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+η-pairing on bipartite and non-bipartite lattices
+Yutaro Misu1, Shun Tamura2, Yukio Tanaka2 and Shintaro Hoshino1
+1Department of Physics, Saitama University, Saitama 338-8570, Japan
+2Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
+(Dated: January 23, 2023)
+The η-pairing is a type of Cooper pairing state in which the phase of the superconducting order
+parameter is aligned in a staggered manner, in contrast to the usual BCS superconductors with a
+spatially uniform phase. In this study, we search for a characteristic η-pairing state in a triangular
+lattice where a simple staggered alignment of the phase is not possible. As an example, we consider
+the attractive Hubbard model on both the square and triangular lattices under strong external
+Zeeman field.
+Using the mean-field approximation, we have identified several η-pairing states.
+Additionally, we have examined the electromagnetic stability of the pairing state by calculating the
+Meissner kernel. Odd-frequency pairing plays a crucial role in achieving diamagnetic response if the
+electrons experience a staggered superconducting phase during the propagation of current.
+I.
+INTRODUCTION
+The diversity of superconducting phenomena has been
+attracting continued attention.
+The superconducting
+state of matter is characterized by the properties of
+Cooper pairs, which can be classified based on their
+space-time and spin structures.
+With regard to their
+space structure, Cooper pairs are typically classified as
+s-wave, p-wave, or d-wave pairs depending on their rel-
+ative coordinate structure. As for their center-of-mass
+coordinate, while it is usually assumed to be zero in
+most superconductors, it is possible to consider the exis-
+tence of a finite center-of-mass momentum. One example
+of this is the Flude-Ferrell-Larkin-Ovchinnikov (FFLO)
+state [1, 2], in which the Cooper pair has a small but finite
+center-of-mass momentum under the influence of a mag-
+netic field. More generally, the magnitude of the center-
+of-mass momentum can be larger and of the order of the
+reciprocal lattice vector ∼ π/a, where a is a lattice con-
+stant. This type of pairing state is known as η-pairing,
+a concept first proposed by C. N. Yang, which forms a
+staggered alignment of the superconducting phase on a
+bipartite lattice [3]. The spatially modulating order pa-
+rameter is known also as the pair density wave, and has
+been discussed in relation to cuprate superconductors [4].
+The actual realization of the η-pairing has been pro-
+posed for the correlated electron systems such as the at-
+tractive Hubbard (AH) model with the magnetic field
+[5], the single- and two-channel Kondo lattices [6, 7], the
+Penson-Kolb model [8], and also the non-equilibrium sit-
+uation [9–14].
+Since the phase of the superconducting
+order parameter can be regarded as the XY spin, the η-
+pairing is analogous to an antiferromagnetic state of the
+XY spin model.
+Hence, the η-pairing state should be
+strongly dependent on the underlying lattice structure
+and we naively expect a variety of the η-pairing state
+if we consider the geometrically frustrated lattice such
+as the triangular lattice since the simple staggered state
+cannot be realized.
+In this paper, we deal with the AH model on the non-
+bipartite lattice in order to search for possible new su-
+perconducting states depending on the feature of the
+non-bipartite lattice structure in equilibrium.
+Already
+in the normal state without superconductivity, it has
+been pointed out that the non-bipartite lattice generates
+a non-trivial state of matter. For example in the Kondo
+lattice, a partial-Kondo-screening, which has a coexisting
+feature of Kondo spin-singlet and antiferromagnetism, is
+realized [15]. Also in the AH model at half-filling, charge-
+density-wave (CDW) is suppressed due to the frustration
+effect [16]. The η-pairing that appears in a photodoped
+Hubbard model on the triangular lattice has been studied
+recently [14]. In the equilibrium situation, the properties
+of the AH model have been studied on bipartite lattices
+[5], but the model on a non-bipartite lattice has not been
+explored.
+As shown in the rest of this paper, there are several
+types of η-pairings on the triangular lattice of the AH
+model under the Zeeman field.
+One of the η-pairing
+states is regarded as a 120◦-N´eel state.
+Since the rel-
+ative phase between the nearest neighbor sites is neither
+parallel nor anti-parallel, the inter-atomic Josephson cur-
+rent is spontaneously generated. This state can also be
+regarded as a staggered flux state, where the flux is cre-
+ated by the atomic-scale superconducting loop current.
+While the staggered flux state has been studied so far
+[17–23], the staggered flux in this paper is induced by
+the Josephson effect associated with superconductivity
+and has a different origin.
+For the analysis of the AH model, we employ the mean-
+field approximation in this paper. It has been suggested
+that a simple η-pairing shows a paramagnetic Meissner
+state [24]. Hence it is necessary to investigate the electro-
+magnetic stability of the solution for superconductivity.
+We evaluate the Meissner kernel whose sign corresponds
+to the diamagnetic (minus) or paramagnetic (plus) re-
+sponse of the whole system, where the physically sta-
+ble state should show diamagnetism. We confirm that
+if the mean-field η-pairing state has the lowest energy
+compared to the other ordered states, the calculation of
+the Meissner kernel shows the diamagnetic response. It
+is also notable that the odd-frequency pairing amplitude,
+which has an odd functional form with respect to the fre-
+quency [6, 25–30], can contribute to the diamagnetism in
+arXiv:2301.08426v1  [cond-mat.supr-con]  20 Jan 2023
+
+2
+the η-pairing state. This is in contrast to the usual super-
+conductivity with the uniform phase where the conven-
+tional even-frequency pairing contributes to the diamag-
+netism. It has been shown that the odd-frequency pairing
+induced at the edge, interface or junctions [31–36] shows
+a paramagnetic response [37–41]. In this paper, by con-
+trast, we consider the odd-frequency pairing realized in
+bulk, which shows a qualitatively different behavior.
+This paper is organized as follows.
+We explain the
+model and method for the AH model in Sec. II, and the
+Meissner kernel in Sec. III. The numerical results for the
+AH model are shown in Sec. IV, and we summarize the
+paper in Sec. V.
+II.
+ATTRACTIVE HUBBARD MODEL
+A.
