted by Brown (2012) as a sign that it is not dominated by methane irradiation products.\n\n Asteroid 2013~RF$_{98}$ was discovered on 2013 September 12 by the Dark Energy Camera (DECam, Flaugher et al. 2015) observing from CTIO \n for the Dark Energy Survey (DES, Abbott et al. 2005) at an apparent magnitude $z$ of 23.5 (Abbott et al. 2016). Its absolute magnitude, \n $H$ = 8.7, suggests a diameter in the range 50--120~km. The orbital solution (2016 August) for this ETNO is rather poor as it is based \n on 38 observations spanning a data-arc of just 56 d, from 2013 September 12 to 2013 November 7, its residual rms amounts to \n 0\\farcs12.\\footnote{Orbit available from \\textsc{Horizons}: $a=307\\pm37$ au, $e=0.882\\pm0.015$, $i=29\\fdg62\\pm0\\fdg08$, \n $\\Omega=67\\fdg51\\pm0\\fdg13$ and $\\omega=316\\degr\\pm6\\degr$, referred to the epoch 2457600.5 JD TDB.} No other data, besides these 38 \n observations, its $H$ value, and its orbital solutions, are known about this ETNO. \n\n Asteroid 474640 has been classified as an extreme detached object by Sheppard \\& Trujillo (2016) and their integrations show that it is \n a stable ETNO within the standard eight-planets-only Solar system paradigm; this result is consistent with that in Brown \\& Batygin \n (2016). In Sheppard \\& Trujillo (2016), 2013~RF$_{98}$ is classified as an extreme scattered object and found to be unstable within the \n standard paradigm over 10 Myr time-scales due to Neptune perturbations. Both ETNOs are rather unstable within some incarnations of the \n Planet Nine hypothesis (de la Fuente Marcos, de la Fuente Marcos \\& Aarseth 2016). The heliocentric orbits available in 2016 \n August$^{1,2}$ have been used to compute the values of the angular separations quoted in the previous section. In spite of the \n limitations of the orbital solution of 2013~RF$_{98}$, the uncertainties mainly affect orbital elements other than inclination, $i$, \n longitude of the ascending node, $\\Omega$, and argument of the perihelion, $\\omega$. These three orbital parameters are the only ones \n involved in the calculation of the directions of perihelia, location of orbital poles, and directions of velocities at perihelion (de \n la Fuente Marcos \\& de la Fuente Marcos 2016). In sharp contrast, the value of $Q$ of 2013~RF$_{98}$ is affected by a significant \n uncertainty (12 per cent).\n\n \\section{Observations}\n\n \\subsection{Spectroscopy}\n Visible spectra of the two targets were obtained using OSIRIS (Cepa et al. 2000) at 10.4~m GTC. The apparent visual magnitude, $V$, \n at the time of observation was 23.3 for (474640)~2004~VN$_{112}$ (heliocentric distance of 47.7 au and phase of 1\\fdg1) and 24.4 for \n 2013~RF$_{98}$ (36.6 au, 1\\fdg3). For each target, acquisition images in the Sloan $r'$ filter were obtained in separate nights in \n order to identify reliably the object in the field of view. This procedure ended up being the most efficient to detect these dim, \n slow-moving (apparent proper motion $<$2\\arcsec\/h) targets. Visible spectra were acquired using the low-resolution R300R grism \n (resolution of 348 measured at a central wavelength of 6635~\\AA\\ for a 0\\farcs6 slit width), that covers the wavelength range from \n 0.49 to 0.92 $\\mu$m, and a 2\\arcsec slit width. Two widely accepted solar analogue stars from Landolt (1992) ---SA93-101 and \n SA115-271--- were observed using the same spectral configuration at an airmass identical to that of the targets to obtain the \n reflectance spectra of the ETNOs. For a given ETNO, the spectrum was then divided by the corresponding spectrum of the solar analog. \n Additional data reduction details are described by de Le\\'on et al. (2016) and Morate et al. (2016). For 474640 we acquired two \n spectra of 1800~s each, while for 2013~RF$_{98}$ we acquired four individual spectra of 1800~s each. Observational details are shown \n in Table \\ref{spec}. The resulting individual reflectance spectra, normalised to unity at 0.55~$\\mu$m and offset vertically for \n clarity, are shown in Fig.~\\ref{specs}.\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{VN112RF98indiv.eps}\n \\caption{Individual visible spectra of (474640)~2004~VN$_{112}$ (top panel) and 2013~RF$_{98}$ (bottom panel) normalised to unity \n at 0.55 $\\mu$m and offset vertically for clarity. Red lines correspond to the linear fitting in the range 0.5--0.9 $\\mu$m \n to compute the spectral slope.\n }\n \\label{specs}\n \\end{figure}\n \\begin{table}\n \\centering\n \\fontsize{8}{11pt}\\selectfont\n \\tabcolsep 0.11truecm\n \\caption{Observational details of the visible spectra (1800~s each) obtained for (474640)~2004~VN$_{112}$ and 2013~RF$_{98}$ (see \n the text for further details).\n }\n \\begin{tabular}{cccccc}\n \\hline\n Target & Spec \\# & Date & UT Start & Airmass & Slope (\\%\/0.1~$\\mu$m) \\\\\n \\hline\n & 1 & 09-03-16 & 04:37 & 1.085 & 10.2$\\pm$0.6 \\\\\n 474640 & 2 & 09-03-16 & 05:07 & 1.072 & 13.3$\\pm$0.6 \\\\\n & & & &\\bf{Mean}& \\bf{12$\\mathbf{\\pm}$2} \\\\\n \\hline\n & 1 & 09-08-16 & 03:07 & 1.234 & 14.9$\\pm$0.8 \\\\\n & 2 & 09-08-16 & 03:37 & 1.181 & 12.3$\\pm$0.8 \\\\ \n 2013~RF$_{98}$ & 3 & 09-08-16 & 04:07 & 1.151 & 17.1$\\pm$0.8 \\\\\n & 4 & 09-08-16 & 04:38 & 1.142 & 16.1$\\pm$0.8 \\\\\n & & & &\\bf{Mean}& \\bf{15$\\mathbf{\\pm}$2} \\\\\n \\hline\n \\end{tabular}\n \\label{spec}\n \\end{table}\n\n In Table \\ref{spec}, the spectral slope (in units of \\%\/0.1~$\\mu$m) has been computed from a linear fitting to the spectrum in the \n wavelength range 0.5--0.9 $\\mu$m. We used an iterative process ---removing a total of 50 points randomly distributed in the spectrum \n and performing a linear fitting--- and obtained a value of the slope for each iteration. The resulting slope value is the mean of a \n total of 100 iterations and the associated error is the standard deviation of this mean. The process of dividing by the spectra of \n the solar analogue stars introduces an error of 0.3~\\%\/0.1~$\\mu$m. The error value shown in the last column of Table \\ref{spec} is \n sum of these two contributions. For each target, we averaged individual spectra to obtain the average visible spectra for 474640 and \n 2013~RF$_{98}$ shown in Fig. \\ref{specBOTH}; the mean spectral slopes in Table \\ref{spec} are the means of the individual spectra \n and their errors, the associated standard deviations. The value of the spectral slope of 474640 is consistent with the one obtained \n by Sheppard (2010). The values in Table \\ref{spec} are similar to those of scattered disc TNOs, Plutinos, high-inclination classical \n TNOs as well as the Damocloids and comets (see e.g. table 5 in Sheppard 2010).\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{newcomparison.eps}\n \\caption{Final visible spectra of the pair of ETNOs (see the text for details). \n }\n \\label{specBOTH}\n \\end{figure}\n\n The spectra in Fig. \\ref{specBOTH} show an obvious resemblance, but the low S\/N makes the identification of absorption features \n difficult. In order to make a better comparison and assuming that the relative reflectance is both slowly varying and corrupted by \n random noise, we applied a Savitzky-Golay filter (Savitzky \\& Golay 1964) to both datasets. Such filters provide smoothing without \n loss of resolution, approximating the underlying function by a polynomial as described by e.g. Press et al. (1992). Fig. \\ref{savgol} \n shows the smoothed spectra after applying a 65 point Savitzky-Golay filter of order 6 to the data in Fig. \\ref{specBOTH}. The \n smoothed spectra, in particular that of 474640 (blue), show some weak features that might be tentatively identified as pure methane \n ice absorption bands (Grundy, Schmitt \\& Quirico 2002). However, the most prominent methane band at 0.73~$\\mu$m is observed neither \n in the spectrum of 474640 nor in that of 2013~RF$_{98}$ (red). The S\/N is insufficient to identify reliably any absorption band, but \n the spectral slopes in the visible of both objects provide some compositional information. Objects with visible spectral slopes in \n the range 0--10~\\%\/0.1~$\\mu$m can have pure ices on their surfaces (like Eris, Pluto, Makemake and Haumea), as well as highly \n processed carbon. Slightly red slopes (5--15~\\%\/0.1~$\\mu$m) indicate the possible presence of amorphous silicates as in the case of \n Trojans (Emery \\& Brown 2004) or Thereus (Licandro \\& Pinilla-Alonso 2005). In any case, the objects will not have a surface \n dominated by complex organics (tholins). Differences between the spectra of 474640 and 2013~RF$_{98}$ might be the result of their \n present-day heliocentric distance. Mechanisms that are more efficient in altering the icy surfaces of these objects at smaller \n perihelion distances include sublimation of volatiles and micrometeoroid bombardment (Santos-Sanz et al. 2009).\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{savgol.eps}\n \\caption{Comparison between the spectra of (474640)~2004~VN$_{112}$ and 2013~RF$_{98}$ smoothed by a Savitzky-Golay filter (see the \n text) and scaled to match at 0.60~$\\mu$m. The most prominent absorption band of pure methane ice at 0.73~$\\mu$m is not \n seen on either spectra. \n }\n \\label{savgol}\n \\end{figure}\n\n \\subsection{Photometry}\n We obtained a total of 3 and 11 acquisition images to identify (474640)~2004~VN$_{112}$ and 2013~RF$_{98}$, respectively. Images \n were taken using the Sloan $r'$ filter and calibrated using the zero point values computed for the corresponding nights. The \n resulting $r'$ magnitudes and their uncertainties are shown in Table \\ref{phot}.\n\n \\subsection{Astrometry}\n We used the acquisition images to compute the celestial coordinates of each target and improve its orbit. This was particularly \n relevant for 2013~RF$_{98}$ that prior to our observations had a rather uncertain orbital determination.$^{2}$ We found \n (474640)~2004~VN$_{112}$ within 1\\arcsec of the predicted ephemerides, but 2013~RF$_{98}$ was found nearly 1\\farcm2 away. Both ETNOs \n were in Cetus. Astrometric calibration of the CCD frames was performed using the algorithms of the {\\it Astrometry.net} system (Lang \n et al. 2010). The quality of high-precision astrometry with OSIRIS at GTC matches that of data acquired with FORS2\/VLT (Sahlmann et \n al. 2016). The collected astrometry is shown in Tables \\ref{astrometry2004VN112} and \\ref{astrometry2013RF98}. The new orbital \n solution for 2013~RF$_{98}$ available from \\textsc{Horizons} (as of 2016 December 18 18:51:59 UT) is based on 51 observations \n spanning a data-arc of 1092 d, its residual rms amounts to 0\\farcs12: $a=349\\pm11$ au, $e=0.897\\pm0.003$, $i=29\\fdg572\\pm0\\fdg003$, \n $\\Omega=67\\fdg596\\pm0\\fdg005$ and $\\omega=311\\fdg8\\pm0\\fdg6$, referred to the epoch 2457800.5 JD TDB; the time of perihelion passage \n is 2455125$\\pm$95 JED (2009 October 20.7289 UT). The time of perihelion passage for 474640 is 2009 August 25.8290 UT. Using the \n new orbit, the directions of the perihelia of this pair are separated by 14\\fdg2$\\pm$0\\fdg6, their orbital poles by \n 4\\fdg056$\\pm$0\\fdg003, and the directions of their velocities at perihelion\/aphelion by 14\\fdg1$\\pm$0\\fdg6. \n\n \\section{Origin of the pair 474640--2013~RF$_\\mathbf{98}$: fragmentation vs. binary dissociation}\n From the spectral analysis discussed above, we have found that the members of the pair (474640)~2004~VN$_{112}$--2013~RF$_{98}$ show \n similar spectral slopes, very different from that of Sedna which has ultra-red surface material (spectral gradient of about \n 26~\\%\/0.1~$\\mu$m according to Sheppard 2010, and 42~\\%\/0.1~$\\mu$m according to Fornasier et al. 2009) but compatible with those of \n (148209)~2000~CR$_{105}$ (spectral gradient of about 14~\\%\/0.1~$\\mu$m, Sheppard 2010) and 2012~VN$_{113}$ (spectral gradient of about \n 13~\\%\/0.1~$\\mu$m, Trujillo \\& Sheppard 2014). These five objects have been included in the group of seven singled out as relevant to \n the Planet Nine hypothesis (Brown \\& Batygin 2016). Such spectral differences suggest that the region of origin of the pair \n 474640--2013~RF$_{98}$ may coincide with that of 148209 and 2012~VN$_{113}$ but not with Sedna's, which is thought to come from the \n inner Oort Cloud (Sheppard 2010). Other ETNOs with values of their spectral gradient in Sheppard (2010) are 2002~GB$_{32}$ \n ($\\sim$17~\\%\/0.1~$\\mu$m) and 2003~HB$_{57}$ ($\\sim$13~\\%\/0.1~$\\mu$m).\n\n Objects with both similar directions of the orbital poles and perihelia could be part of a group of common physical origin (\\\"Opik \n 1971). This particular pair of ETNOs is very unusual and a model analogous to the one used by de la Fuente Marcos \\& de la Fuente \n Marcos (2014) to study the overall visibility of the ETNO population predicts that the probability of finding such a pair by chance is \n less than 0.0002. This model uses the new orbital solutions and assumes an unperturbed asteroid population moving in heliocentric \n orbits. Following \\\"Opik (1971), there are two independent scenarios that could explain this level of coincidence: (1) a large object \n broke up relatively recently at perihelion and these two ETNOs are fragments, or (2) both ETNOs were kicked by an unseen perturber at \n aphelion. Sekanina (2001) has shown that minor bodies resulting from a fragmentation episode at perihelion must have very different \n times of perihelion passage. The fragmentation episode at perihelion can thus be readily discarded as the difference in time of \n perihelion passage for this pair is less than a year. \n\n The second scenario pointed out above implies the presence of an unseen massive perturber, i.e. a trans-Plutonian planet. Close \n encounters between minor bodies and planets can induce fragmentation directly via tidal forces (e.g. Sharma, Jenkins \\& Burns 2006) or \n indirectly by exciting rapid rotation (e.g. Scheeres et al. 2000; Ortiz et al. 2012). Alternatively, wide binary asteroids can be \n easily disrupted during close encounters with planets. The existence of wide binaries among the populations of minor bodies orbiting \n beyond Neptune is well documented (e.g. Parker et al. 2011). Wide binary asteroids have very low binding energies and can be easily \n dissociated during close encounters with planets (e.g. Agnor \\& Hamilton 2006; Parker \\& Kavelaars 2010). Binary asteroid dissociation \n may be able to explain the properties of this pair of ETNOs, but only if there is a massive unseen perturber orbiting the Sun well \n beyond Pluto.\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{angle.eps}\n \\caption{Evolution of the angular separation between the orbital poles of the pair (474640)~2004~VN$_{112}$--2013~RF$_{98}$ for three \n representative test calculations with different perturbers. Red: $a$ = 549 au, $e$ = 0.21, $i$ = 47\\degr, $m$ = 16 \n $M_{\\oplus}$. Blue: $a$ = 448 au, $e$ = 0.16, $i$ = 33\\degr, $m$ = 12 $M_{\\oplus}$. Green: $a$ = 421 au, $e$ = 0.10, \n $i$ = 33\\degr, $m$ = 12 $M_{\\oplus}$.\n }\n \\label{seppoles}\n \\end{figure}\n\n In order to test the viability of this hypothesis, we have performed thousands of numerical experiments following the prescriptions \n discussed by de la Fuente Marcos et al. (2016) and aimed at finding the most probable orbital properties of a putative perturber able \n to tilt the orbital plane of the pair 474640--2013~RF$_{98}$ from an initial angular separation close to zero at dissociation to the \n current value of nearly 4\\degr. These simulations involve $N$-body integrations backwards in time under the influence of an unseen \n perturber with varying orbital and physical parameters (per numerical experiment). Our preliminary results indicate that a planet with \n mass, $m$, in the range 10--20 $M_{\\oplus}$ moving in an eccentric (0.1--0.4) and inclined (20--50\\degr) orbit with semimajor axis of \n 300--600 au, may be able to induce the observed tilt on a time-scale of 5--10 Myr. Perturbers with $m<10\\ M_{\\oplus}$ or $a>600$ au are \n unable to produce the desired effect. Fig. \\ref{seppoles} illustrates the typical outcome of these calculations. A detailed account of \n these numerical experiments will be presented in an accompanying paper (de la Fuente Marcos, de la Fuente Marcos \\& Aarseth, in \n preparation). The orbital parameters of this putative planet are somewhat consistent with those of the object discussed by Holman \\& \n Payne (2016). Super-Earths may form at 125--750 au from the Sun (Kenyon \\& Bromley 2015, 2016). Our analysis favours a scenario in \n which 474640--2013~RF$_{98}$ were once a binary asteroid that became unbound after a relatively recent gravitational encounter with a \n trans-Plutonian planet at hundreds of au from the Sun. An alternative explanation involving an asteroid break-up near aphelion, also \n after a close encounter with a planet, is possible but less probable because it requires an approach at closer range, 20 planetary \n radii versus 0.8~au for binary dissociation.\n\n \\section{Conclusions}\n In this Letter, we provide for the first time direct indication of the surface composition of the ETNOs (474640)~2004~VN$_{112}$ and \n 2013~RF$_{98}$. Both objects are too faint for infrared spectroscopy, but our results show that they are viable targets for visible \n spectroscopy. The analysis of our results gives further support to the trans-Plutonian planets paradigm that predicts the presence of \n one or more planetary bodies well beyond Pluto. Summarizing: \n \\begin{enumerate}[(i)]\n \\item Our estimate of the spectral slope for 474640 is 12$\\pm$2~\\%\/0.1~$\\mu$m and for 2013~RF$_{98}$ is 15$\\pm$2~\\%\/0.1~$\\mu$m. \n These values suggest that the surfaces of these ETNOs can have pure methane ices (like Pluto) and highly processed carbons, \n including some amorphous silicates.\n \\item Although the spectra of the pair 474640--2013~RF$_{98}$ are not perfect matches, the resemblance is significant and\n the disparities observed might be the result of their different present-day heliocentric distance.\n \\item By improving the orbital solution of 2013~RF$_{98}$, we confirm that the pair 474640--2013~RF$_{98}$ has unusual \n relative dynamical properties. The directions of their perihelia are separated by 14\\fdg2 and their orbital poles are 4\\fdg1 \n apart. \n \\item Our numerical analysis favours a scenario in which 474640--2013~RF$_{98}$ were once a binary asteroid that became \n unbound after an encounter with a trans-Plutonian planet at very large heliocentric distance. \n \\end{enumerate}\n\n \\section*{Acknowledgements}\n We thank the anonymous referee for a constructive and detailed report, and S.~J. Aarseth for providing one of the codes used in this \n research and for his helpful comments on binary dissociation. Based on observations made with the Gran Telescopio Canarias; we are \n grateful to all the technical staff and telescope operators for their assistance with the observations. This work was partially \n supported by the Spanish `Ministerio de Econom\\'{\\i}a y Competitividad' (MINECO) under grant ESP2014-54243-R. JdL acknowledges \n financial support from MINECO under the 2015 Severo Ochoa Program MINECO SEV-2015-0548. CdlFM and RdlFM thank A. I. G\\'omez de Castro, \n I. Lizasoain and L. Hern\\'andez Y\\'a\\~nez of the Universidad Complutense de Madrid (UCM) for providing access to computing facilities. \n Part of the calculations and the data analysis were completed on the EOLO cluster of the UCM, and CdlFM and RdlFM thank S. Cano Als\\'ua \n for his help during this stage. EOLO, the HPC of Climate Change of the International Campus of Excellence of Moncloa, is funded by the \n MECD and MICINN. This is a contribution to the CEI Moncloa. In preparation of this Letter, we made use of the NASA Astrophysics Data \n System, the ASTRO-PH e-print server and the MPC data server.\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe discovery of the high-$T_c$ superconductivity in doped\ncuprates,\\cite{Muller} observation of many unconventional properties in doped\nmanganites with their colossal magnetoresistance, bismuthates with\nhigh-$T_c$'s, nickellates and many other oxides \\cite{Imada} shows that we deal\nwith a manifestation of novel strongly correlated states with a local charge\ninstability, mixed valence, \"metal-dielectric\" duality, strong coupling of\ndifferent (charge, spin, orbital, structural) degrees of freedom and non-Landau\nbehaviour of quasiparticles. All this has generated a flurry of ideas,\n models and scenarios of the puzzling transport phenomena and\n stimulated the intensive studies of various correlation effects and charge transfer (CT) phenomena\n in strongly correlated systems derived in either way from insulators unstable\n with regard to the CT fluctuations.\nConventional approach to hotly debated strongly correlated 3d oxides such as cuprates, manganites, and many other similar systems implies making use of a Hubbard model with famous Hamiltonian \n \n\\begin{equation}\n\\hat H = -\\sum _{,\\sigma} t(ij){\\hat c}^{\\dag}_{i\\sigma}{\\hat\nc}_{j\\sigma} +U\\sum _{i,\\sigma \\sigma ^{\\prime}}n_{i\\sigma}n_{i\\sigma\n^{\\prime}}\t\\label{Hubbard}\n\\end{equation}\nwith competing contributions of kinetic and potential terms. Here ${\\hat c}^{\\dag}_{i\\sigma}\/{\\hat\nc}_{j\\sigma}$ are creation\/annihilation operators for low-lying antibonding 3d-O 2p hybridized orbitals. The two-center charge transfer integral $t(ij)$ is often associated with d-d transfer. Mott-Hubbard insulator is believed to arise from a potentially metallic half-filled band as a result of the Coulomb blockade of electron tunnelling ($U\\gg t$) to neighboring sites.\\cite{Mott}\n\nDespite intense effort, the behavior of strongly correlated 3d oxides remain poorly understood and we are still far from a comprehensive understanding of the underlying physics. Moreover, it seems that there are missing qualitative aspects of the problem beyond the simple Hubbard scenario that so far escaped the identification and the recognition. Firstly it concerns strong electron-lattice polarization effects which may be subdivided into\n electron-lattice interaction itself,\\cite{Shluger,Vikhnin} and a contribution of an electronic background that is electronic subsystem which is not incorporated into effective Hubbard model Hamiltonian.\\cite{Hirsch,shift} \n These effects are of great importance for the\nground state electronic and crystalline structure, and can seriously modify\nthe doping response of 3d oxide up to the crucial change of the seemingly\nnatural ground state. This question has not received the attention it deserves. It should be emphasized that traditional Fr\\\"{o}hlich approach to the\nelectron-lattice coupling implies the description of linear effects whereas\nthe charge fluctuations in the insulator do imply strongly nonlinear\nelectron-lattice coupling with the predominance of polarization and relaxation\neffects, and another energy scale. \n \n \n Electron-lattice effects may be directly incorporated into effective Hubbard model. Assuming the coupling with the local displacement (configuration) coordinate $Q$ in the effective potential energy we\narrive at a generalized Peierls-Hubbard model.\\cite{Nasu} From the other hand, the\ntaking account of similar effects in the kinetic energy results in a generalized Su-Schrieffer-Heeger (SSH)\nmodel.\\cite{SSH} The correlation effect of an electronic background was shown \\cite{Hirsch,shift} to be of primary importance for atomic systems with filled or almost filled electron shells. Namely such a situation is realized in oxides with O$^{2-}($2p$^6$) oxygen ions. In particular, the effect results in a correlated character of a charge transfer that seems to be one of the main features for 3d oxides.\n\n\nMany strongly correlated 3d oxides reveal anomalous sensitivity to a small nonisovalent substitution. For example, only 2\\% Sr$^{2+}$ substituted for La$^{3+}$ in La$_2$CuO$_4$ result in a dramatical suppression of long-range copper antiferromagnetism, while it is suppressed with isovalent Cu$^{2+}$ substitution by Zn$^{2+}$ at a much higher concentration close to the site dilution percolating threshold. Simultaneously, the transport properties of La$_{2-x}$Sr$_x$CuO$_{4}$ system reveal unconventional insulator-metal duality starting from very low dopant level.\\cite{Ando} Most likely, all this points to a charge phase instability intrinsic for parent 214 system which somehow evolves with nonisovalent substitution due to a well developed charge potential inhomogeneity and\/or hole doping effect. The problem seems to be closely related with the hidden $multistability$ intrinsic to each\nsolid.\\cite{Toyozawa,Nasu} If the ground state of a solid is\npseudo-degenerate, being composed of true and false ground states with each\nstructural and electronic orders different from others, one might call it {\\it\nmulti-stable}. Below we focus ourselves on a charge degree of freedom and\ncharge (in)stability, rather than orbital or spin degrees of freedom. As an\nilluminating example of such a material with {\\it conceptually simple} but {\\it\nactually false} ground state Toyozawa \\cite{Toyozawa} suggests to address the\nWolfram's red, a quasi-one dimensional material of which the skeleton chain\nconsists of alternate array: (Cl$^-$ - Pt$^{3+}$ -)$^{2n}$ with simple (and\nseemingly metallic), but a false ground state. The real ground state is an\ninsulator with a complicated structure of doubled period: (Cl$^{-}$ - Pt$^{4+}$\n- Cl$^{-}$ - Pt$^{2+}$-)$^n$, which can be reached from the former through the\nPeierls transition with the charge density wave of large amplitude, or\ndisproportionation like reaction. This transition can be considered as the\ncondensation of self-decomposed self-trapped excitons spontaneously generated\non all unit cells.\n\nIn this connection it is worth noting the text-book example of BaBiO$_3$ system\nwhere we unexpectedly deal with the disproportionated Ba$^{3+}$+ Ba$^{5+}$\nground state instead of the conventional lattice of Ba$^{4+}$\ncations.\\cite{Kagan} The bismuthate situation can be viewed also as a result of\na condensation of CT excitons, in other words, the spontaneous\ngeneration of self-trapped CT excitons in the ground state\nwith a proper transformation of lattice parameters.\n At present, a CT instability with regard to disproportionation is believed to be a rather\ntypical property for a number of perovskite 3d transition-metal oxides\n such as SrFeO$_3$, LaCuO$_3$, RNiO$_3$ \\cite{Mizokawa}, moreover, in solid state chemistry one consider tens of\ndisproportionated systems.\\cite{Ionov} \nNew principles must be developed to treat such charge or CT unstable systems with their dramatical non-Fermi-liquid behavior. In particular, we have to change the current paradigm\nof the metal-to-insulator (MI) transition to that of an insulator-to-metal (IM)\nphase transition. These two approaches imply essentially different starting\npoints: the former starts from a rather simple {\\it metallic-like} scenario with inclusion of\ncorrelation effects, while the latter does from strongly correlated {\\it atomic-like}\nscenario with the inclusion of a charge transfer. Electron-lattice polarization effects accompanying the charge transfer appear to be of primary importance to stabilize either phase state. One should emphasize that\nthe theoretical description of such systems is one of the challenging problems in solid state physics.\n \nHereafter, we develop a model approach to describe different charge fluctuations and charge phases in strongly correlated 3d oxides with main focus on the correlated CT effects. As an illustrative model system we address a simple mixed-valence system with three possible stable nondegenerate valent states of a cation-anionic cluster, hereafter $M$: $M^0, M^{\\pm}$, forming the charge (isospin) triplet. The $M^0$ valent state is associated\n with the {\\it conceptually simple} one like CuO$_{4}^{6-}$ in insulating\n copper oxides (CuO, La$_2$CuO$_4$, YBa$_2$Cu$_3$O$_6$, Sr$_2$CuO$_2$Cl$_2$,...)\n or MnO$_{6}^{9-}$ in manganite LaMnO$_3$ or BiO$_{6}^{9-}$ in bismuthates. It is worth noting that such a model is a most relevant to describe different cuprates where novel concepts should compete with a\n traditional Hubbard model approach in a hole representation implying the vacuum state formed by $M^-$ (CuO$_{4}^{5-}$) centers, and some concentration of holes. That is why overall the paper we refer the insulating cuprates to illustrate the main concepts of the approach developed. Our mathematics is based on the $S=1$ pseudo-spin formalism (see e.g. review article Ref.\\onlinecite{pseudo}) to be the effective tool for the\ndescription of the essential physics both of insulators unstable with regard to the CT fluctuations and related mixed-valence systems. Such an approach provides the universal framework for a unified\n description of these systems as possible phase states of a certain {\\it parent\n multi-stable system}. In addition, we may make use of powerful methods developed in\n the physics of spin systems. The model system of $M^{0,\\pm}$ centers is \ndescribed in frames of $S=1$ pseudo-spin formalism by an effective anisotropic non-Heisenberg Hamiltonian which includes two types of correlated two-center hopping:\n$$\nM^0 +M^0 \\leftrightarrow M^{\\pm}+M^{\\mp}\\, \\mbox{and} \\,M^{\\pm} + M^0 \\leftrightarrow M^0 +M^{\\pm},\n$$\nrespectively. It should be noted that we neglect all the intra-center transition, including anion-cation O 2p-3d charge transfer.\n\nOur main goal is to describe different charge phases of the model system and a scenario of evolution of visibly typical insulator to unconventional electron-hole Bose liquid which reveals many unexpected properties, including superconductivity.\nThe paper is organized as follows: In Sec.II we address a metal-oxide cluster model, different mechanisms of correlation effects, and the effects of electron-lattice polarization. In Sec.III we introduce the $S=1$\npseudo-spin formalism to describe the model mixed-valence systems. The effective pseudo-spin \nHamiltonian and possible mean-field phase states of the mixed-valence systems are discussed in Sec.IV. In Sec.V we analyse an eh-representation of different excitations in a monovalent $M^0$ phase, discuss a CT instability, and nucleation of electroh-hole (EH) droplets. Electron-hole Bose liquid is discussed in Sec.VI. Some topological skyrmion-like charge\nfluctuations in the model mixed valence system are described in Sec.VII. Implications for cuprates and manganites\nare discussed in Sec.VIII.