+Hamiltonian
+We consider the Hamiltonian of the AH model with
+magnetic field h which induce Zeeman effect only (Zee-
+man field) :
+H = −t
+�
+⟨i,j⟩σ
+c†
+iσcjσ + H.c. + U
+�
+i
+ni↑ni↓
+− µ
+�
+i
+ni − h ·
+�
+i
+si,
+(1)
+where c†
+iσ and ciσ are the creation and annihilation op-
+erators of the i-th site with spin σ, respectively.
+The
+symbol ⟨i, j⟩ represents a pair of the nearest-neighbor
+sites.
+Here, the parameter t is the nearest-neighbor
+single-electron hopping integral. U (= −|U|) is the on-
+site attractive interaction. The spin operator is defined
+as si =
+1
+2
+�
+σσ′ c†
+iστσσ′ciσ′, where τ is the Pauli ma-
+trix, and the number operator of electrons is denoted as
+ni = ni↑ + ni↓ = �
+σ c†
+iσciσ. The electron concentration
+is controlled by adjusting the chemical potential µ.
+The AH model has been successfully used to elucidate
+several important and fundamental issues in supercon-
+ductors [42]. The model on a bipartite lattice at half fill-
+ing is theoretically mapped onto the repulsive Hubbard
+model by the following partial particle-hole transforma-
+tion [43]
+c†
+i↑ → c†
+i↑, c†
+i↓ → ci↓eiQ·Ri.
+(2)
+The reciprocal vector Q satisfies the condition eiQ·Ri =
+(−1)i that takes ±1 depending on Ri belonging to A or
+B sublattice on the bipartite lattice. Then, the η-pairing
+appears in the region that corresponds to a ferromagnet
+with transverse magnetization in the repulsive model [5].
+In a mean-field theory, the phase diagram for the repul-
+sive Hubbard model without the magnetic field is shown
+in the left panel of Fig. 1 [44]. From this figure, we find
+that the ferromagnet is located in the regime where the
+repulsive interaction U > 0 is large and the electron con-
+centration is not half-filled. Hence, the η-pairing phase
+nc
+t
+|U|
+m
+0
+1
+0
+1
+PM
+AFM
+FM
+FF
+BCS
+-pairing
+η
+Repulsive Hubbard (
+)
+U > 0
+Attractive Hubbard (
+)
+U < 0
+h = 0
+nc = 1.0
+Spin-polarized 
+normal state
+FIG. 1.
+Sketches of the phase diagrams for the repulsive
+Hubbard model [44] (left panel) and AH model (right panel).
+nc is the electron concentration and m is the magnetization.
+When the interaction |U| is large, the ground state in the re-
+pulsive Hubbard model is ferromagnet (FM), while the ground
+state in the AH model is η-pairing.
+is located in the regime where the attractive interaction
+U < 0 is large and the magnetization is finite. The phase
+diagram of the AH model at half filling is shown in the
+right panel of Fig. 1. In principle, an attractive interac-
+tion large enough to realize η-pairing could be realized in
+artificial cold atom systems [45].
+The Cooper pair is formed by the two electrons
+with (k ↑,
+− k + q ↓) where q is the center-of-mass
+momentum. The FFLO state and the η-pairing are dis-
+tinguished by the magnitude of |q|.
+In η-pairing, the
+center-of-mass momentum of the Cooper pair is the or-
+der of the reciprocal lattice vector, while the momentum
+of the FFLO state is much smaller and the spatial mod-
+ulation is slowly-varying compared to the atomic scale.
+Although the large center-of-mass momentum is usually
+not energetically favorable, a strong attractive interac-
+tion can make it stable.
+B.
+Mean-field theory
+By applying the mean-field approximation, we obtain
+the mean-field Hamiltonian
+HMF = −t
+�
+⟨i,j⟩σ
+c†
+iσcjσ + H.c. − µ
+�
+i
+ni − h ·
+�
+i
+si
+−
+�
+i
+�
+vini + Hi · si − ∆ic†
+i↑c†
+i↓ − ���∗
+i ci↓ci↑
+�
+.
+(3)
+
+3
+The order parameters are given by the self-consistent
+equations
+vi ≡ |U|
+2 ⟨ni⟩,
+(4)
+∆i ≡ −|U|⟨ci↓ci↑⟩,
+(5)
+mi = 1
+2
+�
+σσ′
+⟨c†
+iστσσ′ciσ′⟩,
+Hi =
+− 2|U|mi,
+(6)
+where ⟨A⟩ = Tr
+�
+Ae−HMF/T �
+/Tr
+�
+e−HMF/T �
+is a quan-
+tum statistical average with the mean-field Hamiltonian
+and T is temperature.
+∆i is the order parameter for
+s-wave singlet superconductivity (pair potential).
+The
+phase θi ∈ [0, 2π) of the pair potential ∆i = |∆i|eiθi is
+dependent on the site index and will be represented by
+the arrow in a two-dimensional space. The mean-fields
+for the charge and spin are given by vi and Hi, respec-
+tively, at each site. The derivation of the self-consistent
+equations is summarized in Appendix A. We will consider
+the AH model both on the two-dimensional square and
+triangular lattices.
+III.
+MEISSNER KERNEL FOR A GENERAL
+TIGHT-BINDING LATTICE
+A.
+Definition
+As we explained in Sec. I, it is necessary to calculate
+the Meissner kernel to determine whether the mean-field
+solution for η-pairing is electromagnetically stable. In the
+tight-binding model, the electromagnetic field appears as
+Peierls phase:
+Hkin = −t
+�
+⟨i,j⟩σ
+eiAijc†
+iσcjσ + H.c..
+(7)
+The Meissner effect is examined by the weak external or-
+bital magnetic field applied perpendicular to the plane,
+while the η-pairing is stabilized only under a strong Zee-
+man field. In order to make these compatible, we apply
+the Zeeman field parallel to the plane h = (h, 0, 0), which
+does not create the orbital motion of the tight-binding
+electrons.
+Thus, the weak magnetic field that triggers
+the Meissner effect is applied perpendicular to the plane
+in addition to the in-plane magnetic field.
+While the
+out-of-plane Zeeman effect is also induced by the weak
+additional field, it is neglected since the dominant Zee-
+man field already exists by the strong in-plane magnetic
+field.
+Let us formulate the Meissner response kernel on a
+general tight-binding model. We apply the formulation in
+Refs. [46–48] to the present case with sublattice degrees
+of freedom. The current density operator between two
+sites is defined as
+jij = ∂Hkin
+∂Aij
+ˆδij
+= −it
+�
+σ
+�
+c†
+iσcjσeiAij − c†
+jσciσe−iAij�
+ˆδij,
+(8)
+where δij = Ri − Rj is the inter-site lattice vector be-
+tween i-th and j-th sites, and hat (ˆ) symbol means a unit
+vector. In the linear response theory, the current oper-
+ator which appears as a response to the static magnetic
+field in equilibrium is written as
+jij ≃ −it
+�
+σ
+(c†
+iσcjσ − c†
+jσciσ)ˆδij
++ t
+�
+σ
+(c†
+iσcjσ + c†
+jσciσ)ˆδijAij
+≡ jpara
+ij
++ jdia
+ij .
+(9)
+The first term is called the paramagnetic term and the
+second term is diamagnetic.
+The Fourier-transformed
+paramagnetic and diamagnetic current density operators
+are written as jpara(q) and jdia(q). The linear response
+kernel is then defined by ⟨jν(q)⟩ = �
+µ Kνµ(q)Aµ(q),
+where ν, µ = x, y is the direction. We evaluate the ker-
+nel Kνµ(q → 0) ≡ Kνµ when investigating the stability
+of superconductivity. This is called the Meissner kernel,
+which is proportional to the superfluid density.
+The Meissner kernel is separated into paramagnetic
+and diamagnetic terms as Kνµ = (Kpara)νµ + (Kdia)νµ.
+The paramagnetic kernel is given by
+(Kpara)νµ = 1
+N
+� 1/T
+0
+dτ⟨jpara
+ν
+(q = 0, τ)jpara
+µ
+(q = 0)⟩,
+(10)
+where N = �
+i 1 is the number of sites. The Heisenberg
+representation with the imaginary time τ is defined as
+A(τ) = eHτAe−Hτ. The form of the diamagnetic kernel
+is obvious from Eq. (9).
+We note that if the sign of the Meissner kernel K is
+negative, the superconducting state is electromagneti-
+cally stable and is also called a diamagnetic Meissner
+state, which expels magnetic flux. On the other hand, if
+the sign is positive, the superconducting state is called
+the paramagnetic Meissner state, which attracts mag-
+netic flux. For a stable thermodynamic superconducting
+state, the negative value of K is required.
+B.
+Method of evaluation
+The actual evaluation of the kernels is performed based
+on the wave-vector representation.
+Here, the physical
+quantities are described by the operator cα
+kσ where α dis-
+tinguishes the sublattice. Note that the Brillouin zone is
+
+4
+folded by �
+α 1 times. The diamagnetic kernel is rewrit-
+ten as
+(Kdia)νµ = 1
+N
+�
+α,β
+�
+kσ
+�
+m−1
+kαβ
+�
+νµ ⟨cα†
+kσcβ
+kσ⟩.
+(11)
+The inverse mass tensor m−1
+kαβ, which reflects the char-
+acteristics of the lattice shape, are given by
+�
+m−1
+kαβ
+�
+νµ ≡ t
+�
+⟨iα,jβ⟩
+�
+ˆδiαjβ
+�
+ν
+�
+ˆδiαjβ
+�
+µ e−ik·Riαjβ ,
+(12)
+where iα is the i-th unit cell with sublattice α.
+The
+symbol ⟨iα, jβ⟩ represents a pair of the nearest-neighbor
+sites and Riαjβ is the vector between the unit lattice with
+the i-th sublattice α and the unit lattice with the j-th
+sublattice β.
+The paramagnetic term has the form of a current-
+current correlation function. We can calculate this term
+by using the Green’s function matrix
+ˇGk(τ) ≡ −⟨Tτψk(τ)ψ†
+k⟩
+(13)
+where ψk = (cα
+k↑, cα†
+−k↓, · · · )T is the Nambu-spinor. Tτ is
+time-ordering operator regrading τ. Each component of
+the Green’s function matrix is given by the diagonal and
+off-diagonal Green’s functions:
+Gαβ
+σσ′(k, τ) ≡ −⟨Tτcα
+kσ(τ)cβ†
+kσ′⟩,
+(14)
+¯Gαβ
+σσ′(k, τ) ≡ −⟨Tτcα†
+kσ(τ)cβ
+k′σ′⟩,
+(15)
+F αβ
+σσ′(k, τ) ≡ −⟨Tτcα
+kσ(τ)cβ
+−kσ′⟩,
+(16)
+F αβ†
+σσ′ (k, τ) ≡ −⟨Tτcα†
+−kσ(τ)cβ†
+kσ′⟩.
+(17)
+The anomalous part of Green’s function [Eq. (16)] is also
+called the pair amplitude. The paramagnetic kernel in
+Eq. (10) can be divided into the normal (G) and anoma-
+lous (F) Green’s function contributions as
+(Kpara)νµ = − 1
+N
+� � 1/T
+0
+dτ (vkαβ)ν · (vkα′β′)µ ×
+�
+¯Gαβ′
+σσ′(k, τ)Gα′β
+σσ′(k, τ) + ¯Gαβ′
+σσ′(−k, τ)Gα′β
+σσ′(−k, τ)
+�
+− 1
+N
+� � 1/T
+0
+dτ (vkαβ)ν · (v−kα′β′)µ ×
+�
+F βα†
+σ′σ (k, −τ)F α′β′
+σ,σ′ (k, τ) + F βα†
+σ′σ (−k, −τ)F α′β′
+σ,σ′ (−k, τ)
+�
+≡ KG
+para + KF
+para.
+(18)
+The summation � is performed over the indices which appears only in the right-hand side. The velocity vector vkαβ
+is defined by
+(vkαβ)ν ≡ t
+�
+⟨iα,jβ⟩
+�
+ˆδiαjβ
+�
+ν e−ik·Riαjβ .