\n\n\\section{Metal-oxide clusters and correlation effects}\nThe electronic states in strongly correlated 3d oxides manifest\nboth significant correlations and dispersional features. The\ndilemma posed by such a combination is the overwhelming number of\nconfigurations which must be considered in treating strong\ncorrelations in a truly bulk system. One strategy to deal with\nthis dilemma is to restrict oneself to small 3d-metal-oxygen clusters, creating\nmodel Hamiltonians whose spectra may reasonably well represent the\nenergy and dispersion of the important excitations of the full\nproblem. Indeed, such clusters as CuO$_4$ in quasi-2D cuprates, MnO$_6$ in manganite perovskites are basic elements of crystalline and electronic structure.\nDespite a number of principal\nshortcomings, including the boundary conditions, the breaking of\nlocal symmetry of boundary atoms, sharing of common anions for $nn$ clusters etc., the \nembedded molecular cluster method provides both, a clear physical\npicture of the complex electronic structure and the energy\nspectrum, as well as the possibility of quantitative modelling.\nEskes {\\it et al.} \\cite{Eskes}, as well as Ghijsen {\\it et al.}\n\\cite{Ghijsen} have shown that in a certain sense the cluster\ncalculations might provide a better description of the overall\nelectronic structure of insulating 3d oxides than\nband-structure calculations. In particular, they allow to take better into\naccount different correlation effects.\n\nBelow, in the Section we discuss some aspects of electronic structure, energy spectrum, and correlation effects for an illustrative example of CuO$_4$ clusters embedded into an insulating cuprate. \n\n\\subsection{Electronic structure of copper-oxygen clusters}\n\n\nBeginning from 5 Cu $3d$ and 12 O $2p$ atomic orbitals for CuO$_4$\n cluster with $D_{4h}$ symmetry, it is easy to form 17 symmetrized\n $a_{1g},a_{2g},b_{1g},b_{2g},e_{g}$ (gerade=even) and $a_{2u},b_{2u},\n e_{u}(\\sigma),e_{u}(\\pi)$\n (ungerade=odd) orbitals. The even Cu $3d$ $a_{1g}(3d_{z^2}), b_{1g}(3d_{x^2-y^2}),\n b_{2g}(3d_{xy}), e_{g}(3d_{xz},3d_{yz})$\n orbitals hybridize, due to strong Cu $3d$-O $2p$ covalency, with even O\n$2p$-orbitals\n of the same symmetry, thus forming appropriate bonding\n$\\gamma ^{b}$ and antibonding $\\gamma ^{a}$ states. Among the odd\norbitals only $e_{u}(\\sigma)$ and $e_{u}(\\pi)$ hybridize due to\nnearest neighbor $pp$ overlap and transfer thus forming appropriate\nbonding $e_{u}^{b}$ and antibonding $e_{u}^{a}$ purely oxygen\nstates. The purely oxygen $a_{2g},a_{2u},b_{2u}$ orbitals are\nnonbonding.\nAll \"planar\" O $2p$ orbitals in accordance\nwith the orientation of lobes could be classified as $\\sigma$\n($a_{1g},b_{1g},e_{u}(\\sigma)$) or $\\pi$\n($a_{2g},b_{2g},e_{u}(\\pi)$) orbitals, respectively.\n\nBonding and antibonding molecular orbitals in hole representation\ncan be presented, for example, as follows\n$$\n|b_{1g}^{b}\\rangle =\\cos \\alpha _{b_{1g}} |b_{1g}(3d)\\rangle + \\sin\n\\alpha _{b_{1g}}|b_{1g} (2p)\\rangle ,\n$$\n\\begin{equation}\n| b_{1g}^{a}\\rangle =\\sin \\alpha _{b_{1g}} |b_{1g}(3d)\\rangle - \\cos\n\\alpha _{b_{1g}}|b_{1g}(2p)\\rangle ,\n\\end{equation}\nwhere $b_{1g}(3d)=3d_{x^2-y^2}$ and $|b_{1g} (2p)\\rangle$ is a superposition of four O 2p orbitals with $b_{1g}$ symmetry.\n\n\\begin{figure}[t]\n\\includegraphics[width=8.5cm,angle=0]{fig1.eps}\n\\caption{Model single-hole energy spectra for a CuO$_4$ plaquette\nwith parameters relevant for a number of\n insulating cuprates.} \\label{fig1}\n\\end{figure}\nFig. \\ref{fig1} presents a single-hole energy spectrum\nfor a CuO$_4$ plaquette embedded into an insulating cuprate like\nSr$_2$CuO$_2$Cl$_2$ calculated with a reasonable set of parameters.\\cite{CT} For\nillustration we show also a step-by-step formation of the cluster\nenergy levels from the bare Cu $3d$ and O $2p$ levels with the\nsuccessive inclusion of crystalline field (CF) effects, O $2p$-O\n$2p$, and Cu $3d$-O $2p$ covalency. It is worth noting that strong Cu 3d-O 2p overlap and covalency result in a dramatic difference between CuO$_4^{6-}$ center and Cu$^{2+}$ cation as its naive analogue. \n\n\\subsubsection{Two-hole configurations for the CuO$_{4}^{5-}$\ncenter}\n\nStarting from CuO$_{4}^{6-}$ center as a realization of $M^0$ center we arrive at CuO$_{4}^{5-}$ and CuO$_{4}^{7-}$\ncenters as realizations of hole $M^+$ and electron $M^-$ centers, respectively. The electron CuO$_{4}^{7-}$\ncenter has a filled set of Cu 3d and O 2p orbitals, and is often addressed to be a generalization of Cu$^{1+}$ ion. \nThe hole CuO$_{4}^{5-}$ center has actually two-hole configuration with the lowest Zhang-Rice (ZR) spin-singlet $^1A_{1g}$ state \\cite{ZR} formed by the interaction of three \"covalent\" configurations: $(b_{1g}^a)$, $(b_{1g}^b)$, and $(b_{1g}^ab_{1g}^b)$, respectively, or three \"purely ionic\" two-hole configurations\n$|d^{2}\\rangle$, $|pd\\rangle$, and $|p^{2}\\rangle$. Here, $|d\\rangle=|d_{x^2 -y^2}\\rangle$ and\n$|p\\rangle=|p_{b_{1g}}\\rangle$ are the non-hybridized Cu 3$d_{x^2-y^2}$ and O 2$p_{\\sigma}$ orbitals, respectively, with bare energies $\\epsilon_{d}$ and $\\epsilon_{p}$. \n\nLet us present a simple example of the calculation of the\ntwo-hole spectrum in the ZR-singlet sector. \n The matrix of\nthe full effective Hamiltonian within the bare purely ionic basis set has a\nrather simple form\n\\begin{equation}\n{\\hat H} =\\pmatrix{2\\epsilon _{d}+U_{d} & t & 0 \\cr t & \\epsilon\n_{d}+\\epsilon _{p}+V_{pd} & t \\cr 0 & t & 2\\epsilon _{p}\n+U^{*}_{p}\\cr}, \\label{U}\n\\end{equation}\nwhere the effective Coulomb parameter for purely oxygen\nconfiguration incorporates both the intra-atomic parameter $U_p$\nand the oxygen-oxygen coupling to the first and second nearest\nneighbors, respectively\n$$\nU^{*}_{p}=U_{p}+ \\frac{1}{4}V_{pp}^{(1)}+\\frac{1}{8}V_{pp}^{(2)},\n$$\nand the following condition holds:\n$$ U_d \\, > \\,U_p \\, >\\, V_{pd}\n$$\nFor reasonable values of parameters (in eV): $U_{d}= 8.5$,\n$U_{p}=4.0$, $V_{pd}=1.2$, $\\epsilon _{d}=0$, $\\epsilon _{p}=3.0$,\n$t=t_{pd}=1.3$ (see Ref.\\ \\onlinecite{CT}) we obtain for\nthe ZR-singlet energy $E_{ZR}=3.6$, and its wave function\n\\begin{equation}\n|\\Phi_1^{(2)}\\rangle=|b_{1g}^{2};pd\\rangle = 0.25|d^{2}\\rangle\n-0.95|pd\\rangle +0.19|p^{2}\\rangle \\; , \\label{zr1}\n\\end{equation}\nwhere $|pd\\rangle=\\frac{1}{\\sqrt{2}}(|p(1)d(2)\\rangle +|p(2)d(1)\\rangle)$.\nIt reflects the well-known result that the ZR-singlet represents a\ntwo-hole configuration with one predominantly Cu $3d$ and one\npredominantly O $2p$ hole. It is worth noting that the\nhole $CuO_4^{5-}$ center sometimes one naively associate with $Cu^{3+}$ ion, however, such a conclusion is a rather far from reality. Indeed, this center is the mixed valence one, as the $Cu$\nvalence resonates between $+2$ and $+3$.\n\n\nThe two excited states with energies\n$E_{ZR}+5.2$ and $E_{ZR}+6.7$ eV are described by the wave\nfunctions\n\\begin{equation}\n|\\Phi_2^{(2)}\\rangle=|b_{1g}^{2};dd\\rangle = -0.95|d^{2}\\rangle\n-0.21|pd\\rangle +0.22|p^{2}\\rangle , \\label{zr2}\n\\end{equation}\n\\begin{equation}\n|\\Phi_3^{(2)}\\rangle=|b_{1g}^{2};pp\\rangle = 0.17|d^{2}\\rangle\n+0.24|pd\\rangle +0.96|p^{2}\\rangle , \\label{zr3}\n\\end{equation}\nrespectively. Given the ZR-singlet energy one may calculate the minimal\n energy for $M^0 +M^0 \\rightarrow M^{\\pm}+M^{\\mp}$ charge transfer:\n$$\n\\Delta_{CT}=E_{ZR}-2E_{b_{1g}}=(3.6 +0.5) \\mbox{eV} = 4.1\n\\mbox{eV},\n$$\nwhere the stabilization energy for the bonding $b_{1g}^b$ state is\nsimply calculated from matrix (\\ref{U}) at\n$U_{d}=U^{*}_{p}=V_{pd}=0$. It should be emphasized that this quantity plays a\nparticular role as the minimal charge transfer energy which\nspecifies the charge transfer gap. In the general case it is\ndefined as follows:\n$$\n\\Delta _{CT}= E_{N+1}+E_{N-1}-2E_{N},\n$$\nor as the energy required to remove a hole from one region of the\ncrystal and add it to another region beyond the range of excitonic\ncorrelations. The exact diagonalization studies for a series of\nclusters with different size \\cite{Hybertsen,Hybertsen1} show that\n$\\Delta _{CT}$ strongly diminishes with cluster size from $\\approx\n4$eV for small clusters to $\\approx 2.5$eV as extrapolated value\nfor large clusters.\n\n\nOur simple calculation points to a significant role of correlation effect. Indeed, the inter-configurational coupling due to Coulomb repulsion results in a visible deviation of the two-hole ground state wave function from the predictions of simple model of noninteracting particles\n$$\n|(b_{1g}^b)^{2}\\rangle = (-0.8|d(1)\\rangle\n+0.6|p(1)\\rangle ) (-0.8|d(2)\\rangle\n+0.6|p(2)\\rangle )\n$$\n\\begin{equation}\n=0.64|d^{2}\\rangle\n-0.68|pd\\rangle +0.36|p^{2}\\rangle \\,.\n\\label{12}\n\\end{equation}\n In particular, it could give rise to a strong renormalization of the hole transfer integrals. \n\n\\subsection{Electron-lattice polarization effects}\n\n\\subsubsection{Correlation effects of electronic background}\nThe correlation problem becomes of primary importance for atoms\/ions near Coulomb instability when the one-electron gluing cannot get over the destructive effect of the electron-electron repulsion. Such a situation seems to realize in oxides where Hirsch {\\it et al}. \\cite{Hirsch} have proposed an instability of O$^{2-}$(2p$^6$) electronic background. \n The main suggestion in their theory of \"anionic metal\" concerns\nthe occurrence of the non-rigid degenerate structure for a closed\nelectron shell such as O$^{2-}$(2p$^6$) with the internal purely\ncorrelation degrees of freedom. In other words, one should expect\nsizeable correlation effects not only from unfilled 3d- or oxygen 2p\nshells, but from completely filled O 2p$^6$ shell! In order to\nrelevantly describe such a non-rigid atomic background and its\ncoupling to the valent hole one might use a concept of the\nwell-known \"shell-droplet\" model for nuclei after Bohr and\nMottelson.\\cite{Nataf} In accordance with the model a set of\ncompletely filled electron shells which form an atomic background or\nvacuum state for a hole representation is described by certain\ninternal collective degrees of freedom and a number of physical\nquantities such as electric quadrupole and magnetic moments. Valent\nhole(s) moves around this non-rigid background with strong\ninteraction inbetween. Such an approach strongly differs from the\ntextbook one that implies a rigid atomic orbital basis irrespective\nof varying filling number and external potential.\n\n None of the effective many-body Hamiltonians that are\nmost widely used to study the effect of electron correlation in\nsolids such as the Hubbard model, the Anderson impurity and lattice\nmodels, the Kondo model, contain this very basic and fundamental\naspect of electron correlation that follows from the atomic\nanalysis. \\cite{Hirsch1} The Hubbard on-site repulsion U between\nopposite spin electrons on the same atomic orbital is widely\nregarded to be the only important source of electron correlation in\nsolids. It is a clear oversimplification, and we need in a more\nrealistic atomic models to describe these effects, especially for\natoms in a specific external potential giving rise to a Coulomb\ninstability. To this end we have proposed a generalized non-rigid shell model (see Refs.\\onlinecite{shift,shift1,shift2}). The model\nrepresents a variational method for the many-electron atomic\nconfigurations with the trial parameters being the coordinates of\nthe center of the one-particle atomic orbital.\nThe resulting displacement of the atomic orbitals allows a simple\ninterpretation of the electron density redistribution stemmed from taking into account the electron-electron repulsion, and the\nsymmetry of a system can be readily used for the construction of the\ntrial many-electron wave function. As a whole, the model bears a\nstrong resemblance to the conventional well-known shell model by\nDick and Overhauser \\cite{shell} widely used in lattice dynamics. In\nframes of the model the ionic configuration with filled electron\nshells is considered to be constituted of an outer spherical shell\nof 2(2l+1) electrons and a core consisting of the nucleus and the\nremaining electrons. In an electric field the rigid shell retains\nits spherical charge distribution but moves bodily with respect to\nthe core. The polarizability is made finite by a harmonic restoring\nforce of spring constant $k$ which acts between the core and shell.\nThe shells of two ions repel one another and tend to become\ndisplaced with respect to the ion cores because of this repulsion.\nThe respective displacement vector appears to be a simplest\n$collective$ coordinate which specifies the change of the\nelectron-nucleus attraction. It should be noted that such a\ndisplacement does not imply any variation in electron-electron\nrepulsion and respective correlation energy.\n\nHowever, as we shall see below, a simple shell model can be easily\ngeneralized to take account of correlation effects. To this end we\nmust consider the displacements of separate one-electron orbitals to\nform the set of the variational parameters in a correlation\nfunction. Then we can introduce both the displacement of the center\nof \"gravity\" for filled shell and a set of the relative\ndisplacements of separate one-electron orbitals with regard to each\nother. The former form an \"acoustical\" mode and are described in\nframes of conventional shell model, while the latter form different\nnovel \"optical\" modes. Such a seemingly naive non-rigid shell\npicture can provide both the microscopic substantiation of the\nconventional shell model and its generalization. Moreover, this {\\it\nnon-rigid shell model} points to a physically clear procedure to\naccount for the correlation effects. Indeed, the \"optical\"\ndisplacement mode is believed to provide the minimal\nelectron-electron repulsion. The non-rigid shell represents a novel\nspecific atomic state that can be remarkably realized near a Coulomb\ninstability.\n\nThe idea of displaced atomic shells has appeared many years ago\n\\cite{Kozman} in the very beginning of the quantum chemical era,\nand reflected the naive picture of the repelling electrons. However,\nthe physically sound idea did not receive the relevant position in\nthe hierarchy of correlation effects.\n\n\\subsubsection{Electron-lattice relaxation effects}\nAs it is mentioned above, the minimal energy cost of the optically excited\ndisproportionation or electron-hole formation in insulating cuprates is\n$2.0-2.5$ eV. However, the question arises, what is the energy cost for the thermal\nexcitation of such a local disproportionation? The answer implies\nfirst of all the knowledge of relaxation energy, or the energy gain due to the lattice polarization by the\nlocalized charges. The full polarization energy $R$ includes the cumulative\neffect of $electronic$ and $ionic$ terms, associated with the displacement of\nelectron shells and ionic cores, respectively.\\cite{Shluger} The former term $R_{opt}$ is due\nto the {\\it non-retarded} effect of the electronic polarization by the momentarily\nlocalized electron-hole pair given the ionic cores fixed at their perfect crystal positions.\nSuch a situation is typical for lattice response accompanying the Franck-Condon\ntransitions (optical excitation, photoionization). On the other hand, all the\nlong-lived excitations, i.e., all the intrinsic thermally activated states and\nthe extrinsic particles produced as a result of doping, injection or optical\npumping should be regarded as stationary states of a system with a deformed\nlattice structure. These relaxed states should be determined from the condition\nthat the system energy has a local minimum when account is taken of the\ninteraction of the electrons and holes with the lattice deformations. At least,\nit means that we cannot, strictly speaking, make use of the same energy\nparameters to describe the optical (e.g. photoexcited) hole and thermal (e.g.\ndoped) hole.\n\nFor the illustration of polarization effects in cuprates we apply the shell\nmodel calculations to look specifically at energies associated with the\nlocalized holes of Cu$^{3+}$ and O$^-$ in \"parent\" La$_2$CuO$_4$ compound.\nIt follows from these calculations that there is a large difference in the\nlattice relaxation energies for O$^-$ and Cu$^{3+}$ holes. The lattice\nrelaxation energy, -$\\Delta R^{\\alpha}_{th}$, caused by the hole localization\nat the O-site (4.44 eV) appears to be significantly larger than that for the\nhole localized at the Cu-site (2.20 eV). This indicates\nthe strong electron-lattice interaction in the case of the hole localized at the\nO-site and could suggest that the hole trapping is more preferential in the\noxygen sublattice. In both cases we deal with the several eV-effect both for $electronic$ and $ionic$ contributions to relaxation energy. Moreover, such an estimation seems to be typical for different insulators.\\cite{Shluger,Vikhnin}\nIt is worth noting that the electron-lattice interaction is believed to be one of the main sources of correlated particle hopping resulting in different probabilities for two types of a charge transfer.\n\n\n\n\\subsubsection{Generalized Peierls-Hubbard model and \"negative-U\" effect}\n\n Transition metal oxides with strong electron and lattice polarization effects need in a revisit of many conventional theoretical concepts and approaches. In particular, we\n should modify conventional Hubbard model as it is done, for instance, in a \"dynamic\" Hubbard\n model by Hirsch \\cite{Hirsch1} or a modified Peierls-Hubbard model\n \\cite{Nasu} with a classical description of the anharmonic core\/shell displacements.\nHaving in mind the application to insulating cuprates let address a square lattice Hubbard model with a half-filling and a\nstrong on-site coupling of valent hole with core\/shell displacements, which is described by the following Hamiltonian\n$$\n\\hat H = -\\sum _{,\\sigma} t(ij){\\hat c}^{\\dag}_{i\\sigma}{\\hat\nc}_{j\\sigma} +U\\sum _{i,\\sigma \\sigma ^{'}}n_{i\\sigma}n_{i\\sigma\n^{'}}\n$$\n\\begin{equation}\n+ \\sum _{i}v_{an}(q_{i},n_{i}) + \\sum _{}v_{int}(q_{i},q_{j}),\n\\label{HHH}\n\\end{equation}\nwhere ${\\hat c}^{\\dag}_{i\\sigma}$ (${\\hat c}_{j\\sigma}$) are\ncreation (annihilation) operators for valent hole;\n$t(ij)=t(q_{i},q_{j})$ is the transfer integral between two\nneighboring lattice sites which depends on the dimensionless core\/shell\ndisplacement coordinate; $U$ is the on-site repulsion energy; $v_{an}(q_{i},n_{i})$ is the configurational energy that incorporates the coupling between valent holes and the site-localized anharmonic core\/shell mode with\ndimensionless displacement coordinate $q_{i}$;\n\\begin{equation}\nv_{an}(q_{i},n_{i})=\na(n_{i})q_{i}^{2}-b(n_{i})q_{i}^{4}+c(n_{i})q_{i}^{6}, \\label{an}\n\\end{equation}\nwhere $a,b,c$ are the functions of the hole occupation number such\nas\n\\begin{equation}\na(n_{i})= a_{0}+a_{1}n_{i}+a_{2}n_{i}^{2} ,\n\\end{equation}\nIt is clear that $v_{an}(q_{i},n_{i})$ includes the renormalization\nboth of the one-particle energy and the on-site hole-hole repulsion.\nThe last term in (\\ref{H}) represents the intersite configurational\ncoupling. The $q$-dependence of transfer integral implies the\ncorrelated character of the hole hopping, and can be transformed\ninto the effective dependence on hole occupation number\n\\cite{Hirsch1}\n\\begin{equation}\nt(n_{i},n_{j})= t(1+\\alpha (n_{i}+n_{j})+\\beta n_{i}n_{j}) .\n\\end{equation}\nwith $\\alpha ,\\beta$ being the correlated hopping parameters.\n\nThe conventional Hubbard Hamiltonian, or $t$-$U$-model, stabilizes\nthe spin density wave (SDW) electron order with $n_{i}= 1$. In a\nstrongly correlated limit $U\\geq t$ the Hubbard model reduces to a\nHeisenberg antiferromagnetic model. Depending on the parameters of\nthe hole-configurational coupling and correlated hopping the modified\nHubbard Hamiltonian (\\ref{H}) can stabilize the\n``disproportionated'' or charge ordered (CO) electron phase with the\non-site filling numbers $n=0$, and $n=2$ thus leading to the\n``negative-U'' effect. Even simple modified model turns out to be\nvery complicated and leads to a very rich physics.\\cite{Hirsch1}\nDepending on the values of parameters the system yields the SDW\nphase with no core\/shell displacements as a $true$ ground state with a\nglobal minimum of free energy, and CO phase with shell displacements\nas a $false$ ground state with a local minimum, or vice\nversa.\\cite{Nasu} Strong anharmonicity $v_{an}(q_{i},n_{i})$ makes\npossible phase transitions between the phases the first order ones.\n\n\n\\subsubsection{Vibronic reduction of charge transfer integrals}\n\nIn general, charge, spin and vibronic modes are\nstrongly coupled and so we have to do with the hybrid modes. For a weak\nintermode coupling regime the charge transfer is accompanied by the\ninduced local structural fluctuations, that provides the vibronic\nreduction of the charge transfer integral:\n\\begin{equation}\nt_{12}=t_{12}^{(0)}\\,\\ K_{vib}\\,\\quad\nK_{vib}=\\left\\langle \\chi _{1}|\\chi _{2}\\right\\rangle ^{2},\n\\end{equation}\nwhere \\thinspace $K_{vib}$ is a vibronic reduction factor, $\\left\\langle \\chi _{1}|\\chi\n_{2}\\right\\rangle $ is an overlap integral for the local oscillatory states\nwith and without particle transferred. In an opposite regime of the strong intermode coupling one assumes that\ndifferent electronic parameters for the $e$- and $h$-centers are\ndistinguished significantly up to different type of the adiabatic potential\nand appropriate JT mode. This regime favors the charge localization.\n\n\n\nThe vibronic reduction factor $K_{vib}$, or the Franck-Condon factor\n\\cite{Bersuker} may be written as follows\n\\begin{equation}\nK_{vib}=N\\ exp(-\\gamma ),\n\\end{equation}\nwhere $N$ and $\\gamma $ in a complicated manner depend on the vibronic constants, the oxygen and 3d-metal atomic\nmasses. For a simplest one-dimensional single-mode case\n\\begin{equation}\nK_{vib}=\\frac{2\\tau }{1+\\tau ^{2}}\\exp \\left( -\\frac{\\left( \\Delta Q\\right)\n^{2}}{l_{1}^{2}+l_{2}^{2}}\\right) ,\n\\end{equation}\nwhere $l_{1}$ and $l_{2}$ are the effective oscillatory lengths of the $1$-\nand $2$-centers, respectively, $\\tau =l_{1}\/l_{2}$, $\\Delta Q$ is the\ndistance separating the minima of the adiabatic potential for the $1$- and $2$-centers.\n\n\\subsubsection{Spin reduction of charge transfer integrals}\n\nOverall the paper we neglect a spin degree of freedom which can crucially impact on the charge transport. \nThe most part of 3d oxides are characterized by an antiferromagnetic spin background that implies a localization effect due to strong spin reduction of one-particle transfer integrals and the probability amplitude for a polar center transfer $ M^{\\pm} + M^0 \\rightarrow M^0 +M^{\\pm}$ or the motion of the electron (hole) center in the matrix of\n$M^0$-centers. From the other hand, in antiferromagnets there is no problems with another type of the CT which specifies\nthe probability amplitude for a local disproportionation, or the spin-singlet $eh$-pair creation: \n$M^0 +M^0 \\rightarrow M^{\\pm}+M^{\\mp}$, and the inverse process of the spin-singlet $eh$-pair recombination:\n$M^{\\pm}+M^{\\mp}\\rightarrow M^0 +M^0$. In other words, the spin subsystem can strongly affect the correlated character of the charge transfer leading to unconventional situations like that of spin-singlet eh-pairs moving through the lattice freely without disturbing the antiferromagnetic spin background, in contrast to the single particle motion. So, it seems that the situation in antiferromagnetic 3d insulators may differ substantially from that in usual semiconductors or in other bandlike insulators where, as a rule, the effective mass of the electron-hole pair is larger than that of an unbound electron and hole.\n\n\n\n\\section{S=1 pseudospin formalism for model mixed valence system}\n\n\\subsection{Pseudospin operators}\nThe problem of the multi-stability of solids looks rather trivial when one say\nabout the orbital and\/or spin degrees of freedom. Usually in such a case we\nstart from the lattice of coupled orbital and\/or spin momenta described by\nthe relevant (spin-)Hamiltonian that implies the variety of possible collective\norbital and\/or spin orderings that compete with each other under different\nexternal conditions. In other words, the multi-stability accompanies the basic\ndegeneracy inherent to a certain atom, ion, or center with a nonzero orbital\nand\/or spin momentum. Such an outlook is believed to be easily extended to\nsystems with charge degree of freedom which can be represented to be a system\nof either centers which possible charge states form a pseudo-multiplet. Below\nwe address a simple\n model of a mixed-valence system with three possible stable valent states of a cation-anionic\n cluster, hereafter $M$: $M^0, M^{\\pm}$, forming the charge (isospin) triplet. Starting from $M^0$ state as a bare vacuum state, we may address the $M^{\\pm}$ centers as a result of pseudo-spin $\\Delta S_z=\\pm 1$ deviation, or as a hole and electron, respectively.\n Below we intend to concentrate themselves on charge degree of freedom, and\n that is why we neglect the orbital, spin, and lattice degrees of freedom. It implies a renormalization of different parameters, mainly it concerns the charge transfer. \n \nSimilarly to the neutral-to-ionic electronic-structural transformation\n in organic charge-transfer crystals (see paper by T. Luty in Ref.\\onlinecite{Toyozawa})\n the system of charge triplets\ncan be described in frames of the S=1 pseudo-spin formalism.\n To this end we associate three charge states of the $M$-center with different valence:\n $M^0,M^{\\pm}$ with three components of $S=1$ pseudo-spin (isospin)\ntriplet with $M_S =0,+1,-1$, respectively.\n\n The $S=1$ spin algebra includes three independent irreducible tensors\n${\\hat V}^{k}_{q}$ of rank $k=0,1,2$ with one, three, and five components,\nrespectively, obeying the Wigner-Eckart theorem \\cite{Varshalovich}\n\\begin{widetext}\n\\begin{equation}\n\\langle SM_{S}| {\\hat V}^{k}_{q}| SM_{S}^{'}\\rangle=(-1)^{S-M_{S}} \\left(\n\\begin{array}{ccc}S&k&S\n\\\\\n-M_{S}&q&M_{S}^{'}\\end{array}\\right)\\left \\langle S\\right\\| {\\hat\nV}^{k}\\left\\|S\\right\\rangle. \\label{matelem}\n\\end{equation}\n\\end{widetext}\nHere we make use of standard symbols for the Wigner coefficients and reduced\nmatrix elements. In a more conventional Cartesian scheme a complete set of the\nnon-trivial pseudo-spin operators would include both ${\\bf S}$ and a number\nof symmetrized bilinear forms $\\{S_{i}S_{j}\\}=(S_{i}S_{j}+S_{j}S_{i})$, or\nspin-quadrupole operators, which are linearly coupled to $V^{1}_{q}$ and $V^\n{2}_{q}$, respectively\n$$\nV^{1}_{q}=S_{q}; S_{0}=S_{z}, S_{\\pm}=\\mp \\frac{1}{\\sqrt{2}}(S_{x}\\pm iS_{y} ):\n$$\n\\begin{equation}\nV^{2}_{0} \\propto (3S_{z}^{2}-{\\bf S}^2), V^{2}_{\\pm 1}\\propto (S_z S_{\\pm}+\nS_{\\pm}S_z), V^{2}_{\\pm 2}\\propto S_{\\pm}^2 .\n\\end{equation}\nThese pseudo-spin operators are not to be confused with real physical\nspin-operators; they act in an imaginary pseudo-space.\n\nTo describe different types of pseudo-spin ordering in a mixed-valence system\nwe have to introduce eight order parameters: two $diagonal$ order parameters\n$\\langle S_{z}\\rangle$ and $\\langle S_{z}^2\\rangle$, and six {\\it off-diagonal}\norder parameters $\\langle V^{k}_{q}\\rangle$ ($q\\not=0$). Two former order\nparameters can be termed as $valence$ and $ionicity$, respectively. The {\\it\noff-diagonal} order parameters describe different types of the valence mixing.\nIndeed, operators $V^{k}_{q}$ ($q\\not=0$) change the $z$-projection of\npseudo-spin and transform the $|SM_S\\rangle$ state into $|SM_{S}+q\\rangle$\none. In other words, these can change $valence$ and $ionicity$. It should be\nnoted that for the $S=1$ pseudospin algebra there are two operators:\n$V^{1}_{\\pm 1}$ and $V^{2}_{\\pm 1}$, that change the pseudo-spin projection by\n$\\pm 1$, with slightly different properties\n\\begin{equation}\n\\langle 0 |\\hat S_{\\pm} | \\mp 1 \\rangle = \\langle \\pm 1 |\\hat S_{\\pm} | 0\n\\rangle =\\mp 1, \\label{S1}\n\\end{equation}\nbut\n\\begin{equation}\n\\langle 0 |(S_z S_{\\pm}+ S_{\\pm}S_z)| \\mp 1 \\rangle = -\\langle \\pm 1 |(S_z\nS_{\\pm}+ S_{\\pm}S_z)| 0 \\rangle =+1. \\label{S2}\n\\end{equation}\n\n\\subsection{Gell-Mann operators and generalized pseudospin Hamiltonian}\nThree spin-linear (dipole) operators $\\hat S_{1,2,3}$ and five independent\nspin-quadrupole operators $\\{{\\hat S_{i}},{\\hat S_{j}}\\}-\\frac{2}{3} {\\hat\n{\\bf S}}^{2}\\delta _{ij}$ given $S=1$ form eight Gell-Mann operators being the\ngenerators of the SU(3) group. Below we will make use of the appropriate\nGell-Mann $3\\times 3$\n matrices $\\Lambda^{(k)}$, which\ndiffer from the conventional $\\lambda^{(k)}$ only by a\nrenumeration:\\cite{electr} $\\lambda^{(1)}=\\Lambda^{(6)}$,\n$\\lambda^{(2)}=\\Lambda^{(3)}$, $\\lambda^{(3)}=\\Lambda^{(8)}$,\n$\\lambda^{(4)}=\\Lambda^{(5)}$, $\\lambda^{(5)}=-\\Lambda^{(2)}$,\n$\\lambda^{(6)}=\\Lambda^{(4)}$, $\\lambda^{(7)}=\\Lambda^{(1)}$,\n$\\lambda^{(8)}=\\Lambda^{(7)}$. First three matrices $\\Lambda^{(1,2,3)}$\ncorrespond to linear (dipole) spin operators:\n$$\n\\Lambda^{(1)}=S_x ;\\quad \\Lambda^{(2)}=S_y ;\n \\quad \\Lambda^{(3)}=S_z\n$$\nwhile other five matrices correspond to quadratic (quadrupole) spin operators:\n$$\n \\Lambda^{(4)}=-\\{S_zS_y\\} ;\\quad\n \\Lambda^{(5)}=-\\{S_xS_z\\} ;\\quad\n \\Lambda^{(6)}=-\\{S_xS_y\\} ;\n$$\n$$\n\\Lambda^{(7)}=-\\frac{1}{\\sqrt{3}}(S_x^2+S_y^2-2S_z^2) ; \\quad\n\\Lambda^{(8)}=S_y^2-S_x^2 ;\n$$\n$$\nS_x^2+S_y^2+S_z^2=2\\hat E\n$$\nwith $\\hat E$ being a unit $3\\times 3$ matrix.\n\n\nThe generalized spin-1 model can be described by the Hamiltonian bilinear\n on the SU(3)-generators $\\Lambda^{(k)}$\n\\begin{equation}\n\\hat{H}=-\\sum_{i,\\eta}\\sum_{k,m=1}^{8}J_{km}\\hat{\\Lambda}_i^{(k)}\n\\hat{\\Lambda}_{i+\\eta}^{(m)} \\, .\\label{su3}\n \\end{equation}\n Here $i,\\eta$ denote lattice\nsites and nearest neighbors, respectively. This is a $S=1$ counterpart of the\n$S=1\/2$ model Heisenberg Hamiltonian with three generators of the SU(2) group\nor Pauli matrices included instead of eight Gell-Mann matrices.