+(19)
+In order to perform the integral with respect to τ in Eq. (18), we define the Fourier-transformed Green’s function as
+gk(iωn) ≡
+� 1/T
+0
+dτgk(τ)eiωnτ,
+(20)
+where gk represents one of Eqs. (14)-(17) and ωn = (2n + 1)πT is fermionic Mastubara frequency. Moreover, the
+Fourier-transformed Green’s function matrix is given by using the matrix representation of mean-field Hamiltonian
+Eq. (3) as
+ˇGk(iωn) =
+�
+iωnˇ1 − ˇHMF
+k
+�−1 = ˇUk
+�
+iωnˇ1 − ˇΛk
+�−1 ˇU †
+k,
+(21)
+where ˇΛk and ˇUk are, respectively, a diagonal eigenvalue matrix and a unitary matrix satisfying ˇU † ˇHMF
+k
+ˇU = ˇΛk =
+diag(λk1, λk2, . . .). From Eq. (21), Kpara can be calculated as
+(Kpara)νµ = − 1
+N
+� �
+(vkαβ)ν · (vkα′β′)µ Uβ′σ′,ασ
+kp
+Uα′σ,βσ′
+kp′
++ (vkαβ)ν · (v−kα′β′)µ Uβσ′,ασ
+kp
+Uα′σ,β′σ′
+kp′
+� f (λkp) − f (λkp′)
+λkp − λkp′
++ c.c.
+(22)
+where f(λkp) =
+1
+eλkp/T +1 is the Fermi-Dirac distribution function and we have defined the coefficient Uασ,βσ′
+kp
+≡
+� ˇUk
+�
+ασ,p
+�
+ˇU †
+k
+�
+p,βσ′.
+The anomalous part of Eq. (18) KF
+para is further de-
+composed into the contributions KEFP and KOFP from
+
+5
+the even-frequency pair (EFP) and odd-frequency pair
+(OFP) amplitudes defined by
+F EFP(k, iωn) ≡ F(k, iωn) + F(k, −iωn)
+2
+,
+(23)
+F OFP(k, iωn) ≡ F(k, iωn) − F(k, −iωn)
+2
+.
+(24)
+Then, we obtain KEFP and KOFP by using Eqs. (23) and
+(24) as
+KEFP,OFP
+νµ
+= − 1
+2N
+�
+k
+�
+αβα′β′
+(vkαβ)ν · (v−kα′β′)µ
+×
+�
+σσ′
+�
+pp′
+Uβσ′,ασ
+kp
+Uα′σ,βσ′
+kp′
+×
+�f (λkp) − f (λkp′)
+λkp − λkp′
+∓ f (λkp) − f (−λkp′)
+λkp + λkp′
+�
++ c.c.,
+(25)
+where the minus (−) sign in the square bracket is taken
+for EFP contribution and the plus (+) for OFP pairing.
+These quantities are numerically calculated as shown in
+the next section. Note that the cross term of the EFP
+and OFP terms of Green’s functions vanishes after the
+summation with respect to the Matsubara frequency.
+C.
+Paramagnetic Meissner response of a simple
+η-pairing state
+Before we show the results of the AH model, let us show
+that a simple η-pairing state leads to the paramagnetic
+response which would not arise from thermodynamically
+stable states [24, 49]. We consider the simple bipartite
+lattice with staggered ordering vector Q. The anomalous
+contribution to the Meissner kernel may be written as [49]
+KF
+para,xx = −T
+�
+nkk′σσ′
+vx
+kvx
+k′F ∗
+σ′σ(k′, k, iωn)Fσσ′(k, k′, iωn).
+(26)
+This contribution must be negative (diamagnetic re-
+sponse) in order to dominate over the paramagnetic con-
+tribution. For a purely η-pairing state, we assume the
+relation Fσσ′(k, k′) = Fσσ′(k)δk′,−k−Q, and obtain
+KF
+para,xx = −T
+�
+nkσσ′
+(vx
+k)2F ∗
+σ′σ(k, iωn)Fσσ′(k, iωn), (27)
+where we have used vx
+−k−Q = vx
+k valid for square lat-
+tice, which is in contrast to the relation vx
+−k = −vx
+k
+for the uniform pairing with additional minus sign [24].
+We separate the spin-singlet and triplet parts as Fσσ′ =
+Fsiτ y
+σσ′ + Ft · (τiτ y)σσ′, and then obtain
+KF
+para,xx = 2T
+�
+nk
+(vx
+k)2�
+|Fs(k, iωn)|2 − |Ft(k, iωn)|2�
+.
+(28)
+If we consider the simple η-pairing with only spin-singlet
+part (Ft = 0), it leads to the paramagnetic response
+(positive).
+Thus, a simple s-wave spin-singlet η-pairing is unlikely
+realized as a stable state. On the other hand, in the AH
+model with magnetic field, the spin-triplet pair contribu-
+tion is substantially generated by the Zeeman field, which
+plays an important role for the diamagnetic response as
+shown below.
+IV.
+NUMERICAL RESULT FOR AH MODEL
+A.
+Square lattice
+1.
+Prerequisites
+Let us begin with the analysis of the AH model on
+the square lattice. We consider the two-sublattice struc-
+ture to describe the staggered ordered phase such as a
+η-pairing. While the superconducting states in the at-
+tractive model are interpreted in terms of the magnetic
+phases of the repulsive model by the particle-hole trans-
+formation in Eq. (2), the response functions such as the
+Meissner kernel are specific to the attractive model and
+have not been explored.
+In the following, we choose the band width W = 1
+as the unit of energy.
+We fix the value of the attrac-
+tive interaction U = −1.375. The electron concentration
+is fixed as nc = 1, and the temperature is taken to be
+T = 1.0 × 10−3 unless otherwise specified. We will in-
+vestigate the change of the Meissner kernel for η-pairing
+as a function of magnetic field strength h = |h|. In this
+paper, the mean-field solutions are calculated using the
+60 × 60 mesh in k-space. The result of the Meissner ker-
+nel for η-pairings is calculated with the mesh 300 × 300.
+We also checked that the behaviors remain qualitatively
+unchanged when these numbers are increased. The self-
+consistent equations in Eqs. (4)-(6) are computed by
+using an iterative method.
+In the following subsec-
+tion IV A 2, we restrict ourselves to the analysis of two-
+sublattice mean-field solutions, and in IV A 3, we exam-
+ine the solutions when the two-sublattice constraint is
+relaxed.
+2.
+Two-sublattice solution
+Before investigating the electromagnetic stability, we
+clarify the regime where the η-pairing becomes the
+ground state. In this paper, we assume that the inter-
+nal energy in Eq. (1) is approximately equal to the free
+energy in the low temperature region. The upper panel
+of Fig. 2 shows the internal energy of several ordered
+states measured from the normal-state energy as a func-
+tion of the Zeeman field h. Here, the η-pairing solution
+is obtained by solving the self-consistent equation with
+imposing the constraint of the staggered phase of the pair
+
+6
+0.0 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.25 2.5
+h
+−3.0
+−2.5
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+Ei − Enormal
+BCS
+CDW
+Normal
+η-pairing
+0.0 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.25 2.5
+0.00
+0.04
+0.08
+D0
+FIG. 2.
+(Upper panel) Magnetic-field dependence of the
+internal energy for each state measured from the normal state
+in the square lattice model. (Lower plane) Density of state
+(DOS) at zero energy D0 for each state.
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+1.5
+ω
+0.0
+0.1
+0.2
+D(ω)
+h = 1.25
+h = 1.375
+h = 1.5
+FIG. 3. Density of states for the η-pairing around magnetic
+filed h = 1.375 in the square lattice model.
+Here D(ω) is
+normalized as
+�
+dωD(ω) = 1.
+amplitude. A constraint is also used for the calculation
+of the other types of order parameters. Our calculations
+have not found any ordered states other than the types
+shown in Fig. 2 even when a random initial condition is
+employed.
+We determine the thermodynamically stable ground
+state by comparing the internal energies. In low magnetic
+fields, BCS and CDW are degenerated ground states. On
+the other hand, we find that the η-pairing becomes the
+ground state in the magnetic field located in 1.063 < h <
+1.875. The η-pairing solution itself is found in the wider
+regime although the internal energy is not the lowest one.
+It has been known that the attractive Hubbard model
+under a magnetic field also shows the FFLO state [50],
+but this possibility cannot be considered when we take
+the two-sublattice condition. This point will be revisited
+in the next subsection where the two-sublattice condition
+is relaxed.
+The lower panel of Fig. 2 shows the density of
+state (DOS) at the Fermi level for each state. The re-
+sult indicates that there is no energy gap in the η-pairing
+state, in contrast to the conventional BCS pairing state.
+There exists the regime where the DOS at the Fermi
+level for η-pairing is larger than that of normal metal
+(1.25 ≲ h ≲ 1.5). This is due to the van-Hove singular-
+ity of the square lattice model as shown in FIG. 3. We
+also perform the calculation for the cubic lattice where
+the van-Hove singularity is absent at zero energy and con-
+firm in this case that the DOS is smaller than the normal
+state (see Appendix B).
+The stability of the η-pairing depends upon the mag-
+nitude of the magnetic field as seen in the Meissner re-
+sponse kernel K (= Kxx = Kyy) (green symbol) in
+Fig. 4(a).
+The contributions from the paramagnetic
+(Kpara, positive) and diamagnetic (Kdia, negative) parts
+are also separately plotted in the figure. In the regime
+with h ≤ 1.125 and 1.75 ≤ h, the η-pairing is electromag-
+netically unstable, while it is stable in 1.125 < h < 1.75.
+In Fig. 4, the yellow shaded rectangle indicates the regime
+where the η-pairing becomes the ground state as seen
+from Fig. 2. We find a narrow region where η-pairing is
+regarded as the ground state but is not an electromagnet-
+ically stable state around h = 1.125. From these results,
+we see that the η-pairing is not necessarily electromag-
+netically stable even if it becomes the ground state in
+a two-sublattice calculation. As we shall see later, the
+simple η-pairing in this narrow regime does not necessar-
+ily exist if we relax the two-sublattice condition of the
+mean-field solution.
+We also show in Fig. 4(a) the contributions from the
+even- and odd-frequency pairs defined in Eqs. (23) and
+(24). The negative sign of the kernel, which means the re-
+sponse is diamagnetic, is partly due to the odd-frequency
+component of the pair amplitude, (KOFP < 0).
+This
+is in contrast to the FFLO state whose Meissner ker-
+nel is also negative due to the even-frequency component
+[51]. Hence, it implies that the mechanism of the dia-
+magnetism is different between the FFLO and η-pairing
+states.
+In
+addition
+to
+the
+Meissner
+kernel,
+we
+calcu-
+late the local pair amplitudes which are shown in
+FIG. 4(b).
+Here the left- and right-panels represent
+the spin-triplet and spin-singlet components of the lo-
+cal pair amplitude, respectively. The triplet component
+�
+σσ′(τ µiτ y)σσ′Fσσ′(iωn) with µ = x has a finite imagi-
+nary part and zero real part, which represents the odd-
+frequency pair. The other µ = y, z components are zero.
+On the other hand, the singlet component has a finite real
+part and zero imaginary part and is the even-frequency
+pair. We can see that the maximum value of the spin-
+triplet component of the pair amplitude is largest at the
+magnetic field h = 1.375, where the magnitude of KOFP
+is largest. It is also notable that the magnitude of the
+odd-frequency pair amplitude correlates with the magni-
+tude of DOS at zero energy as seen by comparing Figs. 3
+and 4.
+We comment on the singular behavior of KOFP at the
+magnetic field h = 1.375, although it does not affect the
+total Meissner kernel K. This anomalous feature is re-
+lated to the van Hove singularity of the DOS at zero
+energy as shown in FIG. 3, which shows a sharp peak at
+the Fermi level.
+
+7
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+h
+-1.0
+-0.5
+0.0
+0.5
+1.0
+K
+Kdia
+Kpara
+K
+KEFP
+KOFP
+(a)
+(b)
+<latexit sha1_base64="hwzoaSF0Y+Oi/TJGH/A
+8l7Sl9vw=">AD0HichVO5TsNAEH3BnOEIR4NEg4iQKFC0QZxdEA0lVwCJIGSbTbKL9kbFIQioEW08A
++IH+EHKPgEagoaCmY35lIUM5bt2TfvjWd2x1bgiEgy9pLqMrp7ev6B9KDQ8MjmdGx8f3Ir4c2L9q+4e
+HlhlxR3i8KIV0+GEQctO1H5g1TZU/OCMh5HwvT15HvBj16x4oixsUxK0U2qejGZjmbnfysZNFbFv+
+WKqEk7hw0YdLjg8SPIdmIjoOkIeDAFhx7gLCRP6DhHE2nS1onFiWESWqNnhVZHMerRWuWMtNqmrzh0h6
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+6hdqfhPmcAfM24+AbMryxg=</latexit>}Eq. (25)
+°3
+°2
+°1
+0
+1
+2
+3
+!n
+0.0
+0.6
+1.2
+1.8
+2.4
+3.0
+3.6
+4.2
+4.8
+5.4
+6.0
+6.6
+Re[F " #(i!n) ° F # "(i!n)]/
+p
+2
+2.0
+1.875
+1.75
+1.625
+1.5
+1.375
+1.25
+1.125
+1.0
+0.875
+0.75
+0.625
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+h
+-1.0
+-0.5
+0.0
+0.5
+1.0
+K
+Kdia
+Kpara
+K
+KEFP
+KOFP
+OFP
+EFP
+°3
+°2
+°1
+0
+1
+2
+3
+!n
+0.0
+0.6
+1.2
+1.8
+2.4
+3.0
+3.6
+4.2
+4.8
+5.4
+6.0
+6.6
+Im[F # #(i!n) ° F " "(i!n)]/
+p
+2
+FIG. 4.