\n\n\n\\subsection{Generalized mean-field model}\nIn frames of a classical, or mean-field description of the $S=1$ quantum\npseudo-spin system we start from a coherent state approximation with trial\nfunctions \\cite{electr}\n\\begin{equation}\n \\psi=\\prod_{j\\in lattice}c_i(j)\\psi_i=\\prod_{j\\in\nlattice}(a_i(j)+ib_i(j))\\psi_i . \\label{wf}\n\\end{equation}\nHere $j$ labels a lattice site and the spin functions $\\psi_i$ in a Cartesian\n basis are used: $\\psi_z=|10>$ and $\\psi_{x,y}\\sim(|11>\\pm|1-1>)\/\\sqrt 2$.\n The linear (dipole) pseudo-spin operator within $|x,y,z\\rangle$ basis is represented\n by a simple matrix:\n $$\n <\\psi_i|S_j|\\psi_k> =-i\\varepsilon_{ijk},\n $$\n and for the order parameters one easily obtains:\n\\begin{equation}\n <\\hat{\\bf S}> = -2[{\\bf a} \\times {\\bf b}]; \\,\n<\\{\\hat{S_i}\\hat{S_j}\\}>=2(\\delta_{ij}-a_ia_j-b_ib_j) \\label{med}\n\\end{equation}\ngiven the normalization constraint ${\\bf a}^2 +{\\bf b}^2=1$. Thus, for the\ncase of spin-1 system the order parameters are determined by two classical\nvectors (two real components of one complex vector $\\bf c =\\bf a +i\\bf b$\nfrom (\\ref{wf})). The two vectors are coupled, so the minimal number of dynamic\nvariables describing the $S=1$ spin system appears to be equal to four. Along\nwith $\\bf a $, $\\bf b$ vectors one might introduce ${\\bf l}=[\\bf a \\times\n\\bf b]$. Hereafter we would like to emphasize the $director$ nature of the\n${\\bf c}$ vector field: $\\psi ({\\bf c})$ and $\\psi (-{\\bf c})$ describe the\nphysically identical states. It is worth noting that the coherent states\nprovide the optimal way both to a correct mean-field approximation (MFA) and\nrespective continuous models.\\cite{electr}\n\n\\begin{figure}[ht]\n\\includegraphics[width=8.5cm,angle=0]{fig2.eps}\n\\caption{Schematic energy spectrum of two $nn$ M-centers system (see text for details). Arrows mark the dipole-allowed CT transitions. }\n\\label{fig2}\n\\end{figure}\n\n\n\\section{Model mixed-valence system}\n\n\\subsection{Effective pseudo-spin Hamiltonian}\nEffective pseudo-spin Hamiltonian for our model mixed-valence system should\nincorporate a large number of contributions that describe different long- and\nshort-range coupling between $M^{0,\\pm}$ centers, single-ion and two-ion terms.\nSingle-site terms can be subdivided into {\\it single-ion } anisotropy and {\\it\npseudo-Zeeman} interaction. Bilinear and biquadratic two-site terms can be\nsubdivided into $diagonal$ interactions like \"density-density\", and {\\it\noff-diagonal} terms that describe charge fluctuations conserving the total charge of\nthe system, such as one-electron(hole) and two-electron(hole) transport. \n An effective pseudo-spin\nHamiltonian of the model mixed-valence system which takes into\n account the main part of aforementioned contributions can be represented as follows\n$$\n \\hat H = \\sum_{i} (\\Delta _{i}S_{iz}^2\n - h_{i}S_{iz}) + \\sum_{} v_{ij}S_{iz}^2 S_{jz}^2 +\n \\sum_{} V_{ij}S_{iz}S_{jz}+\n$$\n$$\n\\sum_{} [D_{ij}^{(1)}(S_{i+}S_{j-}+S_{i-}S_{j+})+ \nD_{ij}^{(2)}(T_{i+}T_{j-}+T_{i-}T_{j+})]\n$$\n\\begin{equation}\n +\\sum_{} t_{ij}(S_{i+}^{2}S_{j-}^{2}+S_{i-}^{2}S_{j+}^{2}),\n \\label{H}\n \\end{equation}\n where\n $$\n T_{\\pm}= (S_z S_{\\pm}+ S_{\\pm}S_z).\n $$\nTwo first single-ion terms describe the effects of bare pseudo-spin splitting,\nor the local energy of $M^{0,\\pm}$ centers. Interestingly, the parameter $\\Delta$ can be related with correlation Hubbard parameter $U$: $U=2\\Delta$. The second term may be\nassociated with an external, generally speaking, non-uniform pseudo-magnetic\nfield $h_i$, in particular, a real electric field. It is easy to see that it describes an electron\/hole assymetry. The third and fourth terms describe the effects of long- and short-range inter-ionic interaction including screened Coulomb and covalent coupling. \n\nIf to apply the familiar spin terminology, the first term in (\\ref{H})\nrepresents a single-ion anisotropy, the second does the Zeeman term, the fourth\nand fifth do the anisotropic Heisenberg exchange, and the third and sixth do\nthe biquadratic spin-quadrupolar coupling.\nTo illustrate the role of different terms in (\\ref{H}) we present in Fig.1 a schematic energy spectrum of $nn$ pair of $M$ centers provided an eh-symmetry ($h=0$) and $|00\\rangle$ ground state ($\\Delta >0$). It is worth noting the effect of a renormalization of the ground state due to eh-pair creation\/recombination effect ($t_{ij}^{\\prime\\prime}\\not=0$) with a stabilization energy $\\delta \\approx |t_{ij}^{\\prime\\prime}|^2\/2\\Delta$. Two electron-hole states with S- (even) and P- (odd) type symmetry have a very strong\ndipole coupling with the large value of $S-P$ transition dipole matrix element:\n\\begin{equation}\nd = |\\langle S|\\hat{\\bf d}|P\\rangle | \\approx 2eR_{MM} \\approx 2e\\times 4\n\\mbox{\\AA}. \\label{d}\n \\end{equation}\nContrary to $P$-type pair state the $S$-type one is dipole-forbidden and corresponds to\na so-called two-photon state. \n\nOne should note that despite many simplifications, and first the neglect of\norbital and spin degrees of freedom, quenched lattice approximation, the\neffective Hamiltonian (\\ref{H}) is rather complex, and represents one of the\nmost general forms of the anisotropic $S=1$ non-Heisenberg Hamiltonians. For\nthe system there are two classical ($diagonal$) order parameters: $\\langle S_z\n\\rangle = n$ being a valence, or charge density with electro-neutrality\nconstraint $\\sum _{i} n_i = \\sum _{i} S_{iz}=0$, and $\\langle S_{z}^{2} \\rangle\n= n_p$ being the density of polar centers $M^{\\pm}$, or \"ionicity\". In\naddition, there are two unconventional {\\it off-diagonal} order parameters:\n``fermionic'' $\\langle S_+ \\rangle $ and ``bosonic ''$\\langle S_{+}^{2} \\rangle\n$; the former describes a phase ordering for the disproportionation reaction,\nor the single-particle transfer, while the latter does for exchange reaction,\nor for the two-particle transfer. Indeed, the ${\\hat S}_+$ operator creates a\nhole and is fermionic in nature, whereas the ${\\hat S}_{+}^{2}$ does a hole\npair, and is bosonic in nature.\n\n\\subsection{Single and two-particle transport}\nThe last three terms in (\\ref{H}) representing the one- and two-particle hopping, respectively, are of primary importance for the transport properties, and deserve special interest. \n\nTwo types of one-particle hopping are\ngoverned by two transfer integrals $D^{(1,2)}$, respectively.\n In accordance with (\\ref{S1}) and\n (\\ref{S2}) the transfer integral $t_{ij}^{\\prime}=(D_{ij}^{(1)}+ D_{ij}^{(2)})$\n specifies the probability amplitude for a {\\it local disproportionation, or the $eh$-pair creation}\n$$\nM^0 +M^0 \\rightarrow M^{\\pm}+M^{\\mp};\n$$\nand the inverse process of the {\\it $eh$-pair recombination}\n$$\nM^{\\pm}+M^{\\mp}\\rightarrow M^0 +M^0 ,\n$$\nwhile the transfer integral $t_{ij}^{\\prime\\prime}=(D_{ij}^{(1)}- D_{ij}^{(2)})$\n specifies the probability amplitude for a polar center transfer\n $$\n M^{\\pm} + M^0 \\rightarrow M^0 +M^{\\pm},\n $$\n or the {\\it motion of the electron (hole) center in the matrix of\n$M^0$-centers} or motion of the $M^0$-center in the matrix of $M^{\\pm}$-centers.\nIt should be noted that, if $t_{ij}^{\\prime\\prime}=0$ but $t_{ij}^{\\prime}\\not=0$, the eh-pair is locked in two-site configuration.\n\n\nThe two-electron(hole) hopping is governed by transfer integral\n $t_{ij}$, or a probability amplitude for the exchange reaction:\n $$\nM^{\\pm}+M^{\\mp}\\rightarrow M^{\\mp} +M^{\\pm}.\n$$\n or the {\\it motion of the electron (hole) center in the\nmatrix of hole (electron) centers}.\n\nIt is worth noting that in Hubbard-like models\n all the types of one-electron(hole) transport are governed by the same\n transfer integral: $t_{ij}^{\\prime}=t_{ij}^{\\prime\\prime}=t_{ij}$, while our model implies independent parameters for a\n disproportionation\/recombination process and simple quasiparticle motion\n in the matrix of $M^0$-centers. In other words, we deal with a \"correlated\"\n single particle transport.\\cite{Hirsch} \n\n\\subsection{Mean-field approximation:three generic MFA-phases} \nFirst of all we would like to emphasize\nthe difference between classical and quantum mixed-valence systems. Classical (or chemical)\ndescription implies the neglect of the {\\it off-diagonal} purely quantum CT\neffects: $D^{(1,2)}=t=0$, hence the valence of any site remains to be definite:\n$0, \\pm 1$, and we deal with a system of localized polar centers. In quantum systems with a nonzero charge transfer we arrive at {\\it\nquantum superpositions of different valence states} resulting in $indefinite$\non-site valence and ionicity which effective, or mean values $\\langle\nS_{z}\\rangle$ and $\\langle S_{z}^2\\rangle$ can vary from $-1$ to $+1$ and $0$\nto $+1$, respectively.\n\nMaking projection of the effective pseudo-spin Hamiltonian for the system onto\na space of states like (\\ref{wf}), we obtain an energy functional\n which equivalent to a classical energy of the two coupled vector $({\\bf a\n},{\\bf b})$ fields defined on the common lattice. Thus, in the framework of\nthe pseudo-spin $S =1$ centers model when the collective wave\n function is represented to be a product of the site functions\n like (\\ref{wf}), the quantum problem is reduced to a classical\nvariation problem for a minimum of the energy for two coupled vector fields.\n\nIn frames of mean-field approximation (MFA) we may make use of coherent states\n(\\ref{wf}) that provide a physically clear assignment of different phases with\na straightforward recipe of its qualitative and quantitative analysis. All the\nMFA-phases one may subdivide into those with a definite and indefinite\nionicity, respectively. There are two MFA-phases with definite ionicity;\n\n1) {\\bf Insulating monovalent $M^0$-phase with $ \\langle S_{z}^2\\rangle =\n0$:}\n\nThe $M^0$-phase is specified by a simple uniform arrangement of ${\\bf a}$ and ${\\bf \nb}$ vectors parallel to $z$-axis: ${\\bf a}\\parallel {\\bf b}\\parallel O_z$. In\nsuch a case the on-site wave function is specified by unit vector (${\\bf a}$,\nor ${\\bf b}$) parallel to $z$-axis. It is a rather conventional ground state phase for\nvarious charge transfer insulators such as oxides with a positive magnitude of $\\Delta$ parameter ($U>0$). All the centers have the same bare $M^0$ valence state. In other words, the $M^0$-phase\nis characterized both by definite site ionicity and valence.\n So, all the order parameters turn into zero:\n $\\langle S_{z}\\rangle = \\langle S_{z}^2\\rangle =\\langle S_{+}\\rangle =\n \\langle S_{+}^2\\rangle =0$. This is an ``easy-plane'' phase for\nthe pseudo-spins, but an ``easy-axis'' one for the ${\\bf a}$ and\/or ${\\bf \nb}$ vectors.\n\n2) {\\bf Mixed-valence binary (disproportionated) $M^{\\pm}$-phase with $ \\langle\nS_{z}^2\\rangle = 1$:}\n\n This phase usually implies an overall disproportionation $M^0 +M^0 \\rightarrow M^{\\pm}+M^{\\mp}$ that seems to be realizable if $\\Delta$ parameter becomes negative one (negative $U<0$ effect). It is a rather unconventional\n phase for insulators. All the centers have the \"ionized\" valence state, one half the $M^+$ state, and another half the $M^-$ one, though one may in common conceive of deviation from fifty-fifty distribution. A simplified \"chemical\" approach to $M^{\\pm}$-phase as to a classical disproportionated\nphase is widely spread in solid state chemistry.\\cite{Ionov} \n In contrast with the $M^0$ phase the $M^{\\pm}$-phase is specified by a planar\norientation of ${\\bf a}$ and ${\\bf b}$ vectors (${\\bf a}, {\\bf b}\\perp\nO_z$) with a varied angle in between.\n There is no fermionic transport: $\\langle S_{+}\\rangle =\n0$, while the bosonic one may exist, and, in common, $\\langle S_{+}^2\\rangle\n=-\\cos(\\phi _{a}-\\phi _{b})e^{i(\\phi _{a}+\\phi _{b}) }\\not= 0$. This is an\n``easy-axis'' phase for the pseudo-spins, but an ``easy-plane'' one for the\n${\\bf a}$ and ${\\bf b}$ vectors.\n\nThe mixed valence $M^{\\pm}$ phase as a system of strongly correlated electron and hole centers is equivalent to the lattice hard-core Bose system with an inter-site repulsion, or {\\it electron-hole Bose liquid} (EHBL) in contrast with EH liquid in conventional semiconductors like Ge, Si where we deal with a two-component {\\it Fermi-liquid}. Indeed, one may address the electron $M^-$ center to be a system of a local boson ($e^2$) localized on the hole $M^+$ center: $M^- = M^+ + e^2$.\n \n \nIn accordance with this analogy we assign three well known molecular-field uniform phase\nstates of the $M^{\\pm}$ binary mixture:\n\ni) {\\bf charge ordered (CO) insulating state} with $\\langle S_{z}\\rangle = \\pm\n1$, ${\\bf a}\\perp {\\bf b}$, and zero modulus of bosonic off-diagonal order parameter:\n$|\\langle S_{+}^2\\rangle |= 0$;\n\nii) {\\bf Bose-superfluid (BS) superconducting state} with $\\langle S_{z}\n\\rangle\n = 0$, ${\\bf a}$ and $ {\\bf b}$ being collinear, $\\langle S_{+}^2\\rangle = e^{2i\\phi}$;\n\niii) {\\bf mixed Bose-superfluid-charge ordering (BS+CO) superconducting state\n(supersolid) }\n with $0<|\\langle S_{z}\\rangle |<1$, ${\\bf a}$ and ${\\bf b}$ being\noriented in $xy$-plane, but not collinear, $\\langle S_{+}^2\\rangle =\n -\\cos(\\phi _{a}-\\phi _{b})e^{i(\\phi _{a}+\\phi _{b})}\\not= 0$.\n\n In addition, we should mention the high-temperature non-ordered (NO) Bose-metallic phase\n with $\\ll S_z \\gg =0$.\n \n Rich phase diagram of $M^{\\pm}$ binary mixture with unconventional superfluid and supersolid regions looks tempting, however, actually, their stabilization requires strong suppression of Coulomb repulsion between electron (hole) centers. Despite significant screening effect, the stabilization of uniform BS or BS+CO superconducting state as a result of a disproportionation reaction in a bare insulator \\cite{Ionov} seems to be unrealistic.\n\n3) {\\bf Mixed-valence ternary (``under-disproportionated'') $M^{0,\\pm}$-phase:}\n\nFor two preceding cases the order parameter $\\langle S_{z}^2\\rangle $, or\nionicity relates to its limiting values (0 or 1, respectively). For the\nMFA-phase with indefinite ionicity, or mixed-valence ternary\n(``under-disproportionated'') $M^{0,\\pm}$-phase, $0 < \\langle S_{z}^2\\rangle\n<1$, that is we have a mixture of the $M^0,M^{\\pm}$ centers. From the\nviewpoint of the classical ${\\bf a}, {\\bf b}$ vectors formalism the phase\ncorresponds to the arbitrarily space-oriented ${\\bf l}=[{\\bf a}\\times {\\bf \nb}]$ vector. Both off-diagonal order parameters, fermionic $\\langle\nS_{+}\\rangle $ and bosonic $\\langle S_{+}^2\\rangle $ are, in common, non-zero,\nalbeit with some correlation in between. So, for the ternary system one expects\na coexistence of fermionic and bosonic transport.\n\nIt should be noted that a complete pseudo-spin description of the two last\nmodel mixed-valence systems implies a two-sublattice approximation to be a\nminimal model compatible with a sign of Coulomb interaction and a respective\ntendency towards the checkerboard-like charge ordering.\n\n\\section{Insulating monovalent $M^0$-phase}\nInsulating monovalent $M^0$-phase is a typical one for the ground state of insulating transition metal oxides, or Mott-Hubbard insulators. It is worth noting that in frame of conventional band model approach the $M^0$-phase, e.g., in parent cuprates, is associated with a metallic half-filled hole band system. Below we address different types of quasiparticle excitations in such a system focusing on the features of the correlated hopping, governed generally by two transfer integrals $t_{ij}^{\\prime}$ and $t_{ij}^{\\prime\\prime}$ which competition results in unconventional properties of electrons and holes in bare insulating monovalent $M^0$-phase. We show that $M^0$-phase can be unstable with regard to the self-trapping of CT excitons and nucleation of droplets of EH Bose liquid. \n\n\\subsection{Electron, hole, and electron-hole excitations}\n\nStarting from monovalent $M^0$-phase as a vacuum state $\\left|0\\right\\rangle$ we introduce an electron-hole representation where $M^-_i$ center is derived as a result of an electron creation $\\hat a_i^\\dagger \\left|0\\right\\rangle$, and $M^+_i$ center is derived as a result of a hole creation $\\hat b_i^\\dagger \\left|0\\right\\rangle$. Then we transform pseudo-spin Hamiltonian (\\ref{H}) into that of a system of effective electrons and holes\n$$\n\\hat H= \\Delta \\sum _{i}(n^h_i+n^e_i)+\\sum_{\\left\\langle ij\\right\\rangle}t_{ij}(\\hat a_i^\\dagger \\hat a_j+\\hat b_i^\\dagger \\hat b_j)+\n$$\n$$\n\t \\sum_{i,j}[V^{hh}(ij)(n^h_in^h_j+V^{ee}(ij)n^e_in^e_j+V^{eh}(ij)(n^h_in^e_j+n^h_jn^e_i)]\n$$\n\\begin{equation}\n+\\sum_{\\left\\langle ij\\right\\rangle}t_{ij}^{\\prime \\prime}(\\hat a_i^\\dagger \\hat b_j^\\dagger+\\hat a_i \\hat b_j)\n\\label{eh}\n\\end{equation}\nwhere $V_{ii}\\rightarrow\\infty$ to prohibit two-particle occupation of a single site. Here we suppose $h=0$ that provides an electron-hole symmetry. The first line in (\\ref{eh}) represents the single particle (electrons\/holes) terms, the second one does the interparticle coupling, while the third one describes the creation and annihilation (recombination) of eh-pairs. It is worth noting that the latter terms describe some sort of eh-coupling. \n\nIn terms of a pseudo-spin analogy the electrons and holes are associated with pseudo-spin $\\Delta S_z=\\pm 1$ deviations for an easy-plane magnet, localized or delocalized (pseudo-spin wave).\n\nThe behavior of electron\/hole system crucially depends on the relation between two transfer integrals $t_{ij}^{\\prime},t_{ij}^{\\prime \\prime}$\nBelow we address two distinct limiting situations:\n\nI. $t_{ij}^{\\prime \\prime}=0$: Forbidden recombination\/creation regime. In this case we deal with the bands of well defined electrons and holes with a charge gap $E_g^{e,h}=\\Delta - z|t_{nn}^{\\prime}|$ for both types of carriers. Optical gap for unbound eh-pairs is $E_g^{eh}=2E_g^{e,h}$. However, in such a case we may expect for the formation of\nWannier excitons, or eh-pairs bound due to a screened Coulomb eh-coupling. In terms of pseudo-spin formalism the Wannier excitons may be regarded as two pseudo-spin waves having formed a quasilocalized state due to a long-range antiferromagnetic Ising exchange: $V_{ij}S_{iz}S_{jz}$.\n\nII. $t_{ij}^{\\prime}=0$: Regime of localized electrons and holes with a dimerization effect and well defined $nn$ eh-pairs, or CT excitons. In such a case the charge gap is $E_g^{e,h}=\\Delta $ for both types of localized quasiparticles. As it is clearly seen in Fig.1 the $t_{ij}^{\\prime \\prime}\\not=0$ hopping results in a dimerization effect with a quantum renormalization of the vacuum state and indefinite ionicity, the formation of two types of localized eh-pairs, or CT excitons of S- and P-type. In frames of our $nn$ approximation the CT excitons are localized.\nOptical gap is determined by the energy of P-type CT exciton: $E_g^{eh}=2\\Delta+\\delta$. \n\nThus we arrive at two limiting types of monovalent $M^0$ insulators with a dramatic difference in behavior of electrons and holes, as well as electron-hole pairs. In type I insulators ($t_{ij}^{\\prime}\\gg t_{ij}^{\\prime \\prime}$) we deal with well defined bands of electrons and holes forming Wannier excitons, while in type II insulators ($t_{ij}^{\\prime\\prime}\\gg t_{ij}^{\\prime}$) we deal with localized electrons and holes which can form $nn$ eh-pairs, or CT excitons. \n\nThe most part of 3d oxides are characterized by an antiferromagnetic spin background that implies strong spin reduction of one-particle transfer integrals $t_{ij}^{\\prime}$. In other words, these, seemingly, belong to type II insulators, where spin-singlet CT excitons can move through the lattice freely without disturbing the antiferromagnetic spin background, in contrast to the single hole motion. So, it seems that the situation in antiferromagnetic 3d insulators differs substantially from that in usual semiconductors or in other bandlike insulators where, as a rule, the\neffective mass of the electron-hole pair is larger than that of an unbound electron and hole.\n\nThe Wannier excitons are formed due to an eh-coupling, while the CT excitons are formed due to a {\\it kinetic cutoff}, or a specific feature of correlated hopping, in other words, the former have a $potential$, while the latter a $kinetic$ nature. \n\nIt is worth noting that both M-centers within P-type CT excitons have a certain ionicity in contrast to S-type CT excitons which can mix with bare $M^0M^0$ ground state. CT excitons form peculiar quanta of a disproportionation reaction and may be viewed to be a minimal droplet of electron-hole Bose liquid.\n\nIn general, eh-excitations in $M^0$ phase consist of superpositions of pairs of free electrons and holes, and CT excitons. One expects two types of superpositions: CT exciton-like and band-like. The former have a localized character, while the latter have an itinerant one. \n\nThe nature of the optical excitations accompanied by creation of electron-hole\n pairs in 3d oxides is not fully understood.\n One of the central issues in the analysis of electron-hole\nexcitations is whether low-lying states are comprised of free\ncharge carriers or excitons. A conventional approach implies that if\nthe Coulomb interaction is effectively screened and weak, then the\nelectrons and holes are only weakly bound and move essentially\nindependently as free charge-carriers. However, if the Coulomb\ninteraction between electron and hole is strong, excitons are believed to\nform, i.e.\\ bound particle-hole pairs with strong correlation of\ntheir mutual motion. \n \n One of the most popular criteria to discriminate between the states relates to the band\n gap: states below the charge gap correspond to excitons with binding energy\n $ E_b = E_g - E$, and states above the charge\n gap do to free electron-hole pairs. However, this criteria seems to be oversimplified, and the states should be characterized as bound or unbound according to the\n scaling of the average electron-hole separation with system size. Excitons are\n entities with small electron-hole separation which remain finite as the system\n size is increased. By contrast, the average separation between two free charge\n carriers increases indefinitely with system size.\n To distinguish bound and unbound electron-hole states one might use the\n density-density correlation function \\cite{Boman}\n $$\n C(i,j)=\\langle ({\\hat n}_{i}-\\langle {\\hat n}_{i}\\rangle )\n ({\\hat n}_{j}-\\langle {\\hat n}_{j}\\rangle )\\rangle ,\n $$\nwhich measures a correlation of charge fluctuations on site $i$ to a\ncharge fluctuations on site $j$. A negative value correlates an excess\n(deficit) of charge with deficit (excess), or electron-hole distribution. In frame of pseudo-spin approach this \ncorrelation function measures the longitudinal ($\\parallel z$) short-range antiferromagnetic fluctuations \n$$\n C(i,j)=\\langle \\hat S_{iz}\\hat S_{jz}\\rangle \n -\\langle \\hat S_{iz}\\rangle \\langle \\hat S_{jz}\\rangle .\n $$\n The results of the direct\n computations for conjugated polymers in frames of the extended Hubbard model\n \\cite{Boman} show for different symmetry sectors that there exist unbound\n states below the charge gap, and bound states above the charge gap. Thus,\n the charge gap, often used to define the binding energy of excitons, is not a\n decisive criterion by which to decide whether a state is bound or unbound.\n\nIn practice, many authors consider excitons to consist of real-space\nconfigurations with electrons and holes occupying the nearest neighbor sites,\nwhile the electrons and holes are separated from each other in the\nconduction-band states \\cite{Guo}.\n\n The relative energy position and transition intensity of excitons and free unbound\n electron-hole pairs is an issue of large complexity. For instance,\n in polydiacetylenes \\cite{Guo} the large absorption peak at $1.85$ eV is attributed\n to an exciton, since photoconductivity is absent in this energy region. The latter\n has a threshold at around $2.4$ eV. Interestingly, that observed photoconductivity band\n is not visible in the conventional linear (one-photon) absorption spectrum.\n Because of the strong oscillator strength of the exciton, the conduction band has a\n weak oscillator strength and is enveloped by the high-energy tail of the exciton peak.\n\n\n\n\n\\subsection{Coupling with electromagnetic field}\n\nThe electron-hole excitations and optical properties of strongly correlated\n electron systems is of current both experimental and theoretical interest. In particular, the optical conductivity is of fundamental theoretical interest because the spectral weight at low frequencies seems to be the natural order parameter for the Mott transition. \nThe great body of experimental information regarding the eh-excitations in solids is provided by optical and electron energy loss (EELS) spectroscopy, however, its interpretation depends crucially on the theoretical scheme used.\nThe optical conductivity of different model systems has been studied\nby approximate mean field calculations, by analysis of integrable 1D models,\n by exact diagonalization of small\n systems, and by quantum Monte Carlo techniques. The uncertain quantitative\n applicability of the analytic mean field calculations,\n the size limitations of the exact diagonalization and Monte Carlo results\nand the possibility that the integrable 1D models do not exhibit generic\nbehavior one lead to consider other methods for obtaining information about\nthe optical conductivity.\nAn outgoing beyond the effective Hamiltonian methods with restricted basis\nwhich are relevant for description of the lowest energy excitations in the\nextremely limited range of energies, and the elaboration of effective methods\nto describe optically excited states is a challenging problem in solid state\nphysics.\n\n Making use of a standard Peierls transformation in hopping terms in (\\ref{eh}) we arrive at an\neffective Hamiltonian for the coupling with electromagnetic field \n\\begin{eqnarray}\n{\\hat t_{ij}} \\rightarrow {\\hat t_{ij}}e^{i(\\Phi _{j}-\\Phi _{i})},\n\\end{eqnarray}\n\\begin{eqnarray}\n(\\Phi _{j}-\\Phi _{i})=-\\frac{q}{\\hbar c}\\int _{{\\bf R}_{i}}^{{\\bf R}_{j}}{\\bf A}({\\bf r})d{\\bf r},\n\\end{eqnarray}\nwhere ${\\bf A}$ is the vector-potential, and integration runs over line binding the $i$ and $j$ sites. For the nearest neighbours one may use the simplified relation\n\\begin{eqnarray}\n(\\Phi _{j}-\\Phi _{i})=-\\frac{q}{\\hbar c}({\\bf A}({\\bf R}_{i})\\cdot {\\bf R}_{ji}).\n\\end{eqnarray}\n \nThe current density operator one may represent to be a sum of three terms\n\\begin{eqnarray}\n{\\bf j}^{ee}({\\bf R}_{i})=\\frac{iq}{2\\hbar }\\sum _{j}t_{ij}{\\bf R}_{ij}(\\hat a_i^\\dagger \\hat a_j-\\hat a_i\\hat a_j^\\dagger ),\\label{1}\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\bf j}^{hh}({\\bf R}_{i})=\\frac{iq}{2\\hbar }\\sum _{j}t_{ij}{\\bf R}_{ij}(\\hat b_i^\\dagger \\hat b_j-\\hat b_i\\hat b_j^\\dagger ),\\label{2}\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\bf j}^{eh}({\\bf R}_{i})=\\frac{iq}{2\\hbar }\\sum _{j}t_{ij}{\\bf R}_{ij}(\\hat a_i^\\dagger \\hat b_j^\\dagger -\\hat b_i\\hat a_j ),\\label{3}\n\\end{eqnarray}\nwhere the two first terms describe the electron and hole currents, respectively, while the third term describe the current fluctuations due to eh-pair creation\/annihilation. \n\nStandard linear-response theory then yields for optical\nconductivity ($T=0$)\n\\begin{equation}\n{\\mbox Re}\\sigma ({\\bf q},\\omega)=(\\pi \/\\omega) \\sum _{e} |\\mu _{ge}({\\bf q})|^2\n\\delta (E_{e}-E_{g}-\\hbar \\omega),\n\\end{equation}\nwhere\n$$\n\\mu _{ge}({\\bf q}) = \\langle \\Psi _{g}|j_{{\\bf q}}|\\Psi _{e}\\rangle\n$$\nis a transition matrix element, and sum runs over all the excited $\\Psi _{e}$ states. \n\nThe first two current density operators (\\ref{1})-(\\ref{2}) describe electron and hole intraband transition so that\n the optical absorption spectra in $M^0$ phase are specified only by the latter term (\\ref{3}). In other words, the optical response in $M^0$ phase of our model system is determined only through a CT exciton channel. \n \n It should be noted that the electron\/hole current operators in (\\ref{1})-(\\ref{2}) can be expressed through a superposition of antisymmetric pseudospin operators:\n\\begin{equation}\n(\\hat S_{i+}\\hat S_{j-}-\\hat S_{i-}\\hat S_{j+})-\\hat T_{i+}\\hat T_{j-}-\\hat T_{i-}\\hat T_{j+}),\n\\end{equation}\nwhile eh-current fluctuation operator in (\\ref{3}) can be expressed through another superposition of the same antisymmetric pseudospin operators\n\\begin{equation}\n(\\hat S_{i+}\\hat S_{j-}-\\hat S_{i-}\\hat S_{j+})+\\hat T_{i+}\\hat T_{j-}-\\hat T_{i-}\\hat T_{j+}).\n\\end{equation}\n\n\\subsection{Charge transfer instability and CT exciton self-trapping}\nElectron and hole in a CT exciton in type II $M^0$ insulator are strongly coupled both in between and with the lattice.\nIn contrast with conventional wide-band semiconductors where the excitons\ndissociate easily producing two-component electron-hole gas or plasma,\n\\cite{Rice} small CT excitons both free and self-trapped are likely\n to be stable with regard the eh-dissociation. \nTo illustrate the principal features of CT exciton self-trapping effect we address a simplified two-level model of a two-center MM cluster in which a ground state and a CT state are associated with a pseudospin 1\/2 doublet, $|\\uparrow\\rangle$ and $|\\downarrow\\rangle$, respectively. In addition, we introduce some configurational coordinate $Q$ associated with a deformation of the cluster, or respective anionic background.\\cite{shift} Such a model is typical one for so-called (pseudo)Jahn-Teller systems.\nAs a starting point of the model we introduce the effective electron-configurational Hamiltonian as follows\n\\begin{equation}\nH_{s}= -\\Delta {\\hat s}_{z} -t{\\hat s}_{x}- pQ +\\frac{K}{2}Q^2 - aQ{\\hat s}_{z}\\, , \\label{HH}\t\n\\end{equation}\nwhere $s_z = \\frac{1}{2}$\\begin{math}\\left(\n\\begin{array}{clrr}\n1 & 0 \\\\ \n0 & -1 \n \\end{array}\\right)\n \\end{math}, \n $s_x = \\frac{1}{2}$\\begin{math}\\left(\n\\begin{array}{clrr}\n0 & 1 \\\\ \n1 & 0 \n \\end{array}\\right)\n \\end{math}\n are Pauli matrices, and the first term describes the bare energy splitting of \"up\" and\n\"down\" states with energy gap $\\Delta $ (CT energy), while the second term describes the coupling (mixing) between \"up\" and \"down\" pseudospin states.