+(a) Magnetic field dependence of the Meissner ker-
+nel K(= Kxx = Kyy) for the η-pairing on the square lattice.
+The yellow shaded rectangle indicates the range where the
+η-pairing becomes the ground state in two-sublattice calcula-
+tion. The number of the wavenumber k is taken as 300×300.
+(b) Matsubara frequency dependence of the local pair ampli-
+tude at several magnetic fields. The left panel represents the
+imaginary part of [F↓↓(iωn) − F↑↑(iωn)] /
+√
+2, and the right
+panel represents the real part of [F↑↓(iωn) − F↓↑(iωn)] /
+√
+2.
+The values of the pair amplitudes are shifted by 0.6 at each
+magnetic field for visual clarity, and the gray-dotted lines are
+the zero axes for each magnetic field.
+3.
+Beyond two-sublattice
+In order to clarify the stable ordered state where the
+Meissner kernel is positive (paramagnetic), we investi-
+gate mean-field solutions on finite-sized lattice where the
+two-sublattice condition is not imposed.
+We have nu-
+merically solved the Eqs. (4)-(6) self-consistently by us-
+ing the mean-field solutions of the η-pairing obtained for
+two-sublattice as an initial condition.
+Figure 5 shows the spatial distribution of the phase of
+the gap function when the number of sites is 8 × 8. At
+h = 0.5 in (a), where the η-pairing is not a ground state,
+the uniform BCS pairing state is realized as expected.
+With increasing the magnetic field, the longer-periodicity
+structures are found as shown in Figs. 5(b), (c) and (d).
+At h = 1.375 in (c), where the η-pairing solution has the
+lowest energy and the electromagnetic response is well
+diamagnetic, we obtain the staggered alignment of the
+(a) h = 0.5
+(d) h = 1.875
+(c) h = 1.375
+(b) h = 1.125
+FIG. 5.
+Spatial distribution of the phase of the supercon-
+ducting order parameter at several magnetic fields. The cal-
+culation is performed on the finite-sized lattice (8 × 8) with
+open boundary condition. Small black dots are lattice points
+and red arrows indicate the phase of the pair potential for
+each lattice point.
+phases. When the parameters are close to the edges of
+the yellow-highlighted region in Fig. 4, the complex struc-
+tures are formed as shown in (b) and (d). The behavior
+in (b) is interpreted as due to the competing effect where
+the simple uniform and staggered phases are energetically
+close to each other.
+We also investigate the case with the other choice of pa-
+rameters: U = −1.25 and h = 1.25. In this case, we find
+the staggered flux state where the phase of pair poten-
+tial is characterized by 90◦-N´eel ordering as in Fig. 6(a).
+This ordered state cannot be described in the mean-field
+theory with two sublattices.
+Owing to a non-colinear
+90◦-N´eel ordering vector, the spontaneous clockwise or
+counterclockwise loop currents arise by the inter-atomic
+Josephson effect. The current density is calculated by
+jij = −it
+�
+σ
+⟨c†
+iσcjσ − c†
+jσciσ⟩
+(29)
+which is identical to the expression of the paramagnetic
+current in the linear response theory. We can also evalu-
+ate the flux for each plaquette, which is define by
+Φ =
+�
+(i,j)∈plaquette
+jij
+(30)
+This expression is similar to the flux
+�
+C
+j ·ds =
+�
+S
+b·dS
+(j = ∇ × b) defined in a continuum system, where b is
+a flux density. The flux is aligned in a staggered manner
+
+8
+(a)
+(b)
+Current
+Magnetic flux
+FIG. 6. (a) Spatial distribution of the phase of the supercon-
+ducting order parameter for the η-pairing with 90◦-N´eel state
+on the finite-sized lattice under open boundary conditions.
+(b) Spatial distributions of the spontaneous loop current and
+the flux defined on each plaquette. The color of vectors dis-
+plays the magnitude of current, and the color of dots in each
+plaquette indicates the value of the magnetic flux defined in
+Eq. (30).
+on a dual lattice as indicated in Fig. 6(b). The staggered
+flux originating from the normal part has been studied
+before [20–23], while the staggered flux shown in Fig. 6(b)
+has a different origin: it arises from the superconductiv-
+ity associated with the off-diagonal part in the Nambu
+representation.
+We also comment on a feedback effect to the electro-
+magnetic field from the supercurrent.
+Since the char-
+acteristic length scale for the magnetic field in layered
+superconductor becomes long [52], each magnetic flux on
+the plaquette is smeared out with this length. Hence we
+expect that the net magnetic field is not created from the
+staggered superconducting flux.
+B.
+Triangular lattice
+1.
+Mean-field solution
+Now we search for the η-pairing reflecting the charac-
+teristics of a geometrically frustrated triangular lattice
+at the half-filling (nc = 1.0). We choose the parameters
+U = −1.83 and T = 1.0 × 10−3. We consider the cases of
+two- and three-sublattice structures. For a usual antifer-
+romagnet, the typical ordered state in the two-sublattice
+case has a stripe pattern, while in the three-sublattice
+case we expect a 120◦-N´eel state. Below we study the
+superconducting η-pairing phases within the mean-field
+theory.
+We have found the four types of superconducting states
+reflecting the characteristics of the triangular lattice,
+which are referred to as the η-pairing I, II, III, and IV.