\n In terms of a pseudospin analogy the both parameters may be associated with effective fields. The\nthird and fourth terms in (\\ref{H}) describe the linear and quadratic contributions to the configurational energy. Here, the linear term formally corresponds to the energy of an external configurational strain described by an effective strain parameter $p$, while quadratic term with \"elastic\" constant $K$ is associated with a conventional harmonic approximation for configurational energy. The last term describes the electron-configurational (vibronic)\ninteraction with $a$ being a electron-configurational coupling constant. Hereafter we make use of dimensionless configurational variable $Q$ therefore all of the model parameters are assigned the energy units. \nOur model Hamiltonian has the most general form except the simplified form of electron-configurational coupling where we omit the term $\\propto Q {\\hat s}_x$.\nIn frame of adiabatic approximation the eigenvectors for the Hamiltonian can be written as follows:\n$$\n\\Psi _+(Q) =\\cos\\alpha |\\uparrow\\rangle + \\sin\\alpha\n|\\downarrow\\rangle ;\n$$\n \\begin{equation}\n \\Psi _-(Q) =\\sin\\alpha |\\uparrow\\rangle -\n\\cos\\alpha |\\downarrow\\rangle , \\end{equation} where\n$$\n\\tan2\\alpha=\\frac{t}{\\Delta +aQ}\\, .\n$$\n The corresponding eigenvalues\n\\begin{equation}\nE_{\\pm}(Q)=\\frac{K}{2}Q^2 - pQ \\pm\\frac{1}{2}\\left[(\\Delta +aQ)^2 + t^2\n\\right]^{1\/2}\n\\end{equation}\ndefine the upper and lower branches of the configurational, or adiabatic potential (AP),\nrespectively. These potential curves describe the energy of $|\\pm\\rangle$ states as functions of configurational coordinate $Q$. \n\\begin{figure}[t]\n\\includegraphics[width=8.5cm,angle=0]{fig3.eps}\n\\caption{Typical adiabatic potentials of CT center: a) $\\Delta =1.0; t=0.1; p=-0.65; a=1.0; K=2.0$; b)$\\Delta =0.0; t=0.1; p=-0.65; a=1.0; K=2.0$; c)$\\Delta =0.0; t=0.1; p=-0.65; a=2.5; K=2.0$; d)$\\Delta =0.0; t=0.3; p=-0.65; a=2.5; K=2.0$. Arrows mark CT transitions (see text for details).} \\label{fig3}\n\\end{figure}\nThe impact of different model parameters on the shape of AP can be easily understood, if we neglect transfer integral ($t =0$) and deal with two ideal parabolas describing the configurational energy for \"up\" and \"down\" states $|\\uparrow ,\\downarrow\\rangle$, respectively. These identical parabolas with minima at\n$Q_{\\pm}^{(0)}=(p\\mp \\frac{a}{2})\/K$ are shifted relative to each other. \n The shift in\nconfigurational coordinate is $\\Delta Q = Q_{+}^{(0)} -Q_{-}^{(0)}\n=a\/2K$, while the shift in energy is $\\Delta E^{(0)} = E_{+}^{(0)}\n-E_{-}^{(0)} =pa\/K+\\Delta$, where\n$E_{\\pm}^{(0)}=E_{\\pm}(Q_{\\pm}^{(0)})$ or the energy of the bottoms\nof respective parabolas. Interestingly, the energy shift is determined both by \"mechanic\" and \"electronic\" forces.\nIt is worth noting that the $Q$-shift of the \"center of gravity\" of AP is determined by the effective strain $p$. The electron-configurational coupling leads to a pulling apart the parabolas for \"up\" and \"down\" electronic states. \nThe condition \\begin{equation} |\\Delta E^{(0)}| =\n|pa\/K+\\Delta | = a^2\/2K \\label{LU}\\end{equation} defines two specific points, where the minimum of one of the branches\ncrosses another branch, thus specifying the parameters range admissive of a $bistability$ effect. If $E_{+}^0 >E_{-}(Q_{+}^{(0)})$ we arrive at rather conventional situation which is typical for long-lived CT states capable of decaying due to spontaneous Franck-Condon radiative transitions. In opposite case $E_{+}^0 j}t_{ij}(B_{i}^{\\dagger}B_{j}+B_{j}^{\\dagger}B_{i})\n+\\sum\\limits_{i>j}V_{ij}N_{i}N_{j}-\\mu \\sum\\limits_{i}N_{i}, \\label{Bip}\n\\end{equation}\nwhere $N_{i}=B_{i}^{\\dagger}B_{i}$, $\\mu $ is a chemical potential derived from\nthe condition of fixed full number of bosons $N_{l}=\n\\sum\\limits_{i=1}^{N}\\langle N_{i}\\rangle $, or concentration $\\;n=N_{l}\/N\\in\n[0,1]$. The $t_{ij}$ denotes an effective transfer integral, $V_{ij}$ is an\nintersite interaction between the bosons. \\smallskip Here $B^{\\dagger}(B)$ are\nthe Pauli creation (annihilation) operators which are Bose-like commuting for\ndifferent sites $[B_{i},B_{j}^{\\dagger}]=0,$ for $i\\neq j,$ and for the same site $%\nB_{i}^{2}=(B_{i}^{\\dagger})^{2}=0$, $[B_{i},B_{i}^{\\dagger}]=1-2N_i$, $N_i =\nB_{i}^{\\dagger}B_{i}$; $N$ is a full number of sites.\n\nThe disproportionated phase as well as the lattice hard-core Bose-gas is\nequivalent to a system of spins $s=1\/2$ exposed to an external magnetic field in the $z$%\n-direction. Indeed, the charge $(e,h)$, or $M^{\\mp}$-doublet, that is two\ndifferent valence states of $M$-centers, one might associate with two\npossible states of the {\\it charge pseudo-spin} (isospin) $s=\\frac{1}{2}$:\n$|+\\frac{1}{2}\\rangle$ and $|-\\frac{1}{2}\\rangle$ for electron $M^-$ and hole\n$M^+$ centers, respectively. Then the effective Hamiltonian can be written as\nfollows \\cite{RMP,MFA,B}\n\\begin{equation}\nH_{BG}=\\sum_{i>j}J^{xy}_{ij}(s_{i}^{+}s_{j}^{-}+s_{j}^{+}s_{i}^{-})+\\sum\\limits_{i>j}\nJ^{z}_{ij}s_{i}^{z}s_{j}^{z}-h \\sum\\limits_{i}s_{i}^{z}, \\label{spinBG}\n\\end{equation}\nwhere $J^{xy}_{ij}=2t_{ij}$, $J^{z}_{ij}=V_{ij}$, $h =\\mu -\\sum\\limits_{j\n(j\\neq i)}V_{ij}$, $s^{-}= \\frac{1}{\\sqrt{2}}B_ , s^{+}=-\\frac{1}{\\sqrt{2}}\n B^{\\dagger}, s^{z}=-\\frac{1}{2}+B_{i}^{\\dagger}B_{i}$,\n$s^{\\pm}=\\mp \\frac{1}{\\sqrt{2}}(s^x \\pm \\imath s^y)$. Below we make use of\nconventional two-sublattice approach. For the description of the pseudospin\nordering to be more physically clear one may introduce two vectors, the\nferromagnetic and antiferromagnetic ones:\n$$\n{\\bf m}=\\frac{1}{2s}(\\langle{\\bf s}_1 \\rangle +\\langle{\\bf s}_2 \\rangle);\\,\n{\\bf l}=\\frac{1}{2s}(\\langle{\\bf s}_1 \\rangle -\\langle{\\bf s}_2\n\\rangle);\\,\\,{\\bf m}^2 +{\\bf l}^2 =1\\, ,\n$$\nwhere ${\\bf m}\\cdot {\\bf l}=0$.\nThe hard-core boson system in a two-sublattice approximation is described by\ntwo diagonal order parameters $l_z ,m_z$ and two complex off-diagonal\n order parameters: $m_{\\pm}=\\mp \\frac{1}{\\sqrt{2}}(m_x \\pm \\imath m_y)$ and\n$l_{\\pm}=\\mp \\frac{1}{\\sqrt{2}}(l_x \\pm \\imath l_y)$. The complex superfluid\norder parameter $\\Psi ({\\bf r})=|\\Psi ({\\bf r})|\\exp-\\imath\\phi $ is\ndetermined by the in-plane components of ferromagnetic vector: $ \\Psi ({\\bf \nr})=\\frac{1}{2}\\langle (\\hat B_1 +\\hat B_2 )\\rangle\n=sm_{-}=sm_{\\perp}\\exp-\\imath\\phi $, $m_{\\perp}$ being the length of the\nin-plane component of ferromagnetic vector. So, for a local condensate density\nwe get $n_s = s^2 m_{\\perp}^2$. It is of interest to note that in fact all the\nconventional uniform $T=0$ states with nonzero $\\Psi ({\\bf r})$ imply\nsimultaneous long-range order both for modulus $|\\Psi ({\\bf r})|$ and phase\n$\\phi$. The in-plane components of antiferromagnetic vector $l_{\\pm}$ describe\na staggered off-diagonal order.\n\n The model exhibits many fascinating quantum phases and phase\ntransitions. Early investigations predict at $T=0$ charge order (CO), Bose\nsuperfluid (BS) and mixed (BS+CO) supersolid uniform phases with an Ising-type\nmelting transition (CO-NO) and Kosterlitz-Thouless-type (BS-NO) phase\ntransitions to a non-ordered normal fluid (NO) in 2D systems.\\cite{RMP,MFA,B}\n\n\n\\section{Topological charge fluctuations in model mixed-valence system }\nAbove we focused on the homogeneous phase states of the mixed-valence system. \nMain short-length scale charge fluctuations in $M^0$ and $M^{\\pm}$ systems are\nassociated with a thermal exciton creation, or annihilation due to a reaction:\n$(M^0 - M^0) \\leftrightarrow (M^+ - M^- )$. Amongst the long-length scale\ncharge fluctuations in a model system we would like to address the \ntopological defects in quasi-2D systems such as cuprates, in particular, different bubble-like entities like skyrmions, or another out-of-plane vortices. Namely these one can play the main role in a\nnucleation of unconventional charge phases. \n\n\\subsection{Topological defects in $M^0$-phase}\n The most interesting situation concerns the charge fluctuations in\nthe {\\it conceptually simple} monovalent $M^0$-phase which seems to be a\nrepresentative of traditional insulating oxides. From the viewpoint of the\n${\\bf a},{\\bf b}$-vector\n field formalism the $M^0$ system resembles, in a sense, the easy-axis\n (anti)ferromagnet. Indeed, the phase is specified by a simple uniform arrangement\n of ${\\bf a}$ and ${\\bf b}$ vectors parallel to $z$-axis:\n ${\\bf a}\\parallel {\\bf b}\\parallel O_z$, and its energy does not depend on\n the sense of these vectors. The analogy allows us to make use of simple\n physically clear pictures of (anti)ferromagnetic domain structures.\n\nBelow we'll address two types of domain walls in $M^0$-phase. The first would\nillustrate the long-scale fluctuation which conserves the mean on-site valence:\nin other words, $\\langle S_z \\rangle =0$. The second would provide the example\nof a domain wall with a checkerboard charge ordering in its center.\n\n First of all, let note that instead of two parallel vectors\n${\\bf a}$ and ${\\bf b}$ given the normalization condition, the $M^0$-phase\ncan be described by a unified vector ${\\bf n}$: $\\bf a=\\alpha\\bf n, \\bf \nb=\\beta\\bf n$, and $\\alpha+i\\beta= \\exp (i\\kappa)$, $\\kappa\\in R$. Hereafter,\nwe denote $\\bf n=n\\{\\sin\\theta\\cos\\phi, \\, \\sin\\theta\\sin\\phi, \\,\n\\cos\\theta\\}$. Moreover, we may introduce a multitude of phases which differ\nonly by the orientation of the unit vector ${\\bf n}$, or ${\\bf n}$-phases.\nSuch phases may be considered to be the solutions of a purely biquadratic\npseudo-spin Hamiltonian \\cite{electr}\n \\begin{equation}\n {\\hat H}_{bq}=-J_2\\sum_{i,\\eta}\\sum_{k\\geq\nj}^3(\\{\\hat{S}_k\\hat{S}_j\\}_i\\{\\hat{S}_k\\hat{S}_j\\}_{i+\\eta}), \\label{bq}\n \\end{equation}\nwhich can be obtained from (\\ref{su3}) given the only nonzero \"exchange\"\nparameters $J_{kk}$ with $k=4\\div 8$. The quantum Hamiltonian can be mapped\nonto classical Hamiltonian \\cite{electr}\n\\begin{equation}\n H_{bq}= 2J_2|{\\bf n}|^{2}\\int d^2{\\bf r}\\left[\\sum_{i=1}^{3}(\\bf {\\nabla}n_i)^2\\right]\\, .\n\\label{hm4}\n\\end{equation}\nThe constant zero value of the mean on-site valence $\\langle S_z \\rangle =0$\nis the common property of the ${\\bf n}$-phases. Moreover, the mean value of\nall the pseudo-spin components turns into zero: $\\langle {\\bf S} \\rangle =0$.\nLet address the ferromagnetic ${\\bf n}$-phase with $180^o$ domain walls which\nseparate two actually equivalent $M^0$-domains with opposite direction of\n${\\bf n}$ vectors. The picture differs from that of conventional ferromagnets\nwhere $180^o$ domain wall separates the domains with the opposite direction of\nmagnetization. In the center of such a {\\it BS-type} domain wall we may deal\nwith a superfluid BS phase ${\\bf n}\\perp O_z$.\n\nIn 2D systems such as cuprates it appears possible to form skyrmion-like\ntopological defects like bubbles. \\cite{electr} The skyrmion spin texture looks\nlike a bubble domain in ferromagnet and consists of a vortex-like arrangement\nof the in-plane components of spin with the $z$-component reversed in the\ncentre of the skyrmion and gradually increasing to match the homogeneous\nbackground at infinity. The spin distribution within a classical Belavin-Polyakov (BP) skyrmion is\ngiven as follows \\cite{BP}\n\\begin{equation}\n\\phi=q\\varphi +\\varphi\n_0;\\quad\\cos\\theta=\\frac{r^{2q}-\\lambda^{2q}}{r^{2q}+\\lambda^{2q}}, \\label{sk}\n\\end{equation}\nor for the winding number $q=1$\n\\begin{equation}\nn_{x}=\\frac{2r\\lambda }{r^{2}+\\lambda ^{2}}\\cos \\phi ;\\, n_{y}=\\frac{2r\\lambda\n}{r^{2}+\\lambda ^{2}}\\sin \\phi ;\\, n_{z}=\\frac{r^{2}-\\lambda\n^{2}}{r^{2}+\\lambda ^{2}}.\\label{sk1}\n\\end{equation}\nSkyrmions are characterized by the magnitude and sign of its topological\ncharge, by its size (radius) $\\lambda$, and by the global orientation of the\nspin, or U(1) order parameter $\\varphi _0$. The scale invariance of skyrmionic\nsolution reflects in that its energy\nis proportional to winding number and does not depend on radius,\n and global phase $\\varphi _0$.\nAn interesting example of topological inhomogeneity is provided by a\nmulti-center BP skyrmion \\cite{BP} which energy does not depend\non the position of the centers. The latter are believed to be addressed as an\nadditional degree of freedom, or positional order parameter.\n\n\nThe classical Hamiltonian (\\ref{hm4}) has skyrmionic solutions, but instead of\nthe spin distribution in conventional BP skyrmion \\cite{BP} we\ndeal with a zero spin,\n but a non-zero distribution of five spin-quadrupole moments\n $\\langle \\Lambda ^{(4,5,6,7,8)}\\rangle$, or\n $\\langle \\{S_{i}S_{j}\\}\\rangle$ which in turn are determined\nby the skyrmionic texture of the $\\bf n$ vector:\n \\begin{widetext}\n\\begin{equation}\n\\langle (S_x^2 + S_y^{2})\\rangle =1+\\cos^{2}\\theta \\,;\\langle S_z^2\\rangle\n=\\sin^{2}\\theta \\,; \\langle (S_x^2 - S_y^{2})\\rangle =-\\sin^{2}\\theta\n\\cos2(q\\varphi + \\varphi _{0})\\,; \\label{qq}\n\\end{equation}\n$$\n\\langle \\{S_xS_y\\}\\rangle =-\\sin^{2}\\theta \\sin2(q\\varphi + \\varphi _{0});\n\\langle \\{S_xS_z\\}\\rangle =-\\sin2\\theta \\cos(q\\varphi + \\varphi _{0}); \\langle\n\\{S_yS_z\\}\\rangle =-\\sin2\\theta \\sin(q\\varphi + \\varphi _{0}).\n$$\n\\end{widetext}\n\nThus we arrive at the ring-shaped distribution of the effective ionicity\n$\\langle S_z^2\\rangle$ with the maximal value of $1$ ($M^{\\pm}$-phase) at\n$r=\\lambda$, and the minimal value of $0$ at the ring center $r=0$ and far\noutside $r\\rightarrow \\infty$ ($M^0$-phase). The \"bosonic\" off-diagonal complex\norder parameter $\\langle V_{\\pm 2}^2\\rangle \\propto \\langle S_{\\pm}^2\\rangle\n\\propto \\sin^{2}\\theta \\exp(\\pm 2(q\\varphi + \\varphi _{0}))$ has a similar\n$r$-dependence, while the \"fermionic\" off-diagonal complex order parameter\n$\\langle V_{\\pm 1}^2\\rangle \\propto \\sin2\\theta \\exp(\\pm (q\\varphi + \\varphi\n_{0}))$ turns into zero both at the ring center $r=0$, far outside\n$r\\rightarrow \\infty$ ($M^0$-phase), and at $r=\\lambda$, or everywhere, where\nthe ionicity has a strictly definite value.\n\n Despite these skyrmions are derived from the toy model, they\nyield very instructive information as regards the probable spin texture of\nreal solitons with BS-type domain walls and \"superconductive\" fluctuations in\n$M^0$-phase.\n\n The {\\it CO-type} domain walls with a nonzero mean on-site valence given the total\n $\\sum_{i}\\langle S_z \\rangle =0$ can be obtained, in common, in a two-sublattice approximation with\n non-collinear $\\bf a$ and $\\bf b$ vectors.\n Let assume the $M^0$-phase to be divided onto two sublattices, A and B,\n with ${\\bf a}_A ={\\bf b}_A ={\\bf a}_B ={\\bf b}_B $ and ${\\bf l}_A ={\\bf l}_B =0$.\nThen the CO-type domain wall may be represented by a gradual spatial\nnon-equivalent rotation of $\\bf a$ and $\\bf b$ vectors in A and B sublattices\nproviding the nonzero magnitude of $l_z$ components given $l_{zA}=-l_{zB}$ with\nits maximum in the center of the domain wall.\n\n\\subsection{Topological phase separation in 2D EH Bose liquid away from half-filling}\nOne of the fundamental hot debated problems in bosonic physics concerns the\nevolution of the charge ordered (CO) ground state of 2D hard-core BH model\n(hc-BH) with a doping away from half-filling.\nNumerous model studies steadily confirmed the emergence of \"supersolid\" phases\nwith simultaneous diagonal and off-diagonal long range order while Penrose and\nOnsager \\cite{Penrose} were the first showing as early as 1956 that supersolid\nphases do not occur.\nThe most recent quantum Monte-Carlo (QMC) simulations\n\\cite{Batrouni,Hebert,Schmid} found two significant features of the\n2D Bose-Hubbard model with a\n screened Coulomb repulsion: the absence of supersolid\nphase at half-filling, and a growing tendency to phase separation (CO+BS)\nupon doping away from half-filling. Moreover, Batrouni and Scalettar\n\\cite{Batrouni} studied quantum phase transitions in the ground state\nof the 2D hard-core boson Hubbard Hamiltonian and have shown numerically that,\ncontrary to the generally held belief, the most commonly discussed\n\"checkerboard\" supersolid is thermodynamically unstable and phase separates\ninto solid and superfluid phases. The physics of the CO+BS phase separation in\nBose-Hubbard model is associated with a rapid increase of the energy of a\nhomogeneous CO state with doping away from half-filling due to a large\n\"pseudo-spin-flip\" energy cost.\nHence, it appears to be\nenergetically more favorable to \"extract\" extra bosons (holes) from the CO\nstate and arrange them into finite clusters with a relatively small number of\nparticles. Such a droplet scenario is believed to minimize the long-range\nCoulomb repulsion.\n\nMagnetic analogy allows us to make unambiguous predictions as\nregards the doping of BH system away from half-filling. Indeed,\nthe boson\/hole doping of checkerboard CO phase corresponds to the magnetization\nof an antiferromagnet in $z$-direction. In the uniform easy-axis $l_z$-phase of\nanisotropic antiferromagnet the local spin-flip energy cost is rather big. In\nother words, the energy cost for boson\/hole doping into checkerboard CO phase\nappears to be big one due to a large contribution of boson-boson repulsion.\nHowever, the magnetization of the anisotropic antiferromagnet in\nan easy axis direction may proceed as a first order phase transition\nwith a ``topological phase separation'' due to the existence of\nantiphase domains. The antiphase domain walls\nprovide the natural nucleation centers for a spin-flop phase having\nenhanced magnetic susceptibility as compared with small if any\nlongitudinal susceptibility thus providing the advantage of the\nfield energy. Namely domain walls would specify the inhomogeneous\nmagnetization pattern for such an anisotropic easy-axis\nantiferromagnet in relatively weak external magnetic field. As\nconcerns the domain type in quasi-two-dimensional antiferromagnet\none should emphasize the specific role played by the cylindrical, or\nbubble domains which have finite energy and size. These topological\nsolitons have the vortex-like in-plane spin structure and resemble\n classical, or Belavin-Polyakov skyrmions.\\cite{BP} Although some questions were\nnot completely clarified and remain open until now,\n the classical and quantum skyrmion-like topological defects are\nbelieved to be a genuine element of essential physics both of ferro-\nand antiferromagnetic 2D easy-axis systems.\nThe magnetic analogy seems to be a little bit naive, however, it catches the\nessential physics of doping the hc-BH system.\n As regards the checkerboard CO phase of such a system, namely a finite\nenergy skyrmion-like bubble domain \\cite{bubble,bubble1} seems to be the most\npreferable candidate for the domain with antiphase domain wall providing the\nnatural reservoir for extra bosons.\nThe skyrmion spin texture\nconsists of a vortex-like\narrangement of the in-plane components of ferromagnetic ${\\bf m}$ vector with\nthe $l_z$-component reversed\nin the centre of the skyrmion and gradually increasing to match the homogeneous\nbackground at infinity. \n\nRecently \\cite{bubble1} it was shown that the doping, or deviation from half-filling in 2D EH Bose liquid is accompanied by the formation of multi-center topological defect such as charge order (CO) bubble domain(s) with Bose superfluid (BS) and extra bosons both localized in domain wall(s), or a topological CO+BS phase separation, rather than an uniform mixed CO+BS supersolid phase. \n\n The most probable possibility is that every bubble accumulates one or two\nparticles. Then, the number of such entities in a multigranular texture\nnucleated with doping has to nearly linearly depend on the doping.\n Generally speaking, each individual bubble may be characterized by its position, nanoscale size, and the\norientation of U(1) degree of freedom. In contrast with the uniform states the phase of\nthe superfluid order parameter for a bubble is assumed to be unordered. \nIn the long-wavelength limit the off-diagonal ordering can be described by an\neffective Hamiltonian in terms of U(1) (phase) degree of freedom associated\nwith each bubble. Such a Hamiltonian\n contains a repulsive, long-range Coulomb part and a\nshort-range contribution related to the phase degree of freedom. The\nlatter term can be written out in the standard for the $XY$ model form of a\nso-called Josephson coupling \\cite{Kivelson,bubble,CEF}\n\\begin{equation}\nH_J = -\\sum_{\\langle i,j\\rangle}J_{ij}\\cos(\\varphi _{i}-\\varphi _{j}),\n\\end{equation}\nwhere $\\varphi _{i},\\varphi _{j}$ are global phases for micrograins centered at\npoints $i,j$, respectively, $J_{ij}$ Josephson coupling parameter. Namely the\nJosephson coupling gives rise to the long-range ordering of the phase of the\nsuperfluid order parameter in such a multi-center texture. Such a Hamiltonian\nrepresents a starting point for the analysis of disordered superconductors,\ngranular superconductivity, insulator-superconductor transition with $\\langle\ni,j\\rangle$ array of superconducting islands with phases $\\varphi _{i},\\varphi\n_{j}$.\nTo account for Coulomb interaction and allow for quantum corrections we should\nintroduce into effective Hamiltonian the charging energy \\cite{Kivelson}\n$$\nH_{ch}=-\\frac{1}{2}q^2 \\sum_{i,j}n_{i}(C^{-1})_{ij}n_{j}\\, ,\n$$\nwhere $n_{i}$ is a number operator for particles bound in $i$-th micrograin;\nit\nis quantum-mechanically conjugated to $\\varphi$: $n_{i}=-i \\partial \/\\partial\n\\varphi\n_{i}$, $(C^{-1})_{ij}$ stands for the capacitance matrix, $q$ for a particle\ncharge.\n\n Such a system appears to reveal a tremendously rich quantum-critical\nstructure.\\cite{Green,Timm} In the absence of disorder, the\n$T=0$ phase diagram of the multi-bubble system implies either triangular, or\nsquare crystalline arrangements with possible melting transition to a liquid.\n It should be noted that analogy with charged $2D$ Coulomb gas\nimplies the Wigner crystallization of multi-bubble system with Wigner\ncrystal (WC) to Wigner liquid melting transition, respectively. Naturally, that\nthe\nadditional degrees of freedom for a bubble provide a richer physics of\nsuch lattices. For a system to be an insulator, disorder is required, which\npins the multigranular system\nand also causes the crystalline order to have a finite correlation length.\nTraditional approach to a Wigner crystallization implies the formation of a WC\nfor densities lower than a critical density, when the Coulomb energy dominates\nover the kinetic energy. The effect of quantum fluctuations leads to a\n(quantum) melting of the solid at high densities, or at a critical lattice\nspacing. The critical properties of a two-dimensional lattice without any\ninternal degree of freedom are successfully described applying the BKT (Berezinsky-Kosterlitz-Thowless) theory\nto dislocations and disclinations of the lattice, and proceeds in two steps.\nThe first implies the transition to a liquid-crystal phase with a short-range\ntranslational order, the second does the transition to isotropic liquid.\n For such a system provided the bubble positions fixed at all temperatures, the long-wave-length physics would be\ndescribed by an (anti)ferromagnetic $XY$ model with expectable BKT transition and\ngapless $XY$ spin-wave mode.\n\n\nThe low temperature physics in a multi-bubble system is governed by an\ninterplay of two BKT transitions, for the U(1) phase and positional degrees\nof freedom, respectively.\\cite{Timm}\nDislocations lead to a mismatch in the U(1)\ndegree of freedom, which makes the dislocations bind fractional vortices and\nlead to a coupling of translational and phase excitations. Both BKT temperatures\neither coincide (square lattice) or the melting one is higher (triangular\nlattice).\\cite{Timm}\n\n Quantum fluctuations can substantially affect these\nresults. Quantum melting can destroy U(1) order at sufficiently\nlow densities where the Josephson coupling becomes exponentially small. Similar\nsituation is expected to take place in the vicinity of\nstructural transitions in a multigranular crystal. With increasing the\nmicrograin density the quantum effects result in a significant lowering of the melting\ntemperature as compared with classical square-root dependence.\nThe resulting melting temperature can reveal an oscilating behavior as a\nfunction of particle density with zeros at the critical (magic) densities\nassociated with structural phase transitions. \n\nIn terms of our model, the positional order corresponds to an incommensurate\ncharge density wave, while the U(1) order does to a superconductivity. In other\nwords, we arrive at a subtle interplay between two orders. The superconducting\nstate evolves from a charge order with $T_C \\leq T_m$, where $T_m$ is the\ntemperature of a melting transition which could be termed as a temperature of\nthe opening of the insulating gap (pseudo-gap!?).\n\n\n\n\nThe normal modes of a dilute multi-bubble system include the pseudo-spin waves\npropagating in-between the bubbles, the positional fluctuations, or\nquasi-phonon modes, which are gapless in a pure system, but gapped when\nthe lattice is pinned, and, finally, fluctuations in the U(1) order parameter.\n\n The orientational fluctuations of the multi-bubble system are governed by the gapless\n $XY$ model.\\cite{Green} The relevant model description is most familiar as\nan effective theory of the Josephson junction array. An important feature of\nthe model is that it displays a quantum-critical point.\n\nThe low-energy collective excitations of a multi-bubble\nliquid includes an usual longitudinal acoustic phonon-like branch.\nThe liquid crystal phases differ from the isotropic liquid in that they have\nmassive topological excitations, {\\it i.e.}, the disclinations.\nOne should note that the liquids do not support transverse modes, these could\nsurvive in a liquid state only as overdamped modes. So that it is reasonable\nto assume that solidification of the bubble lattice would be accompanied by a\nstabilization of transverse phonon-like modes with its sharpening below melting\ntransition.\nIn other words, an instability of transverse phonon-like modes signals the\nonset of melting.\nThe phonon-like modes in the bubble crystal have much in common with usual phonon\nmodes, however, due to electronic nature these can hardly be detected if any\nby inelastic neutron scattering.\n\nA generic property of the positionally ordered bubble configuration is the\nsliding mode which is usually pinned by the disorder. The depinning of sliding\nmode(s) can be detected in a low-frequency and low-temperature optical\nresponse.\n\n\\section{Implications for strongly correlated oxides}\nIn this Section we suggest some speculations around an unconventional scenario\nof the essential physics of cuprates and manganites that implies their\ninstability with regard to the self-trapping of charge transfer excitons and the\nformation of electron-hole Bose liquid.\n\n\n\\subsection{Cuprates}\n The origin of high-T$_c$\nsuperconductivity is presently still a matter of great controversy.\n The unconventional behavior of\ncuprates strongly differs from that of ordinary metals and merely resembles\nthat of doped semiconductor. Moreover, the history of high T$_c$'s itself shows\nthat we deal with a transformation of particularly insulating state in which\nthe electron correlations govern the essential physics.\n\n\nThe copper oxides start out life as insulators in contrast with BCS\nsuperconductors being conventional metals. It is impossible to understand the\nbehavior of the doped cuprates and, in particular, the origin of HTSC unless\nthe nature of the doped-insulating state is incorporated into the theory. The\nproblem of a doped insulator is sure much more complicated than it is implied\nin oversimplified approaches such as an effective $t$-$J$-model when the\nsituation resembles that of ``throwing the baby out of the bathwater''.\n\n In a case of cuprates we deal with systems which conventional ground state seems\n to be unstable with regard to the transformation into a new phase state with\na variety of unusual properties.\n\nIn our view, the essential physics of the doped cuprates, as well as other\nstrongly correlated oxides, is driven by a self-trapping of the CT excitons,\nboth one-center, and two-center. \\cite{CT,CT1} Such excitons are the result\nof self-consistent charge transfer and lattice distortion with the appearance\nof the ``negative-$U$'' effect.\\cite{Shluger,Vikhnin} The two-center excitons\nare associated with CT transitions between two CuO$_4$ plaquettes, and may be\nconsidered as quanta of the disproportionation reaction\n$$\n\\mbox{CuO}_{4}^{6-}+\\mbox{CuO}_{4}^{6-}\\rightarrow \\mbox{CuO}_{4}^{7-}+\n\\mbox{CuO}_{4}^{5-}\n$$\nwith the creation of electron CuO$_{4}^{7-}$ and hole CuO$_{4}^{5-}$ centers.\nThus, three types of CuO$_4$ centers CuO$_{4}^{5,6,7-}$ should be considered\non equal footing.\n In this connection we would\nlike to draw special attention to the lattice polarization and relaxation\neffects that are of primary importance both for the formation of CT exciton\nitself and its self-trapping. It should be noted that the photo-excited\nelectron-hole pairs or excitons are stabilized into self-trapped excitons (STE)\naccompanied with lattice relaxation within several pico-seconds.\\cite{Toyozawa}\n\nIn contrast with BaBiO$_3$ system where we deal with a spontaneous generation\nof self-trapped CT excitons in the ground state, the parent insulating\ncuprates are believed to be near excitonic instability when\n the self-trapped CT excitons form the candidate relaxed excited states to struggle with the\nconventional ground state. \\cite{Toyozawa} In other words, the lattice relaxed\nCT excited state should be treated on an equal footing with the ground state.\nIf the interaction between STE were attractive and so large that the cohesive\nenergy $W_1$ per one STE exceeds the energy $E_R$ of one STE , the STE's and\/or\nits clusters will be spontaneously generated everywhere without any optical\nexcitation, and be condensed to form a new electronic state on a new lattice\nstructure. \\cite{Toyozawa}\n \nThe minimal energy cost of the optically excited disproportionation or\nelectron-hole formation in insulating cuprates is $2.0\\div 2.5$ eV.\n\\cite{CT,CT1} Interestingly, that this relatively small value of the optical\ngap was nevertheless used by Goodenough \\cite{Good} as argument against the\n``negative-U'' disproportionation reaction 2Cu(II) = Cu(III) + Cu(I), or more\ncorrectly 2[CuO$_{4}^{6-}$]=[CuO$_{4}^{5-}$]+[CuO$_{4}^{7-}$] in cuprates.\nHowever, the question arises, what is the energy cost for the thermal\nexcitation of such a disproportionation? In other words, what is the energy of\nself-trapped two-center CT exciton? It is this quantity rather than its optical\ncounterpart defines the activation energy for such a reaction. The question\nis of primary importance for the self-trapping scenario. The answer implies\nfirst of all the knowledge of electronic and ionic polarization energies for\nelectron and hole. The polarization effects with its typical energy scale of\nthe order of several eV appear to be of primary importance when one considers\ndifferent charge fluctuations in insulators.\n\nSeveral general theories of self-trapping have been proposed starting from\nworks by Landau, Pekar and Toyozawa, however, the stages of the ST process and\ndetailed atomistic and electronic structure of ST-excitons and ST-holes are\nstill unclear even in rather simple and well-studied systems such as\nalkali-metal halides.\n\n\n\n\nAs regards the STE in cuprates, we have some straightforward experimental\nindications. A key characteristic of the STE is its luminescence: STE are\nshort-lived luminescent states of excited crystals. The observation of\nphotoluminescence (PL) near $2.0\\div 2.4$ eV in La$_2$CuO$_4$,\n\\cite{Salamon,Ginder} near $1.3$ and $2.4$ eV in YBa$_2$Cu$_3$O$_{6}$,\n\\cite{Salamon,Denisov,Eremenko} near $1.78,\\,1.95,\\,2.06$ eV in\nPrBa$_2$Cu$_3$O$_{6}$ \\cite{Salamon} is a direct evidence of strongly localized\nlong-lived states related to self-trapped excitons or their derivatives. The\nnear-infrared photoluminescence was observed in many insulating cuprates.\n\\cite{Denisov,Salamon,Sugai} To the best of our knowledge the most detailed PL\nstudy was performed by Denisov {\\it et al}. \\cite{Denisov} in\nYBa$_2$Cu$_3$O$_{6+x}$ in the spectral range $0.7\\div 1.4$ eV. The\nlow-temperature PL in YBa$_2$Cu$_3$O$_6$ consists of three peaks at $0.87$,\n$1.07$, and $\\sim 1.4$ eV, respectively. The PL intensity is much stronger at\nsmall doping level. Moreover, the doping induced PL suppression manifests\nitself more strongly for the low-energy than for the high-energy PL peaks. At\n$x=0.15$ only the high-energy peak located at $1.28$ eV ($T=10 \\,K$) survives\nthat allows us to assume that the STE decay becomes more effective with doping.\nThe high-energy PL peak red-shifts with the lowering the temperature, and its\nintensity decreases.\n\nAll these features can be consistently explained in frames of the STE nature\nof PL. Different PL peak can be assigned both to different STE and its clusters\npointing to the multistage character of the luminescence.\n\n\n\n\nCuprates are believed to be unconventional systems which are unstable with\nregard to a self-trapping of the low-energy charge transfer excitons with a\nnucleation of electron-hole (EH) droplets being actually the system of coupled\nelectron CuO$_{4}^{7-}$ and hole CuO$_{4}^{5-}$ centers having been glued in\nlattice due to a strong electron-lattice polarization effects. Phase transition\nto novel hypothetically metallic state could be realized due to a\n mechanism familiar to semiconductors with filled bands such as Ge and Si where\n given certain conditions one observes\n a formation of metallic EH-liquid as a result of the exciton decay.\n \\cite{Rice}\n The system of strongly correlated electron CuO$_{4}^{7-}$ and hole\nCuO$_{4}^{5-}$ centers appears to be equivalent to an electron-hole\nBose-liquid (EHBL) in contrast with the electron-hole Fermi-liquid in\nconventional semiconductors. A simple model description of such a liquid\nimplies\n a system of local singlet bosons with a charge of $q=2e$ moving in a lattice\n formed by hole centers.\nLocal boson in our scenario represents the electron counterpart of Zhang-Rice\nsinglet, or two-electron configuration $b_{1g}^{2}{}^{1}A_{1g}$.\n Naturally, that conventional electron CuO$_{4}^{7-}$ center represents a\n relaxed state of composite system: \"hole CuO$_{4}^{5-}$ center plus local\n singlet boson\", while the \"non-retarded\" scenario of a novel phase\nis assumed to incorporate the unconventional states of electron CuO$_{4}^{7-}$\ncenter up to its orbital degeneracy.\n\n\nThus we can introduce the concept of insulator-to-EHBL transition as the\nspontaneous condensation of self-trapped excitons and its clusters. However, in\ncuprates we deal with\n the electron\/hole injection to the insulating\nparent phase due to a nonisovalent substitution as in\nLa$_{2-x}$Sr$_x$CuO$_{4}$, Nd$_{2-x}$Ce$_x$CuO$_{4}$, or change in oxygen\nstoihiometry as in YBa$_2$Cu$_3$O$_{6+x}$, La$_{2}$CuO$_{4-\\delta}$,\nLa$_{2}$Cu$_{1-x}$Li$_x$O$_{4}$. Such a substitution provokes the nucleation of\nEH droplets and shifts the phase equilibrium from the insulating state to the\nunconventional electron-hole Bose liquid, or, in other words, induces the\ninsulator-to-EHBL phase transition. Hence, the formation of EHBL in cuprates\ncan be considered as the first order phase transition. The doping in cuprates\ngradually shifts the EHBL state away from half-filing.\n\n\nIt is clear that the EHBL scenario makes the doped cuprates the objects of\n$bosonic$ physics. There are numerous experimental evidence that support the\nbosonic scenario for doped cuprates.\\cite{ASA} In this connection, we would\nlike to draw attention to the little known results of comparative\nhigh-temperature studies of thermoelectric power and conductivity which\nunambiguously reveal the charge carriers with $q=2e$, or two-electron(hole)\ntransport.\\cite{Victor} The well-known relation $\\frac{\\partial\n\\alpha}{\\partial \\ln \\sigma}=const=-\\frac{k}{q}$ with $|q|=2|e|$ is fulfilled\nwith high accuracy in the limit of high temperatures ($\\sim 700\\div 1000 K$)\nfor different cuprates (YBa$_{2}$Cu$_{3}$O$_{6+x}$,\nLa$_{3}$Ba$_{3}$Cu$_{6}$O$_{14+x}$,\n(Nd$_{2\/3}$Ce$_{1\/3}$)$_{4}$(Ba$_{2\/3}$Nd$_{1\/3}$)$_{4}$Cu$_{6}$O$_{16+x}$).\n\n\n \\subsection{Manganites}\n Parent manganites such\nas LaMnO$_3$ are antiferromagnetic insulators with the charge transfer\n gap. Fundamental absorption band in parent manganites is formed both by the\n intracenter O2p-Mn3d transfer \\cite{LMnO3} and by the small intercenter charge transfer\n excitons,\\cite{Khaliullin} which in terms of chemical notions represent somewhat like the disproportionation\n quanta with a rather low threshold of the order of 3 eV, resulting in a formation\n of electron MnO$_6^{10-}$ and hole\n MnO$_6^{8-}$ centers. The CT excitons in LaMnO$_3$ prone to a self-trapping \\cite{Kovaleva}\n and may be considered to be well defined entities only at\n small content, whereas at large densities their coupling is screened and their overlap becomes\n so considerable that they loose individuality, become unstable with regard to the decay\n (the dissociation) to electron and hole centers, and we come to a system of electrons and\n holes, which form an EH Bose liquid. \n\nAn instability of parent manganite LaMnO$_3$ with regard to the overall\ndisproportionation such as\n\\begin{equation}\n\\mbox{MnO}_{6}^{9-}+ \\mbox{MnO}_{6}^{9-} \\rightarrow \\mbox{MnO}_{6}^{10-}+\n\\mbox{MnO}_{6}^{8-} \\label{dispro}\n\\end{equation}\n was strikingly demonstrated recently by Zhou and Goodenough. \\cite{Zhou} The transport\n (thermoelectric power and resistivity) and magnetic (susceptibility) measurements showed that\n LaMnO$_3$ above the cooperative Jahn-Teller orbital-ordering temperature $T_{JT}\\approx 750$ K\n transforms into charge-disproportionated paramagnetic phase with $\\mu _{eff}=5.22 \\mu _{B}$\n and cooperative charge transfer of many heavy vibronic charge carriers. It\n seems rather obvious that with the lowering the temperature we arrive at a\n system with the well developed fluctuations of the EH Bose-liquid phase.\nStrong variation of the LaMnO$_3$ Raman spectra with increasing laser power\n\\cite{Raman} could be related to the photo-induced nucleation and the volume\nexpansion of the EH Bose-liquid, especially, as at a rather big excitation\nwavelength $\\lambda = 514.5$ nm, at $\\lambda = 632.8$\n nm the absorption is considerably stronger in the domains of novel phase than in\n parent lattice.\n\nEffective nucleation of the EH Bose-liquid in manganites could be provoked by a\nnon-isovalent substitution since this strongly polarizable or even metallic\nphase in contrast with parent insulating phase provides an effective screening\nof charge inhomogeneity. Indeed, in thermoelectric power (TEP) experiments with\ndoped manganites such as La$_{1-x}$Sr$_x$MnO$_3$ Hundley and Neumeier\n\\cite{TEP} have found that more hole-like charge carriers or alternatively\nfewer accessible Mn sites are present than expected for the value $x$. They\nsuggest a charge disproportionation model based on the instability of\n Mn$^{3+}$-Mn$^{3+}$ relative to that of Mn$^{4+}$-Mn$^{2+}$. This transformation provides excellent\n agreement with doping-dependent trends exhibited by both TEP and resistivity.\n\n\n A simplified \"chemical\" approach to an EH Bose-liquid as to a\ndisproportionated phase \\cite{Ionov} naively implies an occurrence of Mn$^{4+}$\nand Mn$^{2+}$ ions. However, such an approach is very far from reality. Indeed,\nthe electron MnO$_6^{10-}$ and hole MnO$_6^{8-}$ centers are already the mixed\nvalence centers,\\cite{Mn-ehl} as in the former the $Mn$ valence resonates between $+2$ and\n$+1$, and in the latter does between $+4$ and $+3$.\n\nIn this connection, one should note that in a sense disproportionation reaction\n(\\ref{dispro}) has several purely ionic counterparts, the two rather simple\n\nMn$^{3+}$-O$^{2-}$-Mn$^{3+} \\rightarrow$ Mn$^{2+}$-O$^{2-}$-Mn$^{4+}$,\n\nand\n\nMn$^{3+}$-O$^{2-}$-Mn$^{3+} \\rightarrow$ Mn$^{2+}$-O$^{1-}$-Mn$^{3+}$,\n\nand, finally, one rather complicated\n\nO$^{2-}$-Mn$^{3+}$-O$^{2-} \\rightarrow $ O$^{1-}$-Mn$^{1+}$-O$^{1-}$.\n\nThus, the disproportionation (\\ref{dispro}) threshold energy has to be\nmaximally close to the CT energy parameter $\\Delta _{pd}$. Moreover, namely\nthis is seemingly to be one of the main parameters governing\n the nucleation of EH Bose-liquid in oxides.\n\nSo far, there has been no systematic exploration of exact valence and spin\nstate of Mn in these systems. Park {\\it et al}. \\cite{Park} attempted to\nsupport the Mn$^{3+}$\/Mn$^{4+}$ model, based on the Mn 2p x-ray photoelectron\nspectroscopy (XPES) and O1s absorption. However, the significant discrepancy\nbetween the weighted Mn$^{3+}$\/Mn$^{4+}$ spectrum and the experimental one for\ngiven\n $x$ suggests a more complex doping effect. Subias et al.\\cite{Subias} examined the valence\n state of Mn utilizing Mn $K$-edge x-ray absorption near edge spectra (XANES), however,\n a large discrepancy is found between experimental spectra given intermediate doping and\n appropriate superposition of the end members.\n\nThe valence state of Mn in Ca-doped LaMnO$_3$ was studied by high-resolution Mn\n$K\\beta $ emission spectroscopy by Tyson {\\it et al.} \\cite{Tyson}. No evidence\nfor Mn$^{2+}$ was observed at any $x$ values seemingly ruling out proposals\nregarding the Mn$^{3+}$ disproportionation. However, this conclusion seems to\nbe absolutely unreasonable. Indeed, electron center\n MnO$_{6}^{10-}$ can be found in two configuration with formal Mn valence Mn$^{2+}$ and\n Mn$^{1+}$ (not simply Mn$^{2+}$!), respectively. In its turn, the hole center MnO$_{6}^{8-}$\n can be found in two configurations with formal Mn valence Mn$^{4+}$ and Mn$^{3+}$ (not simply\n Mn$^{4+}$ !), respectively. So, within the model the Mn $K\\beta $ emission spectrum for\n Ca-doped LaMn$O_3$ has to be a superposition of appropriately weighted Mn$^{1+}$, Mn$^{2+}$,\n Mn$^{3+}$, Mn$^{4+}$ contributions (not simply Mn$^{4+}$ and Mn$^{3+}$, as one assumes in\n Ref. \\onlinecite{Tyson}). Unfortunately, we do not know the Mn $K\\beta $ emission spectra for the oxide\n compounds with Mn$^{1+}$ ions, however a close inspection of the Mn $K\\beta $ emission\n spectra for the series of Mn-oxide compounds with Mn valence varying from $2+$ to $7+$\n (Fig.2 of the cited paper) allows to uncover a rather clear dependence on valence, and\n indicates a possibility to explain the experimental spectrum for Ca-doped LaMnO$_3$\n as a superposition of appropriately weighted Mn$^{1+}$, Mn$^{2+}$, Mn$^{3+}$, Mn$^{4+}$\n contributions. It should be noted that an \"arrested\" Mn-valence response to the doping\nin the $x<0.3$ range founded in Ref. \\onlinecite{Tyson} is also consistent with\nthe creation of predominantly oxygen holes.\n\nThis set of conflicting data together with a number of additional data\n\\cite{Croft} suggests the need for an in-depth exploration of the Mn valence\nproblem in this perovskite system. However, one might say, the doped\nmanganites are not only systems with mixed valence, but systems with\n$indefinite$ valence, where we cannot, strictly speaking, unambiguously\ndistinguish Mn species with either distinct valence state.\n\n\\section{Conclusions}\n \n We have developed a model approach to describe charge fluctuations and different charge phases in strongly correlated 3d oxides. In frames of $S=1$ pseudo-spin formalism different phase states of the\nsystem of the metal-oxide $M$ centers with three different valent state $M^{0,\\pm}$ are\nconsidered on the equal footing. Simple uniform mean-field phases include an insulating monovalent $M^0$-phase, mixed-valence binary (disproportionated) $M^{\\pm}$-phase, and mixed-valence ternary (``under-disproportionated'') $M^{0,\\pm}$-phase. We consider two first phases in more details focusing on the problem of electron\/hole states and different types of excitons in $M^0$-phase and formation of electron-hole Bose liquid in $M^{\\pm}$-phase. \n\nOur consideration was focused mainly on a number of issues seemingly being of primary importance for the various strongly correlated oxides such as cuprates, manganites, bismuthates, and other systems with\n CT instability and\/or mixed valence. These includes two types of single particle correlated hopping and the two-particle hopping, CT excitons, electron-lattice polarization effects which are shown to be crucial for the stabilization of either phase, topological charge fluctuations, nucleation of droplets of the electron-hole Bose liquid and phase separation effect. We emphasize an important role of self-trapped CT excitons in typical Mott-Hubbard insulators as candidate \"relaxed excited states\" to struggle for stability with ground state and natural nucleation centers for unconventional electron-hole Bose liquid which phase state include the superfluid.\n \n Pseudo-spin formalism has appeared to be very efficient to reveal and describe different aspects of essential physics for mixed-valence system. We show that the coherent states provide the optimal way both to a correct mean-field approximation and respective continuous models to describe the pseudo-spin system including different topological charge fluctuations, in particular, like domain walls or bubble domains in antiferromagnets. \n All the insulating systems such as $M^0$-phase may be subdivided to two classes: stable and unstable ones with regard to the formation of self-trapped CT excitons. The latter systems appear to be unstable with regard the formation of CT exciton clusters, or droplets of the electron-hole Bose liquid. \nThe model approach suggested is believed to provide a conceptual framework for an in-depth understanding of physics of\n strongly correlated oxides such as cuprates, manganites, bismuthates, and other systems with\n charge transfer excitonic instability and\/or mixed valence. We shortly discuss an unconventional scenario\nof the essential physics of cuprates and manganites that implies their instability with regard to the self-trapping of charge transfer excitons and the formation of electron-hole Bose liquid.\n\n\n\nAuthor acknowledges the stimulating discussions with V. Vikhnin, A.V. Mitin, S.-L. Drechsler, T. Mishonov, R. Hayn, I. Eremin, M. Eremin, Yu. Panov, V.L. Kozhevnikov and support by SMWK Grant, INTAS Grant No. 01-0654, CRDF Grant No. REC-005, RFBR Grant No. 04-02-96077.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe success of many machine learning algorithms with empirical risk minimization (ERM) relies on the independent and identically distributed (i.i.d.) hypothesis that training and test data originate from a common distribution.\nIn practice, however, data in different domains or environments are often heterogeneous, due to changing circumstances, selection bias, and time-shifts in the distributions \\cite{meinshausen2015maximin, rothenhausler2021anchor}. Accessing data from all the domains of interest, on the other hand, is expensive or even impossible. Consequently, the problem of learning a model that generalizes well on the unseen target distributions is a practically important but also challenging task and has gained much research attention in the past decades \\cite{blanchard2011generalizing,blanchard2021domain,fang2013unbiased,muandet2013domain,volpi2018generalizing}.\n\\par\nSince data from some domains are unavailable, assumptions or prior knowledge on the unseen domains are generally required to achieve a guaranteed out-of-distribution (OOD) generalization performance. Recently, causality has become a powerful tool to tackle the OOD problem \\cite{peters2017elements, rojas2018invariant,arjovsky2019invariant,rothenhausler2021anchor}. This is based on the assumption or observation that the underlying causal mechanism is invariant in general, even though the data distributions may vary with domains. It has been shown that a model would perform well across different domains in the minimax sense if such a causal mechanism is indeed captured.\n\n\n\n\n\\par\nTo capture the invariant causal mechanism, existing works have assumed a particular form of the causal diagram \\cite{rothenhausler2021anchor,subbaswamy2019preventing,mitrovic2020representation,heinze2021conditional,mahajan2021CausalMatching}, which may be restrictive in practice and is untestable from the observed data. Other works try to recover the so-called ``causal feature'' from the data to improve the OOD generalization performance \\cite{rojas2018invariant,chang2020invariant,liu2021heterogeneous,gimenez2021identifying}. \nThese works usually assume a linear form of causal feature \\cite{chang2020invariant, rothenhausler2021anchor,liu2021heterogeneous,gimenez2021identifying} or that there are sufficiently many and diverse training domains so that the causal feature could be identified via certain invariant properties \\cite{peters2016causal,rojas2018invariant,arjovsky2019invariant}. In the absence of these assumptions, existing methods such as invariant risk minimization \\cite{arjovsky2019invariant} can fail to capture the invariance or recover the causal feature\neven in simple examples \\cite{kamath2021doesIRM}. In real applications like image classification, the linearity assumption may not hold, and it may be expensive or even impossible to ensure that the available domains are indeed sufficient. As such, the identifiability issue of causal feature can hardly be resolved in practice. \n\\par \nIn this paper, we obviate the aforementioned assumptions and propose a new approach to learn a robust model for OOD generalization under the invariant causal mechanism assumption. We do not try to \\emph{explicitly} recover the causal feature; rather, we directly learn a model that takes advantage of the invariant properties. Our approach is based on the observation that though the explicit functional form of the causal feature is generally unknown and maybe also hard to learn, we often have some prior knowledge on the transformations that the causal feature is invariant to, i.e., transformations that modify the input data but do not change their causal feature. For example, the shape of digit in an image from \\texttt{MNIST} dataset \\cite{lecun1998gradient} can be treated as a causal feature when predicting the digit, while flipping or rotation does not change causal meanings. A detailed discussion on this issue is given in \\Cref{subsec: prior knowledge}. We refer to these transformations as \\emph{causal invariant transformations} (CITs). \n\\par\nTheoretically, we prove that given complete prior knowledge of CITs, it is feasible to learn a model with OOD generalization capability using only single domain data. Specifically, we show that if all the CITs are known, then minimizing the loss over all the causally invariant transformed data, which are obtained by applying the CITs to data from the given single domain, would lead to the desired model that achieves a minimax optimality across all the domains of interest. Noticing that obtaining all CITs may be impractical, we further show that, for the purpose of OOD generalization, it suffices to know only an appropriate subset of CITs, referred to as \\emph{causal essential set} and is formally defined in \\cref{def:causal essential set}. The learned model is then shown to generalize to different domains if it is invariant to transformations in a causal essential set. This is different from existing works \\cite{sokolic2017generalization,sannai2019improved} that demonstrate an improved i.i.d.~generalization capability from invariance properties. \n\nFollowing these theoretical results, we propose to regularize training with the discrepancy between the model outputs of the original data and their transformed versions from the CITs in the causal essential set, to enhance OOD generalization. The CITs can be viewed as data augmentation operations; in this sense, our theoretical results reveal the rationale behind data augmentation in OOD problems.\tExperiments on both synthetic and real-world benchmark datasets, including \\texttt{PACS} \\cite{li2017deeper} and \\texttt{VLCS} \\cite{fang2013unbiased}, verify our theoretical findings and demonstrate the effectiveness of the proposed algorithm in terms of OOD performance. {Noticeably, in some experiments, we use CycleGAN to learn the transformations between different environments, which are then used as our CITs. This is in contrast with \\cite{zhou2020learning} which conjectured that source-to-source transformation could provide little help to domain generation tasks in their approach.}\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\nAs data from some unseen domains are completely unavailable, assumptions or prior knowledge on the data distributions are required to guarantee a good OOD generalization performance. We will briefly review existing domain generalization methods according to these assumptions.\n\n\\paragraph{Marginal Transfer Learning} A branch of works assume that distributions under different domains are i.i.d.~realizations from a superpopulation of distributions and augment the original feature space with the covariate distribution \\cite{blanchard2011generalizing,blanchard2021domain}. This i.i.d.~assumption on the data distribution is akin to the random effect model \\cite{laird1982random,bondell2010joint} or Bayesian approach \\cite{Deely1981EB,ray2020semiparametric}, but may be inappropriate when the difference between domains is irregular, e.g., different styles and backgrounds in the \\texttt{PACS} and \\texttt{VLCS} datasets, respectively.\n\n\\paragraph{Robust Optimization} Existing works also consider OOD data that lie close to the training distribution in terms of probability distance or divergence, e.g., Wasserstein distance \\cite{sinha2018certifying,volpi2018generalizing,lee2017minimax,yi2021improved} or $f$-divergence \\cite{hu2018does,gao2020wasserstein,duchi2021learning}. They proposed to train the model via distributional robust optimization so that the model generalizes well over a set of distributions, the so-called uncertainty set \\cite{ben2013robust,shapiro2017distributionally}. However, picking a suitable probability distance and range of uncertainty set in real scenarios is difficult in practice \\cite{duchi2021learning}. Besides, the distributions in the uncertainty set are actually the distributions of corrupted OOD data such as adversarial sample and noisy corrupted data \\cite{yi2021improved}, while the commonly encountered style-transformed OOD data are not included \\cite{hendrycks2020many}.\n\n\\paragraph{Invariant Feature} \nAnother branch of methods aim to seek a model with features whose (conditional) distributions are invariant across different domains. To this end, it was proposed to learn the feature representation by minimizing some loss functions involving domain scatter \\cite{muandet2013domain,ghifary2016scatter,li2018domain}. Here domain scatter is a quantity characterizing the dissimilarity between (conditional) distributions in different domains, as defined in \\cite{ghifary2016scatter}. \\cite{li2018adversarial} and \\cite{li2018deep} considered to regularize training to reduce the maximum mean discrepancy of the feature distributions of different domains and the Jensen-Shannon divergence of the feature distributions conditional on the outcome, respectively. The rationale behind these methods is to minimize a term that appears in the upper bound of the prediction error in unseen target domains \\cite{ben2007analysis,ben2010theory, ghifary2016scatter}. Theoretically, the success of these methods hinges on the assumption that other terms in the upper bound are small enough \\cite{ghifary2016scatter}. However, the implication of this assumption is usually unclear and provides little guidance for the practitioner \\cite{chen2020domain}.\nAlthough often not stated explicitly, the validity of these methods relies on the \\emph{covariate shift} or \\emph{label shift} assumption that is implausible if spurious correlations occur under certain domains \\cite{chen2020domain,zhou2021domain,kuang2018stable,liu2021heterogeneous}.\n\n\\paragraph{Invariant Causal Mechanism}\nAs in this paper, many existing works also resort to causality to study the OOD generalization problem \\cite{rojas2018invariant,arjovsky2019invariant,chang2020invariant,mitrovic2020representation,ahuja2020invariant,liu2021heterogeneous,heinze2021conditional, gimenez2021identifying}. In the last few years, the relationship between causality, prediction, and OOD generalization has gained increasing interest since the seminal work of \\cite{peters2016causal}. \nThe causality-based methods rest on the long-standing assumption that causal mechanism is invariant across different domains \\cite{peters2017elements}. To utilize the invariant causal mechanism and hence improve OOD generalization, some works impose restrictive assumptions on the causal diagram or structural equations \\cite{subbaswamy2019preventing,heinze2021conditional,rothenhausler2021anchor,mahajan2021CausalMatching}. Another way is through recovering the causal feature \\cite{rojas2018invariant,chang2020invariant, gimenez2021identifying}. For example, \\cite{rojas2018invariant} proposed to select causal variables by statistical tests for equality of distributions, and \\cite{chang2020invariant} leveraged some conditional independence relationships induced by the common causal mechanism assumption. It is worth noting that recovering the causal feature generally relies on restrictive assumptions, e.g., linear structural model or sufficiently many and diverse training domains \\cite{rojas2018invariant,chang2020invariant,gimenez2021identifying,peters2016causal,arjovsky2019invariant,liu2021heterogeneous, krueger2021out}; see \\cite{rosenfeld2020risks} for a further discussion on these two assumptions. Without these assumptions, existing methods such as invariant risk minimization \\cite{arjovsky2019invariant} can fail to choose\nthe right predictor even in simple examples \\cite{kamath2021doesIRM}. In contrast, our approach rests on a more general causal structural model and require less training domains\n\n\\paragraph{Data Augmentation}\nData Augmentation is an important technique to the training pipeline in deep learning \\cite{krizhevsky2014cifar,xie2019unsupervised,jiao2019tinybert,zhu2020freelb,zhang2018mixup,yun2019cutmix}. Commonly used methods include image rotation, cropping, Gaussian blurring, etc. With augmented data involved in training, the model generalization capability can be improved on both in-distribution \\cite{shorten2019survey} and out-of-distribution data \\cite{wang2021generalizing}. Different from the above mentioned works, when domain partition is available, we apply CycleGAN \\cite{zhu2017unpaired} to learn the source-to-source translations to generate corresponding images with different styles, i.e., artificially generated casual invariant transformed data.\n\\par\nNext we clarify the difference between our method and the existing methods that also involve generative models to obtain augmented data \\cite{zhou2020deep,zhou2020learning,kaushikHL20}. Specifically, in \\cite{zhou2020deep,zhou2020learning}, they generate data from inexistent ``novel domains'', instead of the data from known domains---{it was conjectured and empirically shown in \\cite{zhou2020learning} that source-to-source transformation could provide little help to domain generation tasks in their approach}. Roughly speaking, our augmented data represent the same ``causal feature\" under different domains and we leverage the augmented data by contrasting them with the original ones, while existing methods add the average loss on the augmented data from inexistent domains to the objective. Also related is \\cite{kaushikHL20} that artificially generates the counterfactually-augmented data which modify the causal feature but keep the non-causal part. Our work is different in that we use CITs to modify the non-causal feature but keep the causal part unchanged. Moreover, in our experiments, these CITs are obtained from prior knowledge or learned from training data, without the need of \\emph{human manipulations to each training datum}.\n\n\\section{OOD Generalization via Causality}\nIn this section, we consider a general causal structural model for the OOD generalization problem. We prove that the minimax optimality of a model can be obtained via the causal feature, even if we only have access to the data from a single domain. However, as discussed in the introduction, it may be hard to recover the causal feature exactly. We hence proceed with the aid of CITs and show that a learned model can achieve the same guaranteed OOD performance.