+The schematic pictures for these four states are shown
+in Fig. 7(a), where the arrow indicates the phase of the
+superconducting order parameter at each site. We make
+a few general remarks: the three-sublattice structure is
+assumed for I, II, III, while the two sublattice is employed
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+0.0
+0.5
+1.0
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+0.0
+0.5
+1.0
+1.5
+ni, mi
+nA
+nB
+nC
+mA
+mB
+mC
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+0.0
+0.5
+1.0
+ni, mi
+nA
+nB
+mA
+mB
+(a)
+(b)
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+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+h
+°0.05
+0.00
+0.05
+Ei ° E¥°pairing I
+<latexit sha1_base64="HLHSkuiSo7OABq7I/GuoPhNC6Lg=">AD5XichVNa9tAEH20tRx
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+0.0
+1.8
+3.6
+5.4
+7.2
+9.0
+10.8
+12.6
+14.4
+16.2
+18.0
+19.8
+21.6
+23.4
+°3
+°2
+°1
+0
+Ei ° Enormal
+BCS
+normal
+¥-pairing I
+¥-pairing IV
+¥-pairing III
+¥-pairing II
+(c)
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+h
+°0.05
+0.00
+0.05
+Ei ° E¥°pairing I
+0.0
+1.8
+3.6
+5.4
+7.2
+9.0
+10.8
+12.6
+14.4
+16.2
+18.0
+19.8
+21.6
+23.4
+°3
+°2
+°1
+0
+Ei ° Enormal
+BCS
+normal
+¥-pairing I
+¥-pairing IV
+¥-pairing III
+¥-pairing II
+-pairing II
+η
+-pairing IV
+η
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+h
+°0.05
+0.00
+0.05
+Ei ° E¥°pairing I
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+0.0
+0.5
+1.0
+1.5
+ni, mi
+nA
+nB
+nC
+mA
+mB
+mC
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+0.0
+0.5
+1.0
+1.5
+ni, mi
+nA
+nB
+nC
+mA
+mB
+mC
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+0.0
+0.5
+1.0
+ni, mx
+i
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+0.0
+0.5
+1.0
+ni, mx
+i
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+0.0
+0.5
+1.0
+ni, mx
+i
+x
+y
+FIG. 7.
+(a) Schematics for the four η-pairings in the tri-
+angular lattice model. The arrows indicate the phase of the
+pair potential. The size of the circles shows the amount of
+the electron density for each sublattice. (b) Magnetic field
+dependence of the internal energies measured from the nor-
+mal state (upper panel). The lower panel shows the inter-
+nal energy measured from the η-pairing I. (c) Magnetic field
+dependence of the number of electrons and magnetization on
+each sublattice for the η-pairing II (upper panel) and IV (lower
+panel).
+for IV. The type-I has a non-colinear structure, and in the
+other η-pairings the vectors are aligned in a colinear man-
+ner. We also note that CDW accompanies the η-pairings
+II and III, where the number of local filling is indicated
+by the size of the filled circle symbols in Fig. 7(a).
+Figure 7(b) shows the internal energy of the ordered
+states measured from the normal state (Upper panel) and
+from the η-pairing I (Lower panel). From the lower panel
+of Fig. 7(b), we can identify the ground state. With in-
+creasing the magnetic field, the ground state changes as
+BCS → η-pairing II→ η-pairing I → η-pairing IV→ η-
+pairing I → normal. Figure 7(c) shows the particle den-
+
+9
+(a)
+(b)
+Iloop
+h
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+h
+-0.2
+-0.1
+0.0
+0.1
+Iloop
+FIG. 8. (a) Schematic picture of the staggered flux state on
+the triangular lattice. The straight arrows display the phase
+of the pair potential at each site, and the circle arrows indicate
+the staggered loop current. (b) Magnetic field dependence of
+the magnitude of loop current. The yellow shaded rectangle
+indicates the range where the η-pairing I becomes the ground
+state.
+sity and x-direction magnetization mx
+i of each sublattice
+for η-pairing II (Upper panel) and η-pairing IV (Lower
+panel). The values of my
+i and mz
+i are zero because the
+Zeeman field h is applied along the x-direction. Below,
+we explain the characteristic features for each η-pairing
+state.
+η-pairing-I state.— The η-pairing I has 120◦ N´eel or-
+dering vector (Green pentagon in Fig. 7(b)). The spon-
+taneous supercurrent appears in this non-colinear state
+as schematically shown in Fig. 8(a). This superconduct-
+ing state forms a staggered flux state, where the flux is
+aligned on a honeycomb dual lattice, which is similar to
+the η-pairing with 90◦-N´eel ordering vector on the square
+lattice shown in Fig. 6(b). Figure 8(b) displays the val-
+ues of spontaneous loop current density as a function of
+the magnetic field.
+η-pairing-II state.— The η-pairing II has the struc-
+ture with up-up-down colinear phases plus CDW (Red
+hexagon in Fig. 7(b)). There is the relation nA = nB <
+nC for the electron filling at each sublattice shown in
+Fig. 7(c).
+We note that this site-dependent feature is
+characteristic for the II (and IV) state. The phases of the
+pair potential at A and B sublattices are “ferromagnetic”,
+while the phase at C sublattice is “antiferromagnetic”.
+The resulting ordered state is regarded as the emergence
+of the honeycomb lattice formed by equivalent A and B
+sublattices.
+η-pairing-III state.— This is the η-pairing with a stag-
+gered ordering vector and CDW (Magenta square in
+Fig. 7(b)).
+The order parameter ∆ at C sublattice is
+zero, but the others (A,B) are finite. The electron-rich
+sublattices A and B form a simple bipartite η-pairing
+state on an emergent honeycomb lattice. Since this state
+does not become a ground state anywhere for the present
+choice of U = −1.83, we do not further investigate this
+state in the following.