\n\\subsection{Invariant Causal Mechanism}\n\\label{sec:icm3}\nWe begin with a formal definition of the causal structural model used in this paper. In practice, data distributions can vary across domains, but the causal mechanism usually remains unchanged \\cite{peters2017elements}. We consider the following causal structural model to describe the data generating mechanism:\n\\begin{equation}\\label{eq: structural model}\nY = m(g(X),\\eta),\\ \\eta \\Perp g(X)\\ \\text{and}\\ \\eta \\sim F,\n\\end{equation}\nwhere $X$, $Y$ are respectively the observed input and outcome, $g(X)$ denotes the causal feature, $\\eta$ is some random noise, and $m(\\cdot,\\cdot)$ represents the unknown structural function. The relationship $\\eta \\Perp g(X)$ means that the noise $\\eta$ is independent of the causal feature $g(X)$, and $\\eta\\sim F$ indicates that it follows a distribution $F$ that can be unknown. \n\\par\n\n\t{Notice that the structural model (\\ref{eq: structural model}) imposes no assumption on the distribution of the input $X$. Thus, the distributions of the outcome $Y$ can vary with $X$ under different environments, even though $Y$ depends on $X$ \\emph{only} through the causal feature $g(X)$ in the data generating mechanism. Besides, there can be two correlations in the structural model, summarized as follows:\n\t\\begin{enumerate}\n\t \\item Although the causal feature $g(X)$ is assumed to be independent of noise $\\eta$, $X$ can correlate with noise $\\eta$ under a certain domain. To see this, let us consider a toy example with the observed input $X = (X_{1}, X_{2})$. Here noise $\\eta$ is correlated with $X_{1}$ while $\\eta$ is independent of $X_{2}$. Then for the causal structural model $Y = X_{2} + \\eta$, we have $g(X) = X_{2}$ and $g(X)\\Perp \\eta$ while the input $X$ is correlated to $\\eta$. \n\t \\item There may exist correlations between causal feature and other spurious features, e.g., correlation between the objective shape and the image background in image classification tasks.\n\t\\end{enumerate}\n\tUnlike the invariant causal mechanism, these two correlations are supposed to vary across domains and hence are called spurious correlations \\cite{woodward2005making,arjovsky2019invariant}. If not treated carefully, the spurious correlations would deteriorate the performance of ERM-based machine learning methods and make the model perform poorly on the target domain \\cite{shen2020stableDVD,shen2020stableSR,liu2021heterogeneous,arjovsky2019invariant}. For instance, in an image classification task involving horse and camel, it is very likely that in the training data all the horses are on the grass while the camels are in the desert. The spurious correlation between horse\/camel and the background could easily mislead the model to making predictions using the background. Consequently, the trained model would be unreliable on OOD data.}\n\\par\nExisting works consider a similar causal mechanism to (\\ref{eq: structural model}) while more structural assumptions are usually imposed, e.g., $g(\\cdot)$ is linear and the noise is additive \\cite{peters2016causal, pfister2019invariant, rojas2018invariant, gimenez2021identifying, liu2021heterogeneous}. \nOur structural model~(\\ref{eq: structural model}) generalizes existing ones as we get rid of the two structural assumptions. Thus, our model constitutes a more flexible construction that is suitable to tasks in which the assumed linear or separable structural models appear implausible, e.g., the image classification task \\cite{arjovsky2019invariant}. Besides, our algorithm proposed in \\Cref{sec: RICEalgorithm} does not require \\emph{explicitly} learning the causal feature $g(X)$, thus avoids dealing with the identifiability issue of $g(X)$.\n\\subsection{Generalization via Causal Feature}\n\\par\nThroughout the rest of this paper, we focus on the distributions under structural model~(\\ref{eq: structural model}):\n\\begin{equation*}\n{\\mathcal{P}} \\hspace{-2pt}=\\hspace{-2pt} \\{P_{(X,Y)}\\hspace{-1pt}\\mid \\hspace{-1pt}(X, Y)\\hspace{-2pt}\\sim\\hspace{-2pt} P_{(X, Y)}~\\text{under structural model~(\\ref{eq: structural model})}\\}\\hspace{-1pt},\n\\end{equation*}\nwith fixed $g(\\cdot)$, $m(\\cdot,\\cdot)$ and $F$. \nOur goal is to train a model that generalizes well across all distributions $P_{(X,Y)}\\in{\\mathcal{P}}$ which follow the causal mechanism in structural model~(\\ref{eq: structural model}). Particularly, we aim to find a model $h^*(\\cdot)$ such that\n\\begin{equation}\\label{eq: target set}\nh^*(\\cdot)\\in {\\mathcal{H}}_{*} \\coloneqq\\mathop{\\arg\\min}_{h}\\sup_{P\\in {\\mathcal{P}}}\\mathbb{E}_{P}[{\\mathcal{L}}(h(X), Y)],\n\\end{equation}\nwhere ${\\mathcal{L}}(\\cdot, \\cdot)$ denotes a loss function, e.g., mean squared error for regression or cross entropy for classification. A similar minimax formulation appears in many existing works for the OOD generalization problem; see, e.g., \\cite{rojas2018invariant,arjovsky2019invariant,liu2021heterogeneous,Bulhmann2020invariant,gimenez2021identifying}.\n\\par\nIn contrast with methods based on data from sufficiently many domains \\cite{qian2019robust,rojas2018invariant,sagawa2020distributionally,liu2021heterogeneous,krueger2021out}, we next show that if $g(\\cdot)$ is known, we can learn $h^*(\\cdot)$ via single domain data. Let $P_{{{\\rm s}}}$ be the distribution of the source domain from which the training data are collected. Denote the set of optimal models under $P_{{\\rm s}}$ based on causal feature $g(X)$ by\n\\begin{equation}\\label{eq: causal opt}\n{\\mathcal{H}}_{\\rm{s}} \\hspace{-1pt}=\\hspace{-1pt} \\left\\{\\phi\\circ g \\ \\Big | \\ \\phi(w)\\hspace{-1pt}\\in\\hspace{-1pt} \\mathop{\\arg\\min}_{z} \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(z, Y)\\mid g(X) \\hspace{-1pt}=\\hspace{-1pt} w]\\right\\}\\hspace{-2pt},\n\\end{equation}\nwhere $\\circ$ is the composition of functions. In this paper, we do not distinguish two functions of $w$ that are equal to each other ``almost surely\", i.e., are equal for all $w$ except a set of probability zero. Then we have the following result.\n\\begin{theorem}\\label{thm: invariant generalization}\n\tIf $P_{{\\rm s}} \\in {\\mathcal{P}}$, then ${\\mathcal{H}}_{{\\rm s}} \\subseteq {\\mathcal{H}}_{*}$.\n\\end{theorem} \nA proof of \\Cref{thm: invariant generalization} can be found in the supplementary material. \\Cref{thm: invariant generalization} gives a class of models that belong to ${\\mathcal{H}}_{*}$, the set of solutions to the minimax problem defined in \\cref{eq: target set}. A model in ${\\mathcal{H}}_{*}$ makes predictions via the causal feature $g(X)$, and can be learned using the single domain data if the form of $g(\\cdot)$ is known. \\Cref{thm: invariant generalization} generalizes existing results in \\cite{rojas2018invariant, liu2021heterogeneous}, in the sense that it is derived under a more general structural model and also readily includes more loss functions ${\\mathcal{L}}(\\cdot,\\cdot)$ beyond the mean squared loss and cross entropy loss considered in \\cite{koyama2020out}.\n\\subsection{Learning via Causal Invariant Transformation}\\label{subsec: OOD&CIT}\n\\Cref{thm: invariant generalization} shows that it is possible to use only single domain data to learn a class of optimal models ${\\mathcal{H}}_{{\\rm s}}$ in the minimax sense. However, such a result requires an explicit formulation of causal feature $g(X)$, which is somehow impractical \\cite{arjovsky2019invariant}. On the other hand, learning causal mechanism from the observed data may face the issue of identifiability. Thus, in this section, we aim to learn a model of ${\\mathcal{H}}_{{\\rm s}}$ without the requirement of the explicit form of $g(X)$. The idea of our method is to leverage the transformations that do not change the underlying causal feature.\n\\par \nSpecifically, although the explicit form of $g(\\cdot)$ is unknown in general, we can have prior knowledge that the causal feature should remain invariant to certain transformations $T(\\cdot)$. For example, consider the horse v.s.~camel problem in \\Cref{sec:icm3}. For a given image, the shape of a horse\/camel could be the causal feature that determines its category. The \\emph{exact} function w.r.t.~pixels representing the shape may be hard to obtain. Nevertheless, we do know that the shape does not vary with rotation or flipping. We now formally define these transformations. \n\\begin{definition}[Causal Invariant Transformation (CIT)]\\label{def:CIT}\n\tA transformation $T(\\cdot)$ is called a causal invariant transformation if $(g\\circ T)(\\cdot) = g(\\cdot)$. \n\\end{definition}\nHenceforth, ${\\mathcal{T}}_{g} = \\{T(\\cdot): (g\\circ T)(\\cdot) = g(\\cdot)\\}$ denotes the set consisting of all CITs. As shown in \\Cref{lem: invariant characterize} in the supplementary material, the set ${\\mathcal{T}}_{g}$ is quite informative for $g(\\cdot)$ and hence helps resolve the OOD generalization problem according to \\Cref{thm: invariant generalization}. In some cases, knowing ${\\mathcal{T}}_{g}$ may be equivalent to knowing the causal feature or the causal parents of the outcome, e.g., assuming linear causal mechanism. However, in applications like image classification, the causal relationships are complicated and the prior knowledge on CITs can be more accessible compared to that of causal parents, as illustrated at the end of \\Cref{subsec: prior knowledge}.\n\nWith ${\\mathcal{T}}_{g}$, the following theorem states that ${\\mathcal{H}}_{{\\rm s}}$ is available by solving a minimax problem constructed from single domain data, even for unknown $g(\\cdot)$. \t\n\\par\n\\begin{theorem}\\label{thm: alternative problem 1}\n\tIf $P_{{\\rm s}} \\in {\\mathcal{P}}$, then for ${\\mathcal{H}}_{{\\rm s}}$ defined in \\cref{eq: causal opt}\n\t\\begin{equation}\\label{eq: minimax invariant problem}\n\t{\\mathcal{H}}_{{\\rm s}} \\subseteq\n\t\\mathop{\\arg\\min}_{h} \\sup_{T \\in {\\mathcal{T}}_{g}}\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(T(X)), Y)].\n\t\\end{equation}\n\\end{theorem}\n\\par\nA proof of \\Cref{thm: alternative problem 1} is in the supplementary material. If the minimax problem (\\ref{eq: minimax invariant problem}) has a unique minimum (when, e.g., some convexity conditions on the loss function ${\\mathcal{L}}(\\cdot, \\cdot)$ hold), \\Cref{thm: alternative problem 1} implies that the model performs uniformly well over the transformed data obtained from the transformations in ${\\mathcal{T}}_g$ can generalize to distributions in ${\\mathcal{P}}$.\n\\par\nLet ${\\mathcal{P}}_{{\\rm aug}} = \\{P_{(X^{\\prime}, Y)}\\mid (X, Y)\\sim P_{{\\rm s}},\\ X^{\\prime} = T(X), \\ T\\in {\\mathcal{T}}_{g}\\}$, then we can rewrite the minimax problem in (\\ref{eq: minimax invariant problem}) as \n\\begin{equation}\\label{eq: minimax with small range}\n\\mathop{\\min}_{h} \\sup_{P\\in {\\mathcal{P}}_{{\\rm aug}}}\\mathbb{E}_{P}[{\\mathcal{L}}(h(X), Y)],\n\\end{equation}\nProblem~(\\ref{eq: minimax with small range}) has a similar form to problem~(\\ref{eq: target set}). Recalling the structural model (\\ref{eq: structural model}), it can be verified that ${\\mathcal{P}}_{{\\rm aug}}$ is a subset of ${\\mathcal{P}}$. We then have the following two remarks: \n\\begin{enumerate}\n\t\\item ${\\mathcal{P}}_{{\\rm aug}}$ can be a proper subset of ${\\mathcal{P}}$. Thus, the supremum taken in (\\ref{eq: minimax with small range}) is more tractable compared with that in (\\ref{eq: target set}), as we require less information of ${\\mathcal{P}}$. To see this, suppose that $(X, \\eta) \\sim P_{(X, \\eta)} = P_{X}\\times F$ for a distribution $P_{X}$ and $P_{{\\rm s}} = P_{(X, m(g(X),\\eta))}$. Then for any $P\\in {\\mathcal{P}}_{{\\rm aug}}$, we have $Y\\Perp X\\mid g(X)$ if $(X, Y)\\sim P$. However, there can exist $P^{\\prime}\\in {\\mathcal{P}}$ so that $X$ is correlated with $\\eta$ and hence the conditional independence no longer holds. That is, $P^{\\prime}$ lies in ${\\mathcal{P}}$ but not in ${\\mathcal{P}}_{{\\rm aug}}$. \n\t\\item In the horse v.s.~camel example in \\Cref{sec:icm3}, the spurious correlations lead to misleading supervision. The set ${\\mathcal{P}}_{{\\rm aug}}$, on the other hand, is likely to contain distributions that do not have these spurious correlations or even entail opposite correlations. Thus, the model that overfits spurious correlations can not generalize well on these distributions, and can not be a solution to problem~(\\ref{eq: minimax with small range}). For example, the data from some $P\\in{\\mathcal{P}}_{{\\rm aug}}$ can have most horses on the desert while most camels are on grass. Thus, the model that overfits the spurious correlation between animal and background does not perform well on this distribution. \n\\end{enumerate}\n\\par\nAlthough \\Cref{thm: alternative problem 1} provides a way to learn a model with guaranteed OOD generalization, it may be computationally hard to calculate the supremum over ${\\mathcal{T}}_{g}$ when it contains plenty of or possibly infinite transformations. Take image classification tasks for example. Suppose that ${\\mathcal{T}}_{g}$ contains rotations of $\\theta$ degree, with $\\theta=1,\\ldots,360$. Computing the loss over a total of $360$ transformations is computationally expensive. Thus, it is natural to ask a question: can we substitute ${\\mathcal{T}}_{g}$ in (\\ref{eq: minimax invariant problem}) with a proper subset? \n\\subsection{Learning via Causal Essential Set}\n\\label{sec: causal_essential}\nIn this section, we positively answer the question at the end of \\Cref{subsec: OOD&CIT}. We show that it is sufficient to use only a subset of ${\\mathcal{T}}_{g}$, referred to as \\emph{causal essential set}. Next, we first give a formal definition of causal essential set and then prove that it is indeed the desired subset.\n\\begin{definition}[Causal Essential Set]\n\t\\label{def:causal essential set}\n\tFor ${\\mathcal{I}}_{g} \\subseteq {\\mathcal{T}}_{g}$, ${\\mathcal{I}}_{g}$ is a causal essential set if for all $x_{1}$, $x_{2}$ satisfying $g(x_{1}) = g(x_{2})$, there are finite transformations $T_{1}(\\cdot),\\cdots,T_{K}(\\cdot) \\in {\\mathcal{I}}_{g}$ such that $(T_{1}\\circ\\dots\\circ T_{K})(x_{1}) = x_{2}$.\n\\end{definition}\nClearly, there may be multiple causal essential sets, e.g., ${\\mathcal{T}}_{g}$ itself is a causal essential set. In most cases, we believe that there exists ${\\mathcal{I}}_{g}$ that is a proper subset of ${\\mathcal{T}}_{g}$. For example, rotation with one degree itself forms a causal essential set if ${\\mathcal{T}}_{g}$ is the set of rotations with $\\theta=1,\\ldots, 360,$ degrees.\n\\par\nThe next theorem indicates that the prior knowledge on any such causal essential set is sufficient to achieve a guaranteed OOD generalization, using only single domain data. A proof is provided in the supplementary material.\n\\begin{theorem}\\label{thm: alternative problem 2}\n\tIf $P_{{\\rm s}} \\in {\\mathcal{P}}$, then for any ${\\mathcal{I}}_{g}$ that is a causal essential set of $g(\\cdot)$ and ${\\mathcal{H}}_{{\\rm s}}$ defined in (\\ref{eq: causal opt})\n\t\\begin{equation}\\label{eq: invariant feature problem}\n\t\\begin{aligned}\n\t&{\\mathcal{H}}_{\\rm s} = \\mathop{\\arg\\min}_{h}~ \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(X), Y)],\\\\\n\t&~~~~~~~~~~\\text{\\rm subject to}\\ \\ h(\\cdot) = (h\\circ T)(\\cdot), \\ \\forall \\ T(\\cdot)\\in {\\mathcal{I}}_{g}.\n\t\\end{aligned}\n\t\\end{equation}\n\\end{theorem}\t\n\\par\nCompared with \\Cref{thm: alternative problem 1}, ``$\\subseteq$\" related to ${\\mathcal{H}}_{{\\rm s}}$ in (\\ref{eq: minimax invariant problem}) is replaced by ``$=$\" in this theorem, which is a stronger theoretical result. Thus, one can also readily obtain the model that generalizes well on OOD data by minimizing the loss w.r.t. any data distribution induced by structural model~(\\ref{eq: structural model}), but require less CITs. In certain cases, the structure of a causal essential set is simple and is possible to be identified. Due to space limit, this is illustrated by an example in Section~\\ref{app: toy example} in the supplementary material.\n\n\\subsection{Necessity of Prior Knowledge}\\label{subsec: prior knowledge}\nBefore ending this section, we would like to clarify that the prior knowledge on causal invariant transformations can be more manageable, compared with prior knowledge required by many existing approaches.\n\\par\nFirst, assumption or prior knowledge is necessary to get guaranteed causal results from observational data.\nAs previously introduced, some recent works assume \\emph{a prior} particular forms of causal graphs, whose correctness cannot be tested from observational data. Notice that learning causal graphs from the data, or the so-called causal discovery methods \\cite{peters2017elements,spirtes2000causation,zhu2020causal}, faces the same issue. Other works that have theoretical guarantee on OOD generalization require sufficiently many and diverse training domains \\cite{arjovsky2019invariant} and\/or incorporate prior knowledge on causal diagrams and causal mechanisms \\cite{rosenfeld2020risks,heinze2021conditional,Lu2021NonlinearIRM}. Despite the valuable understanding and theoretical guarantee of OOD generalization for these causality based methods, it is not clear whether the assumed conditions or prior knowledge indeed hold in practical applications.\nIn general, only randomized controlled experiments are the golden standard to infer the causal relationship, summarized as ``no causation without manipulation'' \\cite{holland1986statistics}. On the other hand, directly creating ``manipulated'' data can lead to a better causal effect estimation. In \\cite{kaushikHL20}, authors artificially generate counterfactually-augmented data that modify the causal feature but keep the non-causal part for each sentence. Then simple processing (e.g., directly combining the observational and the manipulated data) improves the generalization performance. \n\\par\nThese observations together motivate us to consider: \\emph{how to generate manipulations to achieve a causal guarantee at a lower human cost?} For example, one need not apply human manipulations to each training datum. This consideration leads us to the CITs, on which we often have some prior knowledge. Many augmentation techniques can be used as CITs, such as rotation and Gaussian blurring. When domain partition is available, e.g., in the multi-domain learning setting \\cite{peters2016causal,arjovsky2019invariant}, the CITs can be learned from the data. For example, we employ GAN or other generative models to synthesize ``manipulated'' data in \\Cref{subsec: generalizing on unseen domains}. We believe that such prior knowledge on CITs is more accessible than artificially manipulated data, and is manageable in practical applications. \n\\par\nFinally, we clarify that prior knowledge on CITs can also be more accessible than that of causal feature in many tasks, as illustrated by \\Cref{fig: cow} from \\cite{cloudera2021causality}. In \\Cref{fig: cow}, profiles of cows (highlighted by green lines) are considered as causal feature, and a prediction model \\emph{only} based on such causal feature can generalize well to images with different backgrounds. However, we do not know which pixels or what function of pixels represent the profile; indeed, the profile depends on different pixels in the first and the second images in \\Cref{fig: cow}. In contrast, we readily know that changing the background of the image, e.g., transforming the second image to the third, will not affect the causal feature. This can be treated as our prior knowledge on CITs.\n\\begin{figure}[t!]\n\\centering\n\\begin{subfigure}[t]{0.25\\linewidth}\n \\centering\n \\includegraphics[scale = 0.5]{pic\/cow\/cow1.png}\n \n\\end{subfigure} \n\\hspace{-0.2in}\n\\begin{subfigure}[t]{0.25\\linewidth}\n \\centering\n \\includegraphics[scale = 0.5]{pic\/cow\/cow2.png}\n \n\\end{subfigure}\n\\hspace{-0.2in}\n\\begin{subfigure}[t]{0.25\\linewidth}\n \\centering\n \\includegraphics[scale = 0.5]{pic\/cow\/cow3.png}\n \n\\end{subfigure}\n \\caption{Profiles of cows in these figures are considered as causal features, while the different backgrounds are spurious features.}\\label{fig: cow}\n \\vspace{-0.15in}\n\\end{figure}\n\n\\section{Algorithm}\n\\label{sec: RICEalgorithm}\nWe now propose an algorithm based on the previous analysis w.r.t.~CITs. Let $D(\\cdot,\\cdot)$ denote some measure of discrepancy satisfying $D(v_{1}, v_{2}) = 0$ if $v_{1} = v_{2}$ and $D(v_{1}, v_{2}) > 0$ otherwise. Then for any model $h(\\cdot)$ and transformation $T(\\cdot)$, $\\mathbb{E}_{P_{{\\rm s}}}[D\\big(h(X), h(T(X))\\big)] = 0$ implies $h(\\cdot) = h(T(\\cdot))$ almost surely. Together with \\Cref{thm: alternative problem 2}, we consider the following formulation\n\\begin{equation*}\n\\begin{aligned}\n\\min_{h}&~~\\mathbb{E}_{P_{{\\rm s}}}\\left[{\\mathcal{L}}(h(X), Y)\\right],\\quad\\\\ \n\\textrm{subject to}&~~\\mathbb{E}_{P_{{\\rm s}}}\\left[\\sup_{T\\in {\\mathcal{I}}_{g}}D\\big(h(X), h(T(X))\\big)\\right] = 0,\n\\end{aligned}\n\\end{equation*}\nwhere ${\\mathcal{I}}_{g}$ is a causal essential set. To obviate the difficulty of solving a constrained optimization problem, we further consider to minimize a regularized formulation\n\\begin{equation} \\label{eq: population penalize}\n\\begin{aligned}\n\\mathbb{E}_{P_{{\\rm s}}} [{\\mathcal{L}}(h(X), Y)] \\!+\\! \\lambda_{0}\\mathbb{E}_{P_{{\\rm s}}}\\bigg[\\sup_{T\\in {\\mathcal{I}}_{g}}[D(h(X), h(T(X)))\\bigg]\n\\end{aligned}\n\\end{equation}\nwith a given regularization constant $\\lambda_{0}>0$. Supposing that we have training samples $\\{(x_{i}, y_{i})\\}_{i=1}^{n}$, then we propose to minimize the empirical counterpart of (\\ref{eq: population penalize})\n\\begin{equation}\n\\begin{aligned}\n\\!\\frac{1}{n}\\!\\sum\\limits_{i=1}^{n}\\!{\\mathcal{L}}(h(x_{i}), y_{i})\\!+ \\! \\frac{\\lambda_{0}}{n}\\sum\\limits_{i=1}^{n}\\!\\sup_{T\\in {\\mathcal{I}}_{g}}\\![D(h(x_{i}), h(T(x_{i})))].\n\\end{aligned}\\nonumber\n\\end{equation} \nWe then propose Algorithm \\ref{alg: RICE}, Regularized training with Invariance on Causal Essential set (RICE), to solve the above problem, where the update step in line $7$ can be substituted by other optimization algorithms, e.g., Adam \\cite{kingma2015adam}.\n\\par\nNote that obtaining a complete causal essential set may also be hard in many applications. Nonetheless, we usually have or can learn certain transformations with the desired causal invariance. We will empirically show that the proposed RICE enables an improved OOD generalization, even with a set of only a few CITs. In this case, we can simply replace ${\\mathcal{I}}_g$ with this set in Algorithm \\ref{alg: RICE}.\n\n\\begin{algorithm}[t!]\n\t\\caption{Regularized training with Invariance on Causal Essential set (RICE).}\n\t\\label{alg: RICE}\n\t\\textbf{Input:} Training set $\\{(x_{1}, y_{1}), \\cdots, (x_{n}, y_{n})\\}$, batch size $S$, learning rate $\\eta$, training iterations $N$, model $h_{\\beta}(\\cdot)$ with parameter $\\beta$, initialized parameter $\\beta_{0}$, regularization constant $\\lambda_{0}$, causal essential set ${\\mathcal{I}}_{g}$, and discrepancy measure $D(\\cdot, \\cdot)$.\n\t\\begin{algorithmic}[1]\n\t\n\t\n\t\n\t\t\\For \t{$i=1, \\ldots, n$ } \n\t\t\\State{Generate transformed samples $\\{T(x_{i})\\}_{T\\in {\\mathcal{I}}_{g}}$.}\n\t\n\t\t\\EndFor\n\t\n\t\n\t\t\\For {$t=0, \\cdots ,N$}\n\t\t\\State {Randomly sample a mini-batch ${\\mathcal{S}} = \\{(x_{t_{1}}, y_{t_{1}}),$ $\\cdots, (x_{t_{S}}, y_{t_{S}})\\}$ from training set.\n\t\t\t\\State Fetch the transformed samples $\\{T(x_{t_1})\\}_{T\\in {\\mathcal{I}}_{g}},$ $\\cdots,$ $\\{T(x_{t_{S}})\\}_{T\\in {\\mathcal{I}}_{g}}$.}\t\t\n\t\t\\State {Update model parameters via first-order method e.g., stochastic gradient descent:}\n\t\t\\vspace{-0.1in}\n\t\t {\\small\n\t\t\t\\[\\begin{aligned}\n\t\t\t\\beta_{t + 1} &= \\beta_{t} - \\frac{\\eta}{S}\\sum\\limits_{i=1}^{S}\\nabla_{\\beta}{\\mathcal{L}}(h_{\\beta}(x_{t_{i}}), y_{t_{i}})\\Big|_{\\beta = \\beta_{t}} + \\eta\\nabla_{\\beta}\\Big\\{\\frac{\\lambda_{0}}{S}\\sum\\limits_{i=1}^{S}\\sup\\limits_{T\\in {\\mathcal{I}}_{g}}D\\big(h_{\\beta}(x_{t_{i}}), h_{\\beta}(T(x_{t_{i}}))\\big)\\Big\\}\\Big|_{\\beta = \\beta_{t}}\\hspace{-1pt}.\n\t\t\t\\end{aligned}\\]}\n\t\t\t\\vspace{-0.1in}\n\t\t\\EndFor\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Experiments}\\label{sec:experiment}\nIn this section, we empirically evaluate the efficacy of the proposed algorithm RICE on real-world datasets. We train the model using data from some of the available domains and evaluate the performance on the data from the rest domains that are not used in training. \nAs suggested in \\cite{ye2021ood}, the OOD data can be classified into two categories, namely, data with \\emph{correlation shift} or with \\emph{diversity shift}. \nEmpirical results show that RICE can handle both kinds of OOD data. Due to space limit, part of results, including a toy experiment of synthetic data mentioned in \\Cref{sec: causal_essential} and ablation studies, are given in the supplementary material.\\footnote{Part of the experiments was supported by MindSpore (\\url{https:\/\/www.mindspore.cn}), a deep learning computing framework.}\n\n\\begin{figure*}[h]\n \n\t\\centering\n\t\\includegraphics[scale=0.48]{pic\/algorithm.png}\n \t\\vspace{-0.1in}\n\t\\caption{The proposed algorithm RICE on the \\texttt{PACS} dataset. The training data are from domains \\{art, cartoon, photo\\}, and we want the model to perform well on the sketch data. This figure describes the training procedure of RICE for a training image from the art domain.}\n\t\\label{fig: algorithm}\n\\end{figure*}\n\\subsection{Breaking Spurious Correlation}\\label{subsec: spurious correlation}\nAs we have discussed in \\Cref{sec:icm3}, the spurious correlation in the data may mislead the model to wrong predictions on OOD data, resulting in correlation shift. In this section, we empirically verify that RICE in Algorithm \\ref{alg: RICE} can obviate overfitting such spurious correlations. \n\\vspace{-0.5em}\n\\paragraph{Data} We use the colored \\texttt{MNIST} (\\texttt{C-MNIST}) dataset from \\cite{arpit2019predicting}. As in \\cite{arpit2019predicting}, we vary the colors of both foreground and background of an image.\n\\par\nThe original \\texttt{MNIST} dataset consists of handwritten digits from ten categories, namely, $0$ to $9$. To construct a training set of \\texttt{C-MNIST}, we pick two colors for the foreground of the images in a given category, and then randomly replace the foreground color with one of the two colors assigned to the category. The background color of each image is handled similarly. For the test set, we randomly assign colors to the foreground and background of each image from the \\texttt{MNIST} test set, regardless of its category. Some images from the generated \\texttt{C-MNIST} dataset are visualized in \\Cref{fig: c-mnist} in the supplementary material. The construction introduces a spurious correlation between category and color in the training set, but not in the test set. In the following, we will show that the proposed method RICE will not be affected by this spurious correlation. \n\\vspace{-0.5em}\n\\paragraph{Setup} Our model is a five-layer convolution neural network as in \\cite{arpit2019predicting}. For RICE, we choose the cross entropy loss for ${\\mathcal{L}}(\\cdot, \\cdot)$ and the $\\ell_{2}$-distance for $D(\\cdot, \\cdot)$. The model is updated by Adam \\cite{kingma2015adam}, and other hyperparameters are given in the supplementary material. For a handwritten digit, it is known that the shape of its foreground, rather than the color of either foreground or background, determines its category. Thus, transforming the image background with a color (e.g., black) and its foreground with another color (e.g., white) would be a desired CIT. In our experiment, we simply use the original \\texttt{MNIST} images as the transformed data to show the effectiveness of the proposed method.\n\\par \nAs we use the original \\texttt{MNIST} dataset in training, the training data can be seen from two domains, i.e., the original \\texttt{MNIST} and the \\texttt{C-MNIST} datasets. As such, we compare RICE with several widely used domain generalization algorithms using the same training data, including empirical risk minimization (ERM), ERM with Mixup \\cite[Mixup]{zhang2018mixup}, marginal transfer learning \\cite[MTL]{blanchard2021domain}, group distributionally robust optimization \\cite[GroupDRO]{sagawa2020distributionally}, domain-adversarial neural networks \\cite[DANN]{ganin2016domain} and invariant risk minimization \\cite[IRM]{arjovsky2019invariant}. See the supplementary material for a further introduction of these algorithms. For these baseline algorithms, the hyperparameters are adopted from \\cite{gulrajani2020search}. We remark that the same training data, i.e., \\texttt{MNIST} and \\texttt{C-MNIST}, are used in our method and baseline methods \n\n\\begin{table}[t!]\n \n\t\\caption{Accuracy (\\%) on the \\texttt{C-MNIST} test set.}\n\t\\vspace{-0.1in}\n\t\\label{tbl:c-mnist}\n\t\\setlength\\tabcolsep{2pt}\n\t\\centering\n\t\\scalebox{0.8}{\n\t\t{\n\t\t\t\\begin{tabular}{l|cccccccccc}\n\t\t\t\t\\hline\n\t\t\t\tDataset & ERM & Mixup & MTL & GroupDRO & DANN & IRM & RICE(OURS) \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\texttt{C-MNIST} & 13.3 & 17.5 & 14.7 & 14.1 & 28.1 & 15.8 & \\textbf{96.9} \\\\\n\t\t\t\t\\hline\n\t\\end{tabular}}}\n\t\\vspace{-0.15in}\n\\end{table}\n\n\\vspace{-0.5em}\n\\paragraph{Main Results} The empirical results are reported in \\Cref{tbl:c-mnist}. We observe that only the proposed algorithm RICE works well on the OOD data in this experiment. We speculate that this is because, for the baseline algorithms, the misleading supervision signal from the color is memorized by the models, even though the original \\texttt{MNIST} images are also included in the training set. However, for RICE, the regularizer penalizes the discrepancy between the model outputs of the colored images and the corresponding \\texttt{MNIST} versions, which makes the model insensible to the spurious correlation but more dependent on the invariant causal feature\n\\subsection{Generalizing to Unseen Domains}\\label{subsec: generalizing on unseen domains}\nIn this section, we conduct experiments on two benchmark datasets, \\texttt{PACS} and \\texttt{VLCS}, which are commonly used in domain generalization. The two datasets correspond to the diversity shift that we have mentioned. As in other related works in domain generalization \\cite{zhou2020deep,zhou2021domain,gulrajani2020search}, the domain labels are known in the training set\n\\vspace{-0.5em}\n\\paragraph{Data} \\texttt{PACS} is an image classification dataset consisting of data from four domains of different styles, i.e., \\{art, cartoon, photo, sketch\\}, with seven different categories in each domain. \\texttt{VLCS} is a dataset comprised of four photographic domains: \\{VOC2007, LabelMe, Caltech101, SUN09\\}, and each domain contains five different categories.\n\\vspace{-0.5em}\n\\paragraph{Setup} As in \\cite{gulrajani2020search}, we use the model ResNet50 \\cite{he2016deep} pre-trained on \\texttt{ImageNet} \\cite{deng2009imagenet} as the backbone model, and fine-tune the model with different baseline methods. For RICE, the model is trained by Adam and the used hyperparameters are provided in the supplementary material. To implement RICE, we need to generate casually invariant transformed data. In the \\texttt{PACS} dataset, each domain represents a style of images, e.g., photo or art. Since varying the style of an image does not change its category, we construct the transformations that modify image styles as CITs. To this end, we use CycleGAN \\cite{zhu2017unpaired} to learn the transformations for each pair of domains in the training set, and then implement RICE using the trained CycleGAN models. In \\texttt{VLCS}, the photographic of the image plays a similar role to the style in \\texttt{PACS}, and we also apply CycleGAN to learning the transformations. Other generative models may also be used, e.g., StarGAN \\cite{choi2018stargan}, for data from numerous domains.\n\\par\nThe procedure of RICE is summarized in \\Cref{fig: algorithm}. We also compare the proposed RICE with other commonly used domain generalization algorithms, as in the previous experiment. For better results and a fair comparison with baseline methods, we also provide an ablation study with single domain training data in the supplementary material. \n\\vspace{-0.5em}\n\\paragraph{Main Results}\nThe experimental results on \\texttt{PACS} and \\texttt{VLCS} are summarized in Tables \\ref{tbl:PACS} and \\ref{tbl:VLCS}, respectively. The results of baseline methods are from \\cite{gulrajani2020search}. The proposed RICE exhibits better OOD generalization capability compared with the baseline methods on the \\texttt{PACS} and \\texttt{VLCS} datasets, in terms of the average and particularly the worst-case test accuracies. \nHere we provide an intuitive explanation to the performance of RICE using \\texttt{PACS} as an example. From \\Cref{fig: pacs} in the supplementary material, we can see that the trained CycleGAN model is likely to introduce spurious correlation related to domains, and models which capture the spurious correlation would be penalized in RICE. RICE can achieve an improved performance because it seeks to make predictions on top of the causal feature (e.g., shape of the object in the images), rather than the spurious feature related to the domains (e.g., style for \\texttt{PACS}). Moreover, as also seen from \\Cref{fig: pacs}, some generated images from CycleGAN are indeed not similar to the original images and are also blurring, which indicates that our RICE is robust to the quality of generated data.\n\\par\nFinally, we verify whether the improved performance of RICE originates solely from the augmented data generated by CycleGAN. We conduct ERM training with these augmented data on \\texttt{PACS}. The test accuracies on domains P, A, C, S are respectively 96.2, 84.9, 81.2, 80.5, which are worse than those of RICE. We believe that such a performance, which is consistent with previous observations in \\cite{zhou2020learning}, demonstrates that the regularizer that compares different augmentations with the original data is critical to capture the invariance and to improve the model robustness.\n\\begin{table}[t!]\n\t\\caption{Test accuracy (\\%) of ResNet50 on the \\texttt{PACS} dataset.}\n \\vspace{-0.1in}\n\t\\label{tbl:PACS}\n\t\\centering\n\t\\scalebox{0.85}{\n\t\t{\n\t\t\t\\begin{tabular}{c|*{7}{c}}\n\t\t\t\t\\hline\n\t\t\t\tMethod & P & A & C & S & Avg & Min \\\\\n\t\t\t\t\\hline\n\t\t\t\tERM & 97.2 & 84.7 & 80.8 & 79.3 & 85.5 & 79.3 \\\\\n\t\t\t\tMixup & 97.6 & 86.1 & 78.9 & 75.8 & 84.6 & 75.8 \\\\\n\t\t\t\tMTL & 96.4 & 87.5 & 77.1 & 77.3 & 84.6 & 77.1 \\\\\n\t\t\t\tGroupDRO & 96.7 & 83.5 & 79.1 & 78.3 & 84.4 & 78.3\\\\\n\t\t\t\tDANN & 97.3 & 86.4 & 77.4 & 73.5 & 83.6 & 73.5 \\\\\n\t\t\t\tIRM & 96.7 & 84.8 & 76.4 & 76.1 & 83.5 & 76.1 \\\\\n\t\t\t\tRICE (OURS) & 96.8 & 87.8 & 84.3 & 84.7 & \\bf{88.4} & \\bf{84.3}\\\\\n\t\t\t\t\\hline\n\t\\end{tabular}}}\n\\end{table}\n\n\\begin{table}[t!]\n\t\\caption{Test accuracy (\\%) of ResNet50 on the \\texttt{VLCS} dataset.}\n\t\\vspace{-0.1in}\n\t\\label{tbl:VLCS}\n\t\\centering\n\t\\scalebox{0.85}{\n\t\t{\n\t\t\t\\begin{tabular}{c|*{7}{c}}\n\t\t\t\t\\hline\n\t\t\t\tMethod & V & L & C & S & Avg & Min\\\\\n\t\t\t\t\\hline\n\t\t\t\tERM & 74.6 & 64.3 & 97.7 & 73.4 & 77.5 & 64.3\\\\\n\t\t\t\tMixup & 76.1 & 63.4 & 98.4 & 72.9 & 77.7 & 63.4 \\\\\n\t\t\t\tMTL & 75.3 & 64.3 & 97.8 & 71.5 & 77.2 & 64.3\\\\\n\t\t\t\tGroupDRO & 76.7 & 63.4 & 97.3 & 69.5 & 76.7 & 63.4\\\\\n\t\t\t\tDANN & 77.2 & 65.1 & 99.0 & 73.1 & 78.6 & 65.1\\\\\n\t\t\t\tIRM & 77.3 & 64.9 & 98.6 & 73.4 & 78.5 & 64.9\\\\\n\t\t\t\tRICE (OURS) & 75.1 & 69.2 & 98.3 & 74.6 & \\bf{79.3} & \\bf{69.2}\\\\\n\t\t\t\t\\hline\n\t\\end{tabular}}}\n\t\\vspace{-0.1in}\n\\end{table}\n\n\\section{Concluding Remarks}\nIn this paper, we theoretically show that knowledge of the CITs makes it feasible to learn an OOD generalized model via single domain data. The CITs can be either obtained from prior knowledge or learned from training data, without the need of human manipulations to each training datum. Inspired by our theoretical findings, we propose RICE to achieve an enhanced OOD generalization capability and the effectiveness of RICE is demonstrated empirically over various experiments. \n\\par\nIn our experiments, we focus on image classification tasks for domain generalization. Nevertheless, our theory and the proposed algorithm can apply to other datasets once some CITs are available. For example, in natural language processing (NLP), changing the position of the adverbial or synonym substitution does not change the semantic meaning and hence can be treated as CITs. However, generating such causally invariant sentences via deep learning method is not easy. How to ease the generation and apply our RICE to NLP domain is left as a future work.\n\n\\small{\n\\bibliographystyle{ieee_fullname}\n\n\\section{Proofs}\\label{app: proofs}\n\\subsection{Proof of Theorem \\ref{thm: invariant generalization}}\\label{app: proof of thm i. g.}\n\\paragraph{Restatement of Theorem \\ref{thm: invariant generalization}}\n\\emph{If $P_{{\\rm s}} \\in {\\mathcal{P}}$, then ${\\mathcal{H}}_{{\\rm s}} \\subset {\\mathcal{H}}_{*}$.}\n\\begin{proof}\n\tIt suffices to prove that for any $h_{{\\rm s}} \\in {\\mathcal{H}}_{{\\rm s}} $, we have\n\t\\begin{equation}\\label{eq: single hypo opt}\n\th_{{\\rm s}}(\\cdot) \\in \\mathop{\\arg\\min}_{h} \\sup_{P\\in {\\mathcal{P}}}\\mathbb{E}_{P}[{\\mathcal{L}}(h(X), Y)].\n\t\\end{equation}\n\tTo prove (\\ref{eq: single hypo opt}), we only need to show that for any $h(\\cdot)$ and $P \\in {\\mathcal{P}}$, there exists $Q \\in {\\mathcal{P}}$ such that \n\t\\begin{equation}\\label{eq: sufficient eq 1}\n\t\\mathbb{E}_{Q}[{\\mathcal{L}}(h(X), Y)] \\geq \\mathbb{E}_{P}[h_{{\\rm s}}(X), Y)],\n\t\\end{equation}\n\n\n\n\n\n\tand hence\n\t\\begin{equation}\n\t\\sup_{Q\\in {\\mathcal{P}}}\\mathbb{E}_{Q}[{\\mathcal{L}}(h(X), Y)] \\geq \\sup_{P\\in {\\mathcal{P}}}\\mathbb{E}_{P}[{\\mathcal{L}}(h_{{\\rm s}}(X), Y)]. \\nonumber\n\t\\end{equation}\n\tRecall that \n\t\\begin{equation}\n\t{\\mathcal{H}}_{\\rm{s}} = \\left\\{(\\phi\\circ g)(\\cdot)\\ \\Big | \\ \\phi(w) \\in \\mathop{\\arg\\min}_{z} \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(z, Y)\\mid g(X) = w], \\quad a.s.\\right\\}.\\nonumber\n\t\\end{equation}\n\tSince $h_{{\\rm s}}(\\cdot) \\in {\\mathcal{H}}_{{\\rm s}}$, there is some $\\phi_{{\\rm s}}(\\cdot)$ satisfying $h_{{\\rm s}}(\\cdot) = (\\phi_{{\\rm s}}\\circ g)(\\cdot)$ and $\\phi_{{\\rm s}}(w) \\in \\mathop{\\arg\\min}_{z} \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(z, Y)\\mid g(X) = w]$ for almost every $w$.\n\tSuppose $(X, \\eta) \\sim P_{X}\\times F$ and $(m(g(X), \\eta), X)\\sim Q$ where $P_{X}$ is the marginal distributions of $X$ under $P$. Let ${\\mathcal{U}}$ be the support of noise $\\eta$. Then \n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\mathbb{E}_{Q}[{\\mathcal{L}}(h(X), Y)\\mid X = x] & = \\int_{{\\mathcal{U}}} {\\mathcal{L}}(h(x), m(g(x), u))P_{\\eta}(du)\\\\\n\t& \\geq \\int_{{\\mathcal{U}}} {\\mathcal{L}}(\\phi_{{\\rm s}}(g(x)), m(g(x), u))P_{\\eta}(du) \\\\\n\t& = \\mathbb{E}_{P}[{\\mathcal{L}}(h_{{\\rm s}}(X), Y)\\mid g(X) = g(x)] \\quad a.s. \\nonumber\n\t\\end{aligned}\n\t\\end{equation}\n\tHere the first equation follows from the fact that $X$ and $\\eta$ are independent under $Q$. The inequality is from the fact that\n\t\\[\\int_{\\mathcal{U}} \\mathcal{L}(h(x), m(g(x), u))P_{\\eta}(du) = \\mathbb{E}_{P_{\\rm s}}[\\mathcal{L}(h(x), Y) \\mid g(X) = g(x)]\\]\n\tand\n\t\\[\\phi_{{\\rm s}}(w) \\in \\mathop{\\arg\\min}_{z} \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(z, Y)\\mid g(X) = w] = \\mathop{\\arg\\min}_{z} \\int_{{\\mathcal{U}}}{\\mathcal{L}}(z, m(w, u))P_{\\eta}(du)\\]\n\tfor almost every $w$. The last equation is due to $P \\in {\\mathcal{P}}$.\n\tThen equation (\\ref{eq: sufficient eq 1}) follows by taking expectation and the law of iterated expectation.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm: alternative problem 1}}\\label{app: proof of thm minimax and restriction}\nTo begin with, we establish two useful lemmas regarding CITs. The first lemma states that $g(\\cdot)$ is determined up to an invertible transformation by the transformation that it is invariant to. \n\nFor a given function $h(\\cdot)$, let ${\\mathcal{T}}_{h} = \\{T(\\cdot): (h\\circ T)(\\cdot) = h(\\cdot)\\}$. Then we have the following lemma.\n\\begin{lemma}\\label{lem: invariant characterize}\n\tFor any $h_{1}(\\cdot)$ and $h_{2}(\\cdot)$,\n\t${\\mathcal{T}}_{h_{1}} \\subset {\\mathcal{T}}_{h_{2}}$ if and only if there exists a function $v(\\cdot)$ such that $h_{2}(\\cdot) = (v \\circ h_{1})(\\cdot)$, and ${\\mathcal{T}}_{h_{1}} = {\\mathcal{T}}_{h_{2}}$ if and only if there is an invertible function $v(\\cdot)$ such that $h_{2}(\\cdot) = (v \\circ h_{1})(\\cdot)$.\n\\end{lemma}\n\\par\n\\begin{proof}\n\tWe only prove the former statement as the latter can be obtained as a corollary of the former. The ``if\" direction is obvious. \n\t\\par\n\tHere we prove the ``only if\" direction. Let ${\\mathcal{R}}_{1}$ and ${\\mathcal{R}}_{2}$ be the range of $h_{1}(\\cdot)$ and $h_{2}(\\cdot)$, respectively. For any $w_{1}\\in {\\mathcal{R}}_{1}$ and $w_{2} \\in {\\mathcal{R}}_{2}$, define ${\\mathcal{D}}_{h_{1},w_{1}} = \\{x: h_{1}(x) = w_{1}\\}$ and ${\\mathcal{D}}_{h_{2},w_{2}} = \\{x: h_{2}(x) = w_{2}\\}$. Then $h_{2}(\\cdot) = (v \\circ h_{1})(\\cdot)$ if and only if for any $w_{2} \\in {\\mathcal{R}}_{2}$, there is some $w_{1}\\in{\\mathcal{R}}_{1}$ such that ${\\mathcal{D}}_{h_{1},w_{1}} \\subset {\\mathcal{D}}_{h_{2},w_{2}}$. Thus, the former claim holds if we can show the following: ${\\mathcal{T}}_{h_{1}} \\subset {\\mathcal{T}}_{h_{2}}$ implies that there is some $w_{2}\\in{\\mathcal{R}}_{2}$ such that ${\\mathcal{D}}_{h_{1},w_{1}} \\subset {\\mathcal{D}}_{h_{2},w_{2}}$ for any $w_{1} \\in {\\mathcal{R}}_{1}$. We will prove this by contraction. \n\t\\par\n\tSuppose there exists $w_{1}$ such that ${\\mathcal{D}}_{h_{1},w_{1}} \\not\\subset {\\mathcal{D}}_{h_{2},w_{2}}$ for any $w_{2}\\in{\\mathcal{R}}_{2}$. Because $\\bigcup_{w_{2}\\in{\\mathcal{R}}_{2}}{\\mathcal{D}}_{h_{2},w_{2}}$ constitutes the whole space, there is some $w_{2}$ such that ${\\mathcal{D}}_{h_{1}, w_{1}}\\bigcap{\\mathcal{D}}_{h_{2},w_{2}} \\neq \\varnothing$ and ${\\mathcal{D}}_{h_{1}, w_{1}}\\not\\subset{\\mathcal{D}}_{h_{2},w_{2}}$. Thus, ${\\mathcal{D}}_{h_{1}, w_{1}}\\setminus {\\mathcal{D}}_{h_{2},w_{2}} \\neq \\varnothing$. Let $x^{\\dag}$ denote a point in\n\t${\\mathcal{D}}_{h_{1},w_{1}}\\setminus {\\mathcal{D}}_{h_{2},w_{2}}$ and let $x^{\\prime}$ a point in ${\\mathcal{D}}_{h_{2},w_{2}}\\bigcap{\\mathcal{D}}_{h_{1},w_{1}}$. Define $T_{*}$ as the transformation such that $T_{*}(x^{\\prime}) = x^{\\dag}$, $T_{*}(x^{\\dag}) = x^{\\prime}$ and $T_{*}(x) = x$ for $x \\neq \\{x^{\\prime}, x^{\\dag}\\}$. Then it is straightforward to verify that $T_{*}\\in {\\mathcal{T}}_{h_{1}}$ but $T_{*}\\notin {\\mathcal{T}}_{h_{2}}$, which is a contradiction.\n\\end{proof}\nThus $g(\\cdot)$ can be characterized by ${\\mathcal{T}}_{g}$ up to an invertible transformation. Define ${\\mathcal{C}}_{g} = \\{g^{\\prime}(\\cdot): g^{\\prime}(\\cdot) = (v\\circ g)(\\cdot) \\ \\text{for some invertible transformation $v(\\cdot)$}\\}$. For any $g^{\\prime} \\in {\\mathcal{C}}_{g}$, by defining \n\\begin{equation}\n{\\mathcal{H}}_{\\rm{s}}^{\\prime} = \\left\\{(\\phi\\circ g)(\\cdot)\\ \\Big | \\ \\phi(w) \\in \\mathop{\\arg\\min}_{z} \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(z, Y)\\mid g^{\\prime}(X) = w], \\quad a.s.\\right\\}, \\nonumber\n\\end{equation}\nsimilar arguments as in the proof of Theorem \\ref{thm: invariant generalization} can show ${\\mathcal{H}}_{{\\rm s}}^{\\prime} \\subset {\\mathcal{H}}_{*}$.\nTo train a model that generalizes well on all the data distributions following the same causal mechanism, any $g^{\\prime}(\\cdot)\\in {\\mathcal{C}}_{g}$ is sufficient. Thus, if ${\\mathcal{T}}_{g}$ is known, to find a model belongs to ${\\mathcal{H}}_{*}$, one may firstly find an invariant feature map $g^{\\prime}(\\cdot)$ such that ${\\mathcal{T}}_{g^{\\prime}} = {\\mathcal{T}}_{g}$ and then obtain the model according to Theorem \\ref{thm: invariant generalization}. However, finding a $g^{\\prime}(\\cdot)$ such that ${\\mathcal{T}}_{g^{\\prime}} = {\\mathcal{T}}_{g}$ is sometimes still a hard task.\n\nFor any function $h(\\cdot)$, define ${\\mathcal{I}}_{h}$ in the same way as ${\\mathcal{I}}_{g}$ with $g(\\cdot)$ replaced by $h(\\cdot)$ in the definition. We then have the following lemma.\n\\begin{lemma}\\label{lem: essential invariant set}\n\tFor any $h_{1}(\\cdot)$ and $h_{2}(\\cdot)$, if ${\\mathcal{I}}_{h_{1}} \\subset {\\mathcal{T}}_{h_{2}}$, then there exists a function $v(\\cdot)$ such that $h_{2}(\\cdot) = (v \\circ h_{1})(\\cdot)$.\n\\end{lemma}\n\\par\n\\begin{proof}\n\tLike in the proof of Lemma (\\ref{lem: invariant characterize}), it suffices to show that ${\\mathcal{I}}_{h_{1}} \\subset {\\mathcal{T}}_{h_{2}}$ implies for any $w_{1} \\in {\\mathcal{R}}_{1}$, there is some $w_{2}\\in{\\mathcal{R}}_{2}$ such that ${\\mathcal{D}}_{h_{1},w_{1}} \\subset {\\mathcal{D}}_{h_{2},w_{2}}$. We prove this by contraction. \n\t\n\tSuppose there is some $w_{1}$ such that ${\\mathcal{D}}_{h_{1},w_{1}} \\not\\subset {\\mathcal{D}}_{h_{2},w_{2}}$ for any $w_{2}\\in{\\mathcal{R}}_{2}$. Because $\\bigcup_{w_{2}\\in{\\mathcal{R}}_{2}}{\\mathcal{D}}_{h_{2},w_{2}}$ is the whole space, there is some $w_{2}$ such that ${\\mathcal{D}}_{h_{1},w_{1}}\\bigcap{\\mathcal{D}}_{h_{2},w_{2}} \\neq \\emptyset$ and ${\\mathcal{D}}_{h_{1},w_{1}}\\not\\subset{\\mathcal{D}}_{h_{2},w_{2}}$. Thus, ${\\mathcal{D}}_{h_{1},w_{1}}\\setminus {\\mathcal{D}}_{h_{2},w_{2}} \\neq \\emptyset$. Let $x^{\\dag}$ be a point in\n\t${\\mathcal{D}}_{h_{1},w_{1}}\\setminus {\\mathcal{D}}_{h_{2},w_{2}}$ and let $x^{\\prime}$ be a point in ${\\mathcal{D}}_{h_{2},w_{2}}\\bigcap{\\mathcal{D}}_{h_{1},w_{1}}$. According to the definition of essential invariant subset, because $h_{1}(x_{1}) = h_{2}(x_{2})$, there are finite transformations $T_{1}(\\cdot),\\dots,T_{K}(\\cdot) \\in {\\mathcal{I}}_{g}$ such that $\\bar{T}(x^{\\prime}) = x^{\\dag}$ where $\\bar{T}(\\cdot) = (T_{1}\\circ\\dots\\circ T_{K})(\\cdot)$. It can be verified that ${\\mathcal{T}}_{h_{2}}$ is closed with respect to function composition. Hence, $\\bar{T}(\\cdot) \\in {\\mathcal{T}}_{h_{2}}$. However, $h_{2}(\\bar{T}(x^{\\prime})) = h_{2}(x^{\\dag}) \\neq w_{2} = h_{2}(x^{\\prime})$, which is a contradiction. \n\\end{proof}\n\\paragraph{Restatement of Theorem \\ref{thm: alternative problem 1}}\n\\emph{If $P_{{\\rm s}} \\in {\\mathcal{P}}$, then\n\t\\begin{equation}\n\t{\\mathcal{H}}_{{\\rm s}} \\subset\n\t\\mathop{\\arg\\min}_{h} \\sup_{T \\in {\\mathcal{T}}_{g}}\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(T(X)), Y)],\\nonumber\n\t\\end{equation}\n\twhere ${\\mathcal{H}}_{{\\rm s}}$ is defined in (\\ref{eq: causal opt}).\n}\n\\begin{proof}\n\tIt suffices to show that for all $h_{{\\rm s}}(\\cdot)\\in {\\mathcal{H}}_{{\\rm s}}$, we have\n\t\\begin{equation}\n\th_{{\\rm s}}(\\cdot) \\in\n\t\\mathop{\\arg\\min}_{h} \\sup_{T \\in {\\mathcal{T}}_{g}}\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(T(X)), Y)].\n\t\\end{equation} \n\tNote that $h_{{\\rm s}}(\\cdot) = (\\phi_{{\\rm s}}\\circ g)(\\cdot)$ for some $\\phi_{{\\rm s}}(\\cdot)$ and hence is invariant to any transformation $T(\\cdot)\\in {\\mathcal{T}}_{g}$. We then have $\\sup_{T \\in {\\mathcal{T}}_{g}}\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{{\\rm s}}(X), Y)] = \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{{\\rm s}}(T(X)), Y)]$. Thus, it suffices to prove that for all $h(\\cdot) $, there exists $T(\\cdot) \\in {\\mathcal{T}}_{g}$ such that \n\t\\begin{equation}\\label{eq: sufficient eq 2}\n\t\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(T(X)), Y)] \\geq \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{{\\rm s}}(X), Y)].\n\t\\end{equation}\n\tAccording to axiom of choice, there is a choice function $a$ such that $a(w) \\in {\\mathcal{D}}_{g,w}$ for almost every $w$. Define $\\tilde{T}$ to be a transformation such that $\\tilde{T}(x) = a(w)$ for $x\\in {\\mathcal{D}}_{g,w}$. Then $\\tilde{T}(\\cdot) \\in {\\mathcal{T}}_{g}$ and we have\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(\\tilde{T}(X)), Y)\\mid g(X) = w] & = \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(a(w)), Y)\\mid g(X) = w] \\\\\n\t& \\geq \\mathbb{E}_{P_{{\\rm s}}}[\\phi_{{\\rm s}}(w), Y)\\mid g(X) = w] \\\\\n\t& = \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{{\\rm s}}(X), Y)\\mid g(X) = w] \\quad a.s. \n\t\\end{aligned}\n\t\\end{equation}\n\tBy taking expectation on both sides, we can obtain equation~(\\ref{eq: sufficient eq 2}).\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{thm: alternative problem 2}}\\label{app: proof of thm: alternative problem 2}\n\\paragraph{Restatement of Theorem \\ref{thm: alternative problem 2}}\n\\emph{If $P_{{\\rm s}} \\in {\\mathcal{P}}$,\n\tthen \n\t\\begin{equation}\\label{eq: invariant feature problem-copy}\n\t\\begin{aligned}\n\t{\\mathcal{H}}_{\\rm s} = \\mathop{\\arg\\min}_{h} \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(X), Y)]\\quad\n\t\\text{\\rm subject to}\\ \\ h(\\cdot) = (h\\circ T)(\\cdot), \\ \\forall \\ T(\\cdot)\\in {\\mathcal{I}}_{g}.\n\t\\end{aligned}\n\t\\end{equation}\n\twhere ${\\mathcal{I}}_{g}$ is any causal essential set of $g(\\cdot)$ and ${\\mathcal{H}}_{{\\rm s}}$ is defined in (\\ref{eq: causal opt}).}\n\n\\begin{proof}\n\n\tWe first show\n\t\\begin{equation}\n\t\\begin{aligned}\n\t{\\mathcal{H}}_{\\rm s} \\subset &\\mathop{\\arg\\min}_{h} \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(X), Y)]\\\\\n\t&~\\text{subject to}~\\ \\ h(\\cdot) = (h\\circ T)(\\cdot), \\ \\forall \\ T(\\cdot)\\in {\\mathcal{I}}_{g}.\\nonumber\n\t\\end{aligned}\n\t\\end{equation}\n\tNote that the restriction in (\\ref{eq: invariant feature problem-copy}) is equivalent to ${\\mathcal{I}}_{g} \\subset {\\mathcal{T}}_{h}$.\n\tIt suffices to show that \n\t\\begin{equation}\\label{eq: proof-constrained opt}\n\t\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(X), Y)] \\geq \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{{\\rm s}}(X), Y)]\n\t\\end{equation}\n\tfor any $h(\\cdot)$ with ${\\mathcal{I}}_{g} \\subset {\\mathcal{T}}_{h}$ and for any $h_{{\\rm s}}(\\cdot) \\in {\\mathcal{H}}_{{\\rm s}}$. If ${\\mathcal{I}}_{g} \\subset {\\mathcal{T}}_{h}$, according to Lemma \\ref{lem: essential invariant set}, there exists $v(\\cdot)$ such that $h(\\cdot) = (v \\circ g)(\\cdot)$. By the definition of $h_{{\\rm s}}(\\cdot)$, there also exists $\\phi_{{\\rm s}}(\\cdot)$ satisfying $h_{{\\rm s}}(\\cdot) = (\\phi_{{\\rm s}}\\circ g)(\\cdot)$ and $\\phi_{{\\rm s}}(w) \\in \\mathop{\\arg\\min}_{z} \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(z, Y)\\mid g(X) = w]$ for almost every $w$. Thus, we have\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h(X), Y)\\mid g(X) = w] & = \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(v(w), Y)\\mid g(X) = w]\\\\\n\t& \\geq \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(\\phi_{{\\rm s}}(w), Y)\\mid g(X) = w] \\\\\n\t& \\geq \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{{\\rm s}}(X), Y)\\mid g(X) = w] \\quad a.s.\n\t\\end{aligned}\\nonumber\n\t\\end{equation}\n\tThen (\\ref{eq: proof-constrained opt}) follows by taking expectation.\n\t\n\tNext we show the opposite inclusion to prove (\\ref{eq: invariant feature problem-copy}). Suppose $h_{*}(\\cdot)$ is a solution to the optimization problem in (\\ref{eq: invariant feature problem-copy}). Then according Lemma~\\ref{lem: essential invariant set}, there is some $v_{*}(\\cdot)$ such that $h_{*}(\\cdot) = (v_{*}\\circ g)(\\cdot)$. Let $h_{{\\rm s}}(\\cdot) = (\\phi_{{\\rm s}}\\circ g)(\\cdot) \\in {\\mathcal{H}}_{{\\rm s}}$. Then \n\t\\begin{equation}\\label{eq: lower bound}\n\t\\begin{aligned}\n\t\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{*}(X), Y)\\mid g(X) = w]\n\t&= \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(v_{*}(w), Y)\\mid g(X) = w] \\\\\n\t& \\geq \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(\\phi_{{\\rm s}}(w), Y)\\mid g(X) = w] \\\\\n\t& = \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{{\\rm s}}(X), Y)\\mid g(X) = w] \\quad a.s.,\n\t\\end{aligned}\n\t\\end{equation}\n\tby definition. \n\tBecause $h_{*}(\\cdot)$ is a solution to the minimization problem, we have\n\t\\begin{equation}\n\t\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{*}(X), Y)] = \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{{\\rm s}}(X), Y)]. \\nonumber\n\t\\end{equation} \n\tCombining this with (\\ref{eq: lower bound}), we have\n\t\\begin{equation}\n\t\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{*}(X), Y)\\mid g(X) = w] \\leq \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(h_{{\\rm s}}(X), Y)\\mid g(X) = w] \\quad a.s.\n\t\\end{equation} \n\tThis implies \n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(v_{*}(w), Y)\\mid g(X) = w]\n\t& \\leq \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(\\phi_{{\\rm s}}(w), Y)\\mid g(X) = w]\\\\\n\t& = \\min_{z} \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(z, Y)\\mid g(X) = w]\n\t\\quad a.s.\\nonumber\n\t\\end{aligned}\n\t\\end{equation} \n\tThus, we conclude that $v_{*}(w) \\in \\mathop{\\arg\\min}_{z} \\mathbb{E}_{P_{{\\rm s}}}[{\\mathcal{L}}(z, Y)\\mid g(X) = w]$.\n\\end{proof}\n\\section{More Experimental Results}\n\\subsection{Toy Example and Simulation}\\label{app: toy example}\nIn the following toy example, we are able to construct an explicit formulation of the causal essential invariant set. \n\\begin{example}\\label{eg: toy} \\emph{\n\t\tLet $X$ be a non-singular $2\\times 2$ matrix and $X^{(j)}$ be the $j$-th column of $X$ for $j=1,2$. Suppose that $g(X)$ is the area of the triangle formed by the two points $X^{(1)}$, $X^{(2)}$ and the origin. Then it is not hard to show that $\\{T_{R,\\theta}(\\cdot), T_{S,a}(\\cdot),T_{M}(\\cdot), T_{P}(\\cdot), T_{I}(\\cdot)\\mid \\theta \\in \\ [0,\\pi\/4], \\ a\\in [2\/3,3\/2]\\}$ is an essential invariant set of $g(\\cdot)$, where \n\t\t\\begin{equation}\n\t\t\\begin{array}{ll}\n\t\tT_{R,\\theta}(X) = \\left(\n\t\t\\begin{array}{cc}\n\t\t\\cos \\theta & -\\sin\\theta\\\\\n\t\t\\sin\\theta & \\cos\\theta \n\t\t\\end{array}\n\t\t\\right)X , \n\t\t&T_{S,a}(X) = X\\left(\n\t\t\\begin{array}{cc}\n\t\ta & 0\\\\\n\t\t0 & a^{-1} \n\t\t\\end{array}\n\t\t\\right),\\\\ \n\t\tT_M(X) = X\\left(\n\t\t\\begin{array}{cc}\n\t\t-1 & 0\\\\\n\t\t0 & 1 \n\t\t\\end{array}\n\t\t\\right), \n\t\t&T_{P}(X) = X\\left(\n\t\t\\begin{array}{cc}\n\t\t1 & 1\\\\\n\t\t0 & 1 \n\t\t\\end{array}\n\t\t\\right), \\\\ \n\t\tT_{I}(X) = X\\left(\n\t\t\\begin{array}{cc}\n\t\t-1 & 0\\\\\n\t\t0 & -1 \n\t\t\\end{array}\n\t\t\\right). &\n\t\t\\end{array} \\nonumber\n\t\t\\end{equation}\n\t\tHere $T_{R,\\theta}(\\cdot)$ rotates the triangle with $\\theta$ degree clockwise, and $T_{S,a}(\\cdot)$ scales the two edges (one connects $X^{(1)}$ to the origin and the other connects $X^{(2)}$ to the origin) of the triangle with $a$ and $a^{-1}$ times, respectively. $T_{M}(\\cdot)$ mirrors the triangle with respect to the x-axis. $T_{P}(\\cdot)$ transforms the triangle to another triangle with same base and height, and $T_{I}(\\cdot)$ transforms the the triangle to another one that is symmetric with respect to the origin. All these transformations are known to keep the triangle area unchanged based on elementary geometry.}\n\\end{example}\n\\par\nNow we verify the effectiveness of the proposed method in the main body using this example. \n\\paragraph{Data.} We consider the following data generation process:\n\\begin{equation} \\label{eq: DGP}\n\\begin{aligned}\n&X^{(1)}\\sim N(0, I_{2}), \\ X^{(2)}\\sim N(0, 2I_{2}),\\ X = (X^{(1)}, X^{(2)}),\\\\\n& \\epsilon \\sim N(0,1), \\ \\eta = \\frac{a\\Phi^{-1}(\\pi^{-1}\\alpha) + \\epsilon}{\\sqrt{a^{2} + 1}},\\\\\n& Y = |\\det(X)| + \\eta, \n\\end{aligned}\n\\end{equation}\nwhere $I_{2}$ is the identity matrix of order $2$, $\\Phi(\\cdot)$ is the cumulative distribution function of standard normal distribution. \nIn this data generation process, $|\\det(X)|$ is the area of the triangle formed by $X^{(1)}$, $X^{(2)}$ and the origin, and is the causal feature in this example. Here $\\alpha$ is the angle between $(X^{(1)} + X^{(2)})\/2$ and x-axis, and is correlated with $Y$ in certain domains, with $a$ a parameter that reflects this correlation. However, this correlation is a spurious correlation that changes across domains, i.e., $a$ is set to be different in different domains.\nIn the training population, we pick $a=-3$. We then generate i.i.d.~samples of size $1,000$, denoted by $\\{(Y_{i}, X_{i})\\}_{i=1}^{1000}$, and train a model $h(X,\\beta)$ with parameter $\\beta$ to predict $Y$ based on these generated samples.\n\\paragraph{Model.} For any $2\\times 2$ matrix\n\\begin{equation}\nX = \\left(\\begin{array}{cc}\nX_{11}&X_{12}\\\\\nX_{21}&X_{22}\n\\end{array}\\right), \\nonumber\n\\end{equation}\nlet \n\\begin{equation}\n\\begin{aligned}\nv(X)& = (1, X_{11}, X_{21}, X_{12}, X_{22}, X_{11}^{2}, X_{21}^{2}, X_{12}^{2}, X_{22}^{2},\\\\\n&\\quad~~~X_{11}X_{21}, X_{11}X_{12}, X_{11}X_{22}, X_{21}X_{12}, X_{21}X_{22}, X_{12}X_{22})^{{\\rm T}}.\\nonumber\n\\end{aligned}\n\\end{equation}\nThe model is\n\\begin{equation}\nh_{\\beta}(X) = {\\rm ReLU}(\\beta_{[1]}^{{\\rm T}}v(X)) + \\beta_{[2]}^{{\\rm T}}v(X), \\nonumber\n\\end{equation}\nwhere $\\beta = (\\beta_{[1]}^{{\\rm T}}, \\beta_{[2]}^{{\\rm T}})^{{\\rm T}}$ is the model parameter. We pick this model because we have known that $|\\det(X)|$ is a function of $v(X)$, and there is some $\\beta^{*}$ such that $h_{\\beta^{*}}(X) = |\\det(X)|$.\n\n\\paragraph{Method.} Based on the essential invariant set given in Example \\ref{eg: toy}, we define five invariant transformations\n\\begin{equation}\n\\begin{array}{ll}\nT_{1}(X) = \\left(\n\\begin{array}{cc}\n\\cos\\frac{\\pi}{12} & -\\sin\\frac{\\pi}{12}\\\\\n\\sin\\frac{\\pi}{12} & \\cos\\frac{\\pi}{12}\n\\end{array}\n\\right)X , \n&T_{2}(X) = X\\left(\n\\begin{array}{cc}\n1.1 & 0\\\\\n0 & 1.1^{-1} \n\\end{array}\n\\right),\\\\ \nT_3(X) = X\\left(\n\\begin{array}{cc}\n-1 & 0\\\\\n0 & 1 \n\\end{array}\n\\right), \n&T_{4}(X) = X\\left(\n\\begin{array}{cc}\n1 & 1\\\\\n0 & 1 \n\\end{array}\n\\right), \\\\ \nT_{5}(X) = X\\left(\n\\begin{array}{cc}\n-1 & 0\\\\\n0 & -1 \n\\end{array}\n\\right). &\n\\end{array}\\nonumber\n\\end{equation}\nFor ease of notation, we let $T_{0}(X) = X$ be the identity transformation.\nWe learn the model parameter by minimizing four different loss functions, namely, the empirical risk\n\\begin{equation}\n\\frac{1}{n}\\sum_{i=1}^{n}(Y_{i} - h_{\\beta}(X_{i}))^{2}, \\nonumber\n\\end{equation}\nthe average risk over different transformations\n\\begin{equation}\n\\frac{1}{n}\\sum_{k=0}^{5}\\sum_{i=1}^{n}(Y_{i} - h_{\\beta}(T_{k}(X_{i})))^{2},\n\\nonumber\n\\end{equation}\nthe maximal risk over different transformations\n\\begin{equation}\n\\max_{k=0,\\dots,5}\\left\\{\\frac{1}{n}\\sum_{i=1}^{n}(Y_{i} - h_{\\beta}(T_{k}(X_{i})))^{2}\\right\\},\\nonumber\n\\end{equation}\nand the RICE loss function\n\\begin{equation}\n\\frac{1}{n}\\sum_{i=1}^{n}(Y_{i} - h_{\\beta}(X_{i}))^{2}+\\lambda \\max_{k=0,\\ldots,5}\\left\\{\\frac{1}{n}\\sum_{i=1}^{n}(h_{\\beta}(X_{i}) - h_{\\beta}(T_{k}(X_{i}))^{2}\\right\\},\\nonumber\n\\end{equation}\nwhere $n=1000$. For the given quantities $l_{0}, \\cdots, l_{5}$, we replace the maximum $\\max_{k=0,\\dots,5}\\{l_{k}\\}$ in the above losses with the softmax weighting quantity \n$\\sum_{k=0}^{5}\\exp(0.2l_{k})l_{k}\/\\sum_{k=0}^{5}\\exp(0.2l_{k})$\nfor ease of computation in the implementation of RICE.\n\\paragraph{Results.} The resulting model is evaluated on i.i.d.~sample generated following the data generation process (\\ref{eq: DGP}) with different $a$. The following figure plots the squared prediction error of the four methods on test data with different values of $a$. Each reported value is the average over $200$ simulations.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[scale=0.4]{.\/pic\/toy-example.pdf}\n\t\\caption{Squared prediction error on test data from distributions with different values of $a$.}\n\\end{figure}\n\nIt can be seen that when the test distribution has similar spurious correlations as the training population, minimizing the empirical risk performs the best among the four methods. However, it performs the worst if an opposite spurious correlation appears in the test population. The RICE algorithm has the best worst-case performance, which is consistent with our theoretical analysis. Moreover, the RICE algorithm seems successfully capture the invariant causal mechanism across different environments, as its prediction errors under different test distributions are stable and close to the variance of the intrinsic error $\\eta$. \n\\subsection{Hyperparameters}\\label{app: hyperparameters}\nWe summarize the hyperparameters of the proposed RICE for \\texttt{C-MNIST}, \\texttt{PACS}, and \\texttt{VLCS} datasets in Table \\ref{tbl:hyper}. The learning rate is decayed by $0.2$ at epoch $6, 12$, and $20$. \n\\begin{table*}[t!]\n\t\\caption{Hyperparameters of the proposed RICE on \\texttt{C-MNIST}, \\texttt{PACS}, and \\texttt{VLCS}.}\n\n\t\\label{tbl:hyper}\n\t\\centering\n\n\t{\n\t\t{\n\t\t\t\\begin{tabular}{c c c c}\n\t\t\t\t\\hline\n\t\t\t\tDataset & \\texttt{C-MNIST} & \\texttt{PACS} & \\texttt{VLCS} \\\\\n\t\t\t\t\\hline\n\t\t\t\tLearning Rate & 0.1 & 5e-5 & 5e-5 \\\\\n\t\t\t\tBatch Size & 128 & 32 & 32 \\\\\n\t\t\t\tWeight Decay & 5e-4 & 0 & 0 \\\\\n\t\t\t\tDrop Out & 0 & 0.1 & 0.1 \\\\\n\t\t\t\tEpoch & 20 & 20 & 20 \\\\\n\t\t\t\t$\\lambda_{0}$ & 0.25 & 0.5 & 0.5 \\\\\n\t\t\t\t$\\beta_{1}$ & 0.9 & 0.9 & 0.9 \\\\\n\t\t\t\t$\\beta_{2}$ & 0.999 & 0.999& 0.999 \\\\ \n\t\t\t\t\\hline\n\t\\end{tabular}}}\n\\end{table*}\n\\subsection{Ablation Study}\\label{app: ablation}\nIn Section \\ref{sec:experiment}, for the experiments on \\texttt{PACS} and \\texttt{VLCS}, we collect training data from several domains for the proposed RICE. However, our theory in Section \\ref{subsec: OOD&CIT} requires only a single domain. Thus, in this subsection, we study the performance of RICE with single domain training data. \n\\par\nOur experiments are conducted on both \\texttt{PACS} and \\texttt{VLCS}. All the hyperparameters are set to be same with those in Section \\ref{sec:experiment}, except the number of training domains---we only use single domain data and hence less training samples for each single experiment. For example, for \\texttt{PACS}, if the test domain is sketch, then we run RICE on training data from one of the three other domains (photo, art and cartoon) and report the accuracy on the test domain. To run RICE, the data generated by CycleGAN are used as augmented data and in the regularization term. For a fair comparison, we do not use the CycleGAN that transfer from training domain to test domain and adopt similar experimental settings for ERM.\n\\par\nThe results are summarized in Figure \\ref{fig: ablation1}. We can see that RICE performs much better than the baseline method ERM, which verifies our theoretical conclusions in Theorem \\ref{thm: alternative problem 2}. Besides, the test accuracy on the target domain can be quite high even when the model is trained using data from a single domain. For example, on \\texttt{VLCS} dataset, when test data is from SUN09 domain, the model trained on VOC2007 domain even exhibits a better OOD generalization than the model trained on data from three domains. This implies that, for OOD generalization problem, the number of domains may not be crucial to the performance as long as some representative CITs are available.\n\n\\begin{figure*}[t!]\\centering\n\t\\subcaptionbox{Photo}{\n\t\t\\includegraphics[width=0.43\\textwidth]{.\/pic\/ablation\/pacs_photo.png}}\n\t\\subcaptionbox{Art}{\n\t\t\\includegraphics[width=0.43\\textwidth]{.\/pic\/ablation\/pacs_art.png}} \\\\\n\t\\vspace{-1em}\n\t\\subcaptionbox{Cartoon}{\n\t\t\\includegraphics[width=0.43\\textwidth]{.\/pic\/ablation\/pacs_cartoon.png}}\n\t\\subcaptionbox{Sketch}{\n\t\t\\includegraphics[width=0.43\\textwidth]{.\/pic\/ablation\/pacs_sketch.png}}\\\\\n\t\\vspace{-1em}\n\t\\subcaptionbox{VOC2007}{\n\t\t\\includegraphics[width=0.43\\textwidth]{.\/pic\/ablation\/vlcs_pas.png}}\n\t\\subcaptionbox{LabelMe}{\n\t\t\\includegraphics[width=0.43\\textwidth]{.\/pic\/ablation\/vlcs_label.png}}\n\t\\\\\n\t\\vspace{-1em}\n\t\\subcaptionbox{Caltech101}{\n\t\t\\includegraphics[width=0.43\\textwidth]{.\/pic\/ablation\/vlcs_cal.png}}\n\t\\subcaptionbox{SUN09}{\n\t\t\\includegraphics[width=0.43\\textwidth]{.\/pic\/ablation\/vlcs_sun.png}}\n\n\n\n\n\n\n\n\n\n\t\\caption{Performance of RICE and ERM on the \\texttt{PACS} (a-d) and \\texttt{VLCS} (e-h) datasets with training data from single domains. Figure title indicates the test domain, and the blue dashed line represents the test accuracy when the training data are from three domains, as reported in Section~\\ref{sec:experiment}.}\n\t\\label{fig: ablation1}\n\\end{figure*}\n\n\n\n\n\n\n\n\\subsection{Generated Data}\\label{app: generated data}\nOur experiments in the main body involve generating causally invariant images. In this subsection, we present visualizations of some generated images for a better understanding of the proposed algorithm.\n\\paragraph{\\texttt{C-MNIST}} Figure \\ref{fig: c-mnist} shows some \\texttt{C-MNIST} images. As seen from the training set, there exist spurious correlations between the colors of the foreground or background and the category. However, the correlation disappears in the test set, as the foreground and background colors are randomly assigned. \n\\paragraph{\\texttt{PACS}} We also present some transformed data from \\texttt{PACS} dataset generated by CycleGAN. The CycleGAN is used to simulate CITs as we have clarified the main body of this paper. As the data in \\texttt{PACS} come from $7$ categories, for each category we pick $4$ pictures respectively from domains \\{photo, art, cartoon, sketch\\}. The transformed images are shown in Figure \\ref{fig: pacs}, where the columns correspond to the styles of \\{photo, art, cartoon, sketch\\}, respectively. \n\\par\nLet us look at these generated data over different domains. For the generated images of the photo domain (the first column), the trained CycleGAN tends to alter its color of foreground and add a background, especially when the original images are from the cartoon and sketch domains. Similar trends exhibit in the generated data of the art domain (the second column). In contrast to the two aforementioned domains, the generated cartoon data in the third column remove the background (if exists) while keep or alter the color of the foreground. The generated sketch data (the fourth column) are more likely to be a grayscale view of the original images. However, for each generated image, the shape of its foreground (i.e., the casual feature to decide the category) does not change when we vary the domains.\n\\par\nThe proposed algorithm RICE regularizes the model to encourage the model to be invariant under the CITs, i.e., invariant to the changes of spurious features. This enables the model to be robust to the misleading signal from spurious features and to make predictions via the casual feature. For example, for the dog images in the last row of Figure \\ref{fig: pacs-c}, which are generated from the images of cartoon style (the third column), the generated dog image of photo style (the first column) has a grass background. However, RICE requires the model to exhibit similar outputs for the two images, hence breaking the spurious correlation between dog and grass. \n\\paragraph{\\texttt{VLCS}} Similar to \\texttt{PACS}, we present some of the domain transformed data from \\texttt{VLCS} dataset generated by CycleGAN. We pick $4$ pictures respectively from domains \\{VOC2007, LabelMe, Caltech101, SUN09\\} for each of the $5$ categories in \\texttt{VLCS}. Then we vary the domains of these picked data using the trained CycleGAN models. The transformed data are visualized in Figure \\ref{fig: vlcs}. \n\\par\nThe generated \\texttt{VLCS} images exhibit similar behaviors as \\texttt{PACS}. Specifically, for a given image from a certain domain, the CycleGAN model tends to deterministically vary the color of the background according to the domains. Thus, the reasoning about the effectiveness of RICE on \\texttt{PACS} also applies here. \n\n\\begin{figure*}[t!]\\centering\n\t\\subcaptionbox{Training data in \\texttt{C-MNIST}}{\n\t\t\\includegraphics[width=0.35\\textwidth]{.\/pic\/c-mnist\/colored.jpg}}\n\t\\hspace{0.2in}\n\t\\subcaptionbox{Test data in \\texttt{C-MNIST}}{\n\t\t\\includegraphics[width=0.35\\textwidth]{.\/pic\/c-mnist\/random.jpg}}\n\t\\caption{Images of the \\texttt{C-MNIST} dataset.}\n\t\\label{fig: c-mnist}\n\\end{figure*}\n\\begin{figure*}[t!]\\centering\n\t\\vspace{-0.1in}\n\t\\subcaptionbox{original domain: photo}{\n\t\t\\includegraphics[width=0.35\\textwidth]{.\/pic\/pacs\/pic.jpg}}\n\t\\hspace{0.2in}\n\n\t\\subcaptionbox{original domain: art}{\n\t\t\\includegraphics[width=0.35\\textwidth]{.\/pic\/pacs\/art.jpg}}\n\t\\\\\n\t\\subcaptionbox{original domain: cartoon\\label{fig: pacs-c}}{\n\t\t\\includegraphics[width=0.35\\textwidth]{.\/pic\/pacs\/cartoon.jpg}}\n\t\\hspace{0.2in}\n\t\\subcaptionbox{original domain: sketch}{\n\t\t\\includegraphics[width=0.35\\textwidth]{.\/pic\/pacs\/sketch.jpg}}\n\t\\caption{Synthetic data of \\texttt{PACS} generated by CycleGAN. Columns from left to right correspond to domains of \\{photo, art, cartoon, sketch\\}, respectively. Figure title indicates the domain of original data, based on which the data of the rest domains in the figure are generated by CycleGAN.}\n\t\\label{fig: pacs}\n\\end{figure*}\n\\begin{figure*}[t!]\\centering\n\t\\subcaptionbox{original domain: VOC2007}{\n\t\t\\includegraphics[width=0.35\\textwidth]{.\/pic\/vlcs\/pas.jpg}}\n\t\\hspace{0.2in}\n\t\\subcaptionbox{original domain: LabelMe}{\n\t\t\\includegraphics[width=0.35\\textwidth]{.\/pic\/vlcs\/label.jpg}}\n\t\\\\\n\t\\subcaptionbox{original domain: Caltech101}{\n\t\t\\includegraphics[width=0.35\\textwidth]{.\/pic\/vlcs\/cal.jpg}}\n\t\\hspace{0.2in}\n\t\\subcaptionbox{original domain: SUN09 }{\n\t\t\\includegraphics[width=0.35\\textwidth]{.\/pic\/vlcs\/sun.jpg}}\n\n\t\\caption{Synthetic data of \\texttt{VLCS} generated by CycleGAN. Columns from left to right correspond to domains of \\{VOC2007, LabelMe, Caltech101, SUN09\\}, respectively. Figure title indicates the domain of original data, based on which the data of the rest domains in the figure are generated by CycleGAN.}\n\t\\label{fig: vlcs}\n\\end{figure*}\n\\subsection{Benchmark algorithms}\\label{app: benchmark algo}\n\\begin{itemize}\n\t\\item Empirical Risk minimization (ERM) pools together the data from all the domains and then minimizes the empirical loss to train the model. Notice that here an \\texttt{ImageNET} pre-trained model is used.\n\t\\item Marginal Transfer Learning \\cite{blanchard2021domain} use the mean embedding of the feature distribution in each domain as an input of the classifier.\n\t\\item Group Distributionally Robust Optimization (GroupDRO) \\cite{sagawa2020distributionally} minimizes the largest loss across different domains.\n\t\\item Domain-Adversarial Neural Networks (DANN) \\cite{ganin2016domain} use adversarial networks to match the feature distribution in different domains.\n\t\\item Invariant Risk Minimization (IRM )\\cite{arjovsky2019invariant} learns a feature representation such that the optimal classifiers on top of the representation is the same across the domains.\n\\end{itemize}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{The problem}\n\\label{sec-intro}\n\nBrownian motion is a martingale with centred Gaussian marginals, and \nvariance $t$. Hamza and Klebaner~\\cite{HamzaKlebaner:06} asked, are \nthere other real-valued martingale processes with the same univariate \nmarginals as Brownian motion? Since Dupire~\\cite{Dupire:97} has shown \nthat Brownian motion is the unique \nmartingale diffusion with $\\mathcal N(0,t)$ marginals, finding alternative \nmartingales with the same marginals involves relaxing either the \ncontinuity assumption, or the Markov property.\n\nHamza and Klebaner~\\cite{HamzaKlebaner:06} answered their question in \nthe positive, but it turns out that Madan and Yor~\\cite{MadanYor:02} had \nalready provided a construction based on the Az\\'{e}ma-Yor solution of \nthe Skorokhod embedding problem. Both the Madan and \nYor~\\cite{MadanYor:02} and Hamza and Klebaner~\\cite{HamzaKlebaner:06} \nconstructions are discontinuous processes. However, \nAlbin~\\cite{Albin:07} produced an ingenious solution to the problem with \ncontinuous paths, based on products of Bessel process. Subsequently, \nOleszkiewicz~\\cite{Oleszkiewicz:08} produced a simpler construction \nwhich is extremely natural and is based on mixing and the Box-Jenkins \nsimulation of a Gaussian variable. Oleszkiewicz also introduced the \nevocative term {\\em fake Brownian motion} to describe a (continuous) \nmartingale with the same univariate marginal distributions as Brownian \nmotion.\n\nNow that there has been serious study of fake Brownian motion it seems \nreasonable to ask if there exist fake versions of other canonical martingales. In a \nrelated context, Hirsch et \nal~\\cite{HirschProfetaRoynetteYor:11} give several general methods of constructing \nprocesses with given marginals, but none is quite \nsuited to the problem we consider here.\nIn particular, in this article we provide a family of fake exponential \nBrownian motions: we give a martingale stochastic process with \ncontinuous paths and with lognormal marginals matching those of exponential \nBrownian motion, which is not exponential Brownian motion. The ideas \napply to other time-homogeneous diffusions.\n\nOne reason for interest in fake processes in this sense comes from mathematical finance, \nsee Hobson~\\cite{Hobson:11}. In finance knowledge of a continuum of option prices of a \nfixed maturity (one for each possible strike) is equivalent to knowledge of the marginal \ndistribution of the underlying financial asset under the pricing measure. This result \ndates back to Breeden and Litzenberger~\\cite{BreedenLitzenberger:78}. Knowledge of the \nprices of options for the continuum of strikes and maturities is equivalent to knowledge \nof all the marginals of the underlying financial asset, but implies nothing about the \nfinite-dimensional distributions. (However, general theory tells us that the asset \nprice, suitably discounted, must be a martingale, again under the pricing measure.) In \nfinance we often have knowledge of option prices, but we have little reason to believe \nin any particular model. The canonical model in finance is the \nOsborne-Samuelson-Black-Scholes exponential Brownian motion model, under which vanilla \noption prices are given by the Black-Scholes formula. The fact that we can provide \nalternative martingale models with the same marginals, and therefore which support the \nsame option prices, shows that there is no unique model which is consistent with those \nprices.\n \nLet $\\delta_x$ denote the unit mass at $x$. Suppose \n$P=(P_t)_{t \\geq 0}$ is exponential Brownian motion, scaled such that \n$P_0=1$, and with drift parameter chosen such that $P$ is a martingale. \nSetting volatility equal to one we may assume that $P$ solves $dP_t\/P_t \n= dB_t$. Let $(\\nu_t)_{t \\geq 0}$ be the family of laws of $(P_{t})_{t \n\\geq 0}$ with lognormal densities $(p_t(\\cdot))_{t > 0}$ where\n\\begin{equation}\n\\label{eqn:ebmdensity} p_t(x) = \\frac{1}{\\sqrt{2 \\pi}}\n\\frac{e^{-t\/8}}{t^{1\/2}}\n\\frac{e^{-(\\ln x)^2\/2t}}{x^{3\/2}} . \n\\end{equation}\nWe write $\\nu_t \\sim {\\mathcal L}(P_t) \\sim p_t(\\cdot)$. Following Hamza \nand Klebaner we ask {\\em does there exist a continuous martingale \nwith marginals $(\\nu_t)_{t \\geq 0}$, which is distinct from \nexponential Brownian motion?} The main goal of this note is to show that \nthe answer to this question is positive.\n\n\nFor the general diffusion case, let $X=(X_t)_{t \\geq 0}$ be a time-homogeneous \nmartingale diffusion so that $X$ solves $dX_t = \\sigma(X_t)dB_t$ subject to $X_0 =x_0$. \nLet $I\n\\subseteq \\mathbb R$ be the state space of $X$ and let \n$\\mu^X_t$, with associated density $f_t(\\cdot)$, be the law of $X_t$. \nAssume that the boundary points of $I$ are not \nattainable so that $I = (\\ell, r)$ where $-\\infty \\leq l < x_0 < r \\leq \\infty$ and that \n$\\sigma$ is positive and continuously differentiable on the interior of $I$. Further, we \nsuppose \nthat \n$f_t(x)$ is continuously differentiable in both $x$ and $t$. This is automatically \nsatisfied for Brownian motion and exponential Brownian motion and by results of \nRogers~\\cite{Rogers:85} is also satisfied if \n$\\sigma$ is thrice continuously differentiable and $\\int_\\ell \n\\frac{1}{\\sigma(x)} dx = \\infty = \\int^r \\frac{1}{\\sigma(x)} dx$.\n\nOur goal is to describe a fake version of $X$, where a fake version of \nthe martingale $X$ is a continuous martingale $\\tilde{X} = \n(\\tilde{X}_t)_{t \\geq 0}$ such that ${\\mathcal L}(\\tilde{X}_t) \\sim \n\\mu^X_t$, which is distinct from $X$.\n\nUsing Jensen's inequality and the martingale property of $X$ we can \ndeduce that the family $(\\mu^X_t)_{t \\geq 0}$ must be increasing in \nconvex order. Further, the results of Kellerer~\\cite{Kellerer:72} imply \nthat the fact that a family of measures is increasing in convex order is \nnecessary and sufficient for there to be a martingale with those \nmarginals. Of course, such a process need not be a diffusion. For a \ntime-homogeneous diffusion we have the following result, which follows from Tanaka's \nformula (Revuz and\nYor~\\cite[Theorem 6.1.2]{RevuzYor:99}), the fact that $\\int_0^t I_{\n\\{ X_s \\geq x \\} }dX_s$ is a martingale, and an application of Fubini's\nTheorem.\n\n\\begin{lem}[Carr and Jarrow~\\cite{CarrJarrow:87}, Klebaner~\\cite{Klebaner:02}]\n\\label{lem:mgdiff}\nSuppose $\\sigma(.)$ is continuous. \nThen\n\\[ \\mathbb E[(X_T - x)^+] = (x_0-x)^+ + \\frac{\\sigma(x)^2}{2} \\int_0^T f_t(x) \ndt \\]\n\\end{lem}\n\n\n\\section{Fake solutions based on Skorokhod embeddings}\nGiven a stochastic process $X$ on a state space $E$, and a probability law $\\mu$ on $E$, \nthe Skorokhod embedding problem~\\cite{Skorokhod:65} is to find a stopping time $\\tau$ \nsuch that $X_\\tau \\sim \\mu$. In the simplest Brownian case, the problem becomes, given \nBrownian motion null at zero and a centred, square-integrable probability measure $\\mu$ \non $\\mathbb R$, to find a stopping time $\\tau$ such that $B_\\tau \\sim \\mu$ and $\\mathbb E[\\tau]$ is \nfinite (and then necessarily $\\mathbb E[\\tau]= \\int x^2 \\mu(dx)$).\n\nThere are a multiplicity of solutions of the Skorokhod embedding problem \nfor Brownian motion. Many of these solutions are based on considering \nthe bivariate process $(B_t,A_t)_{t \\geq 0}$ where $A$ is some \nincreasing additive functional, null at 0. Then the solution of the \nembedding problem is to take $\\tau \\equiv \\tau^A_\\mu = \\inf\\{t>0 : (B_t, \nA_t) \\in \\mathcal D^A_\\mu \\}$ where $\\mathcal D_\\mu$ is some appropriate domain in $\\mathbb R \n\\times \\mathbb R_+$. In particular the solutions of \\{Root~\\cite{Root:69}, \nRost~\\cite{Rost:76}, Az\\'{e}ma-Yor~\\cite{AzemaYor:79a}, \nVallois~\\cite{Vallois:83, Vallois:92}\\} are based on the choices $A_\\cdot$ \nequals \\{time, time, the maximum process, the local time\\} \nrespectively.\n\nThese constructions are suggestive of a direct construction of a \ndiscontinuous fake martingale diffusion $\\tilde{X}= (\\tilde{X}_t)_{t \\geq 0}$. \nFix an additive functional $A$ and let $\\mathcal D_t \\equiv \n\\mathcal D^A_{\\mu^X_t}$ denote the domain such that $\\tau_t \\equiv \n\\tau^A_{\\mu^X_t}$ is a solution of the Skorokhod embedding problem for \n$\\mu^X_t$ in Brownian motion. Then provided the stopping times $\\tau_t$ \nare non-decreasing in $t$, or equivalently provided the stopping domains \n$\\mathcal D_t$ are decreasing in $t$, the process $\\tilde{X}$ given by \n$\\tilde{X}_t = B_{\\tau_t}$ is a martingale with the required \ndistributions. All that remains is to check that the regions $\\mathcal D_t$ are \nindeed decreasing in $t$.\n\nIn general, the fact that a family of distributions $(\\mu^X_t)_{t \\geq 0}$ \nis increasing in convex order is not sufficient to guarantee that \nthe associated domains $\\mathcal D_t$ are decreasing in $t$, and for any \nspecific additive functional and associated construction of the solution \nto the Skorokhod embedding problem this needs to be checked.\n\nMadan and Yor~\\cite{MadanYor:02} use this approach and the Az\\'ema-Yor \nsolution to describe a discontinuous fake diffusion. The condition that \n$\\mathcal D_t$ is decreasing in $t$ is restated as the fact that the marginal \ndistributions are increasing in residual mean life order \n(equivalently, the barycentre functions are increasing in $t$). It can \nbe checked that this property holds for the lognormal family of \ndistributions so that the Madan and Yor~\\cite{MadanYor:02} construction \ngives a fake exponential Brownian motion. Similarly, the Root, R\\\"{o}st, \nand Vallois constructions all extend from the univariate case for a \nsingle marginal, to give fake exponential Brownian motions. However, in \nall cases the resulting process is discontinuous.\n\n\n\\section{Continuous Fake exponential Brownian motion} \n\nLet $X = (X_t)_{t \\geq 0}$ be the solution of an It\\^{o} stochastic \ndifferential equation, and let $\\mu^X_t \\sim \\mathcal L(X_t)$. Then \nGy\\\"{o}ngy~\\cite{Gyongy:86} showed that there is a diffusion process $Y= \n(Y_t)_{t \\geq 0}$ such that $\\mathcal L(Y_t) \\sim \\mu^X_t$. In a related \nresult, Dupire~\\cite{Dupire:97} showed that if the family of \ndistributions $(\\mu_t)_{t \\geq 0}$ is increasing in convex order, and \nif the associated call price functional $C(t,x) = \\int_{\\mathbb R}(y-x) \n\\mu_t(dy)$ is sufficiently regular (in particular, $C$ is $C^{1,2}$ and \nhas certain limiting properties for large $x$) then there is a \ntime-inhomogeneous martingale diffusion $Y$ solving $dY_t = \\eta(t,Y_t) dW_t$ such \nthat $\\mathcal L(Y_t) \\sim \\mu_t$. The coefficient $\\eta$ is given by\n\\( \\eta(t,y)^2 = 2\\frac{\\partial C(t,y)}{\\partial t} \/\n\\frac{\\partial^2 C(t,y)}{\\partial^2 y}\n. \\)\nWe call the process $Y$ associated with the family $(\\mu_t)_{t \\geq 0}$ \nthe Dupire diffusion. \n\n\nGiven a process whose univariate marginals coincide with those of \nexponential Brownian motion, both Gy\\\"ongy and Dupire show how to \nconstruct a diffusion process whose marginals are lognormal. However, in \nboth cases the resulting process is exponential Brownian motion itself.\nInstead, in this note we give a family of continuous martingales \nwhich are not exponential Brownian motion but which share the \nlognormal univariate marginal distributions of exponential Brownian \nmotion. The process we construct is a mixture of two well-chosen \nmartingales.\n\nLet $X=(X_t)_{t \\geq 0}$ be some time-homogeneous martingale diffusion \nsuch that $X_0=x_0$ and for $t>0$, $\\mathcal L(X_t) \\sim f_t(\\cdot)$. \nThe idea is to try to write $f_t(x) = c g_t(x) + (1-c)h_t(x)$ \nfor a constant $c \\in (0,1)$ and a pair of families \n$\\{(g_t)_{t > 0}, (h_t)_{t > 0}\\}$ such that for each $t$, $g_t$ \nand $h_t$ are the densities of random variables with mean $x_0$, and such that \nif $\\mu^G_t$ (respectively $\\mu^H_t$) is the law of a random variable \nwith density $g_t$ (respectively $h_t$), and if $\\mu^G_0 = \\delta_{x_0} = \\mu^H_0$, \nthen \nthe family $(\\mu^G_t)_{t \n\\geq 0}$ (respectively $(\\mu^H_t)_{t \\geq 0}$) is increasing in convex \norder. Then, if $G=(G_t)_{t \\geq 0}$ (respectively $H=(H_t)_{t \\geq 0}$) \nis the Dupire diffusion associated with\nthe family of laws $(\\mu^G_t)_{t \\geq 0}$ (respectively \n$(\\mu^H_t)_{t \\geq 0}$) and if $\\tilde{X} = (\\tilde{X})_{t \\geq 0}$ is defined by\n\\( \\tilde{X}_t = G_t I_{ \\{ Z^c = 1 \\} } + H_t I_{ \\{ Z^c=0 \\} } \\)\nthen $\\tilde{X}$ is a fake version of $X$. Here $Z^c$ is a Bernoulli \nrandom variable which is independent of $G$ and $H$, is known at time 0, \nand is such that $\\mathbb P(Z^c=1)=c$, where $c \\in (0,1)$.\n\nIt remains to show that we can construct the families $\\{(g_t)_{t > 0},(h_t)_{t \n> 0} \\}$ \nand that the resulting process is not $X$ itself. For this we need \nthe family $(\\mu^G_t)_{t \\geq 0}$ to be increasing (in convex order), \nbut not too \nquickly as that will mean there is no `room' for the family \n$(\\mu^H_t)_{t \\geq 0}$ \nto be increasing. \n\nThe final simplifying idea is to suppose that $G$ is a time change of \n$X$, $G_t = X_{a(t)}$. By choosing $a$ to be an increasing, \ndifferentiable process we automatically get the existence \nof $G$. If further we require that $\\dot{a}<1$ we can hope that the \nfamily $(\\mu^H_t)_{t \\geq 0}$ is also increasing in convex order. \n\n\\begin{prop}\n\\label{prop:convexorder}\nSuppose $a(t)$ is a strictly increasing, twice continuously differentiable \nfunction, null at \n0, with\n$\\dot{a}(t) < 1$ for $t>0$, and define \n\\[ K = \\inf_{t > 0} \\inf_{y \\in I} \n\\frac{f_t(y)}{f_{a(t)}(y)}. \\]\nSuppose $K>0$. Fix $c \\in (0,K)$\nand for $t > 0$ set \n\\[ h_t(y)= \\frac{1}{1-c} \\left\\{ f_t(y) - cf_{a(t)}(y) \\right\\}. \n\\]\nThen $(h_t)_{t > 0}$ is a family of densities which is \nincreasing in convex order.\n\nMoreover, there exists a martingale diffusion $H$ such that $\\mathcal L(H_t) \\sim h_t(\\cdot)$.\n\\end{prop}\n\n\\begin{proof}\nFrom the definition of $c$ we have that $h_t$ is non-negative and it is\nimmediate that $h_t(y)$ integrates to one so that $h_t$ is the density\nof a continuous random variable. Further, since $f_t$ corresponds to a\nmean $x_0$ random variable, so does $h_t$.\n\nLet $(\\tilde{H}_t)_{t > 0}$ be a family of random variables such that $\\mathcal L(\\tilde{H}_t) \n\\sim h_t( \\cdot)$.\nBy Lemma~\\ref{lem:mgdiff} and the hypotheses of the proposition,\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t} \\mathbb E[(\\tilde{H}_t - x)^+] & = & \n\\frac{1}{1-c} \\frac{\\partial}{\\partial t} \\mathbb E \\left[(X_t - x)^+ - c(X_{a(t)} - x)^+ \\right]\n\\label{eqn:Cdot}\n\\\\\n& = & \\frac{1}{1-c} \\sigma(x)^2 \\left( f_t(x) - c \\dot{a}(t) f_{a(t)}(x) \\right)\n> 0 . \\nonumber \n\\end{eqnarray} \nHence the family \n$(h_t)_{t > 0}$ is a family of densities which is\nincreasing in convex order.\n\nDefine\n\\begin{equation} \n\\label{eqn:etadef}\n\\eta(t,y)^2 = \\sigma(y)^2 \\frac{ f_t(y) - c \\dot{a}(t)\nf_{a(t)}(y)}{f_t(y) - c\nf_{a(t)}(y) } .\n\\end{equation}\nand let $H$ be a weak solution of $dH_t = \\eta(t,H_t) dB_t$ subject to $H_0=x_0$. By our \nassumptions the \nsolution to this SDE is unique in law. Moreover we have\n\\[ 1 \\leq \\frac{\\eta(t,y)^2}{\\sigma(y)^2}\n< \\frac{f_t(y)}{f_t(y) - cf_{a(t)}(y) } \\leq L^2 \\]\nwhere $L^2 = K\/(K-c)$.\nFurther, $X^L$ given by $X^L_t = X_{L^2 t}$ is a\nmartingale and\nsolves $dX^L_{t} =\n\\sigma_L(X^L_{t}) dB_t$ where $\\sigma_L(x) = L \\sigma(x)$.\nThen, $\\eta(t,y) \\leq \\sigma_L(y)$ and by Theorem~3 of Hajek~\\cite{Hajek:85}, we can\nfind an increasing time-change $\\Gamma$ with $\\Gamma_t \\leq t$ such that\n$\\hat{H}_t = X^L_{\\Gamma_t}$ solves $d\\hat{H}_t = \\eta(t,\\hat{H}_t) dB_t$. \nHence, by the uniqueness in law of solutions to this equation, $H$ is a martingale.\n\nSuppose that $X$ is non-negative (the case where $X$ is bounded above or below reduces \nto this case after a reflection and\/or a shift.)\nDefine $C_h:[0,\\infty) \\times [0,\\infty) \\mapsto \\mathbb R$ via $C_h(0,y)=(x_0-y)^+$ and for \n$t > 0$, \n$C_h(t,y) = \\int (z-y)^+ h_t(z) dz$. \nThen, using (\\ref{eqn:Cdot}) we deduce that $C_h$ solves Dupire's equation\n\\begin{equation}\n\\label{eqn:dupire}\n \\frac{1}{2} \\eta(t,y)^2 C''(t,y) = \\dot{C}(t,y) \n\\end{equation}\nsubject to\n\\begin{equation} \n\\label{eqn:bcR+}\n \\mbox{$C(0,y) = (x_0 - y)^+$, $C(0,t) = x_0$, $0 \\leq C \\leq x_0$.}\n\\end{equation} \nIndeed, by an argument similar to that in \nEkstr\\\"{o}m and\nTysk~\\cite[Step 5 of the proof of Theorem 2.2]{EkstromTysk:12} $C_h$ is the unique \nsolution of (\\ref{eqn:dupire}) satisfying (\\ref{eqn:bcR+}). \n\nHowever, by the results of Dupire~\\cite{Dupire:97} and \nKlebaner~\\cite{Klebaner:02} $C_H(t,y) = \n\\mathbb E[(H_t-y)^+]$ also solves (\\ref{eqn:dupire}) and (\\ref{eqn:bcR+}) and \nhence, $\\mathcal L(H_t) \\sim h_t(\\cdot)$.\n\nNow consider the case where $X$ the range of $X$ is the whole real line. Let $\\eta$ and \n$H$ be as before. Fix $\\overline{T} \\in (0,\\infty)$.\nLet $C_h: [0,\\overline{T}] \\times \\mathbb R \\mapsto \\mathbb R$ be the \nsolution of (\\ref{eqn:dupire}) subject to\n\\begin{equation}\n\\label{eqn:bcR} \n \\mbox{$C(0,y) = (x_0 - y)^+$, $\\sup_{t \\leq \\overline{T}} \\lim_{y \\uparrow \\pm \\infty} \nC_t(t,y) - (x_0-y)^+ = 0$, $(x_0-y)^+ \\leq C_h(t,y) \\leq (x_0 - y)^+ + J$, }\n\\end{equation}\nwhere $J = J(\\overline{T}) = \\frac{\\mathbb E[(X_{L^2 \\overline{T}} - x_0)^+]}{1-c}$.\nBy a small\nmodification of the arguments of Ekstr\\\"{o}m and\nTysk~\\cite{EkstromTysk:12} to allow for the fact that we are working with real-valued \nprocesses, it follows that the solution to this equation is unique.\nBut $C_H(t,y) =\n\\mathbb E[(H_t-y)^+]$ also solves (\\ref{eqn:dupire}) and $(x_0 - y)^+ \\leq C_H(t,y) \\leq \n\\mathbb E[(X_{L^2 t} - y)^+]$ and hence $\\mathcal L(H_t) \\sim h_t$ for $t \\leq \\overline{T}$. Since \n$\\overline{T}$ is arbitrary, the result follows for all $t$.\n\\end{proof}\n\n\\begin{thm}\nLet $G$ be given by $G_t = X_{a(t)}$ \nand let $H$ \nbe the Dupire diffusion associated with the \nfamily of densities $(h_t)_{t > 0}$. Then\n\\[ \\tilde{X}_t = G_t I_{ \\{ Z^c = 1 \\} } + H_t I_{ \\{ Z^c=0 \\} } \\]\nis a fake version of $X$.\n\\end{thm}\n\n\\begin{proof}\nIf $\\tilde{f}_t(x)$ denotes the time-$t$ density of $\\tilde{X}$ we have that\n\\[ \\tilde{f}_t(x) = c g_{t}(x) + (1-c) h_t(x) =c f_{a(t)}(x) + (1-c) h_t(x) = f_t(x). \n\\]\nand $X_t$ has the same distribution as $X$. \n\nSince $G$ and $H$ are martingales, so is $\\tilde{X}$.\nBut the quadratic variations of $X$ and $\\tilde{X}$ are different and hence\n$\\tilde{X}$ is a fake version of $X$.\n\\end{proof}\n\n\n\\begin{eg}[Fake Brownian motion]\nIn this case \n\\[ \\frac{f_t(y)}{f_{a(t)}(y)} = \\sqrt{ \\frac{a(t)}{t} } e^{y^2\/2 [ \n1\/a(t)-1\/t]} \\geq \\sqrt{ \\frac{a(t)}{t} }. \\]\nFix $c \\in (0,1)$ and choose $a(t)\/t > c^2$ with $\\dot{a}(t) \\leq 1$, \nfor \nexample ${a}(t)=K^2 t$ for some $K \\in (c,1)$.\n\\end{eg}\n\n\\begin{eg}[Fake exponential Brownian motion]\nRecall the density given in (\\ref{eqn:ebmdensity}). Then\n\\[ \\min_{x>0} \\frac{f_t(x)}{f_{a(t)}(x)} = \n\\min_{x>0} \\sqrt{ \\frac{a(t)}{t} } e^{[a(t)-t]\/8} e^{(\\ln x)^2[ \n1\/a(t)-1\/t]\/2} = \\sqrt{ \\frac{a(t)}{t} } e^{[a(t)-t]\/8} = \n\\frac{\\psi(a(t))}{\\psi(t)} \\]\nwhere $\\psi$ is the increasing function $\\psi(t) = \\sqrt{t}e^{t\/8}$. \nFix $K \\in (0,1)$ \nand set $a(t)= \\psi^{-1}(K \\psi(t))$. Then $a(t)0$.\n\\end{eg}\n\n\n\\begin{rem}\nThe argument can be extended to cover the case where the original \nprocess $X$ is a time-inhomogeneous martingale diffusion $dX_t \n= \\sigma(t,X_t) dB_t$ provided $k>0$ where\n\\[ k := \\inf_{t > 0} \\inf_{y \\in I} \\left\\{ \\frac{f_t(y)}{f_{a(t)}(y)} \\wedge \n\\frac{f_t(y) \\sigma(t,y)^2}{f_{a(t)}(y) \\sigma(a(t),y)^2} \\right\\} .\n\\]\n\\end{rem}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}