+η-pairing-IV state.— This is the η-pairing with a sim-
+ple stripe alignment (Cyan rhombus in Fig. 7(b)). This
+η-pairing is accompanied by CDW around h = 1.9 shown
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+h
+°1.5
+°1.0
+°0.5
+0.0
+0.5
+1.0
+1.5
+Kxx, Kyy
+(a) -pairing Ⅰ
+η
+Eq. (25)
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+h
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+Kxx, Kyy
+(b) -pairing Ⅱ
+η
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+h
+°1.5
+°1.0
+°0.5
+0.0
+0.5
+1.0
+1.5
+Kyy
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+h
+°1.5
+°1.0
+°0.5
+0.0
+0.5
+1.0
+1.5
+Kxx
+(c1) -pairing Ⅳ
+η
+(c2) -pairing Ⅳ
+η
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
+h
+°1.5
+°1.0
+°0.5
+0.0
+0.5
+1.0
+1.5
+Kyy
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+h
+-1.0
+-0.5
+0.0
+0.5
+1.0
+K
+Kdia
+Kpara
+K
+KEFP
+KOFP
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+h
+-1.0
+-0.5
+0.0
+0.5
+1.0
+K
+Kdia
+Kpara
+K
+KEFP
+KOFP
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+h
+-1.0
+-0.5
+0.0
+0.5
+1.0
+K
+Kdia
+Kpara
+K
+KEFP
+KOFP
+<latexit sha1_base64="hwzoaSF0Y+Oi/TJGH
+/A8l7Sl9vw=">AD0HichVO5TsNAEH3BnOEIR4NEg4iQKFC0QZxdEA0lVwCJIGSbTbKL9kbFIQio
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+FIG. 9. Magnetic field dependence of the Meissner kernels
+Kxx and Kyy for the η-pairings I, II, IV on the triangular lat-
+tice. The yellow shaded rectangle indicates the regime where
+each η-pairing becomes the ground state. The symbols are
+the same as those in Fig. 4(a). For the η-pairing IV, Kxx and
+Kyy are separately plotted in (c1) and (c2).
+in Fig. 7(c). As shown below, this stripe phase show an
+anisotropic behavior in linear response coefficients, while
+the other η-pairing states are isotropic.
+2.
+Meissner response
+Now we discuss the Meissner response. Figure 9(a,b,c)
+shows the Meissner kernels Kxx, Kyy for the η-pairing I,
+II and IV. The yellow-highlighted parts indicate the re-
+gion where each η-pairing becomes the ground state as
+identified from Fig. 7(b).
+The result for the η-pairing
+III is not shown because it does not become a ground
+state at U = −1.83. We confirm that the Meissner re-
+sponse is basically diamagnetic if the η-pairing becomes
+the ground state as shown in Figs. 9(a,b,c). Thus the en-
+ergetic stability and diamagnetic response are reasonably
+correlated. In the following, we discuss the properties of
+the Meissner kernel for each state.
+The Meissner kernels for both η-pairing I and η-pairing
+II shown in Figs. 9(a) and (b) satisfy the relation Kxx =
+Kyy, which means an isotropic linear response. For the η-
+pairing I, the Meissner kernel becomes positive in the re-
+gions h < 1.2, 1.95 < h < 2.12, while the kernel becomes
+negative in the ground state region (Fig. 9(a)). Although
+the local current density is finite for the η-pairing I state,
+it does not affect the expression of the Meissner kernel in
+Eq. (10) since the total current j(q = 0) is zero.
+Next we disucuss the η-pairing IV state. The Meiss-
+
+10
+ner kernel jumps at h = 1.8 due to the emergence of
+the CDW order parameter as shown in Fig. 9(c1,c2). It
+is notable that the η-pairing IV with the stripe pattern
+shows a difference between x and y directions as shown
+in Figs. 9(c1,c2), respectively. This characteristic behav-
+ior can be intuitively understood from Fig. 7(a), where
+the current along the x-axis flows with experiencing a
+staggered pair potential, whereas the current in the y-
+direction feels an uniform pair potential. In the Meissner
+response, Kxx shows a characteristic behavior of the η-
+pairing, while Kyy is qualitatively the same as the kernel
+of BCS. Thus, as shown in Fig. 9(c1), the diamagnetic
+response in the x-axis direction is related to to the odd-
+frequency pair, whereas the diamagnetic response in the
+y-axis direction, shown in Fig. 9(c2), is related to even-
+frequency pair.
+V.
+SUMMARY AND OUTLOOK
+We have studied the square and the triangular lattice
+of the attractive Hubbard model by using the mean-field
+theory.
+Several types of η-pairing have been found in
+the triangular lattice where a simple bipartite pattern
+is not allowed.
+Using the formulation of the Meissner
+kernel for a general tight-binding lattice, we have inves-
+tigated the electromagnetic stability of η-pairings. We
+have confirmed that the electromagnetic stability of the
+η-pairing correlates with the internal energy. In a narrow
+parameter range, we also find that the η-pairing state can
+show an unphysical paramagnetic response if we assume
+the two or three sublattice structure in the mean-field
+calculation.
+In this case, another solution with longer
+periodicity needs to be sought.
+When the current path experiences the staggered
+phase of the superconducting order parameter, the odd-
+frequency component of the pair amplitude contributes
+to the diamagnetic response. This is in contrast to the
+conventional BCS case in which the even-frequency com-
+ponent of the pair amplitude contributes to the diamag-
+netism.
+We have further clarified that one of the η-
+pairing states on the triangular lattice has a stripe pat-
+tern and shows an anisotropic Meissner response. In this
+case, the odd-frequency pair contributes diamagnetically
+or paramagnetically depending on the direction of cur-
+rent.
+We comment on some issues which are not explored in
+this paper. We expect that the η-pairing without a simple
+staggered phase will appear on pyrochlore, kagome and
+quasicrystalline lattice, whose phase-alignment could be
+qualitatively different from the triangular lattice. In ad-
+dition, there is another model that shows η-pairing in
+equilibrium. A two-channel Kondo lattice (TCKL) is an
+example of a model in which η-pairing appears even in
+the absence of a Zeeman field [24]. Our preliminary cal-
+culation for the TCKL shows a number of ordered states
+which have similar energies.
+These additional studies
+provide more insight into the exotic superconductivity
+(a)
+(b)
+(c)
+°1
+0
+1
+!n
+0.0
+0.6
+1.2
+1.8
+2.4
+3.0
+3.6
+4.2
+4.8
+5.4
+6.0
+Re[F " #(i!n) ° F # "(i!n)]/
+p
+2
+h=1.417
+h=1.375
+h=1.333
+h=1.25
+h=1.167
+h=1.083
+h=1.0
+h=0.917
+h=0.833
+h=0.75
+h=0.667
+°0.2°0.1 0.0
+0.1
+0.2
+!n
+0.0
+0.6
+1.2
+1.8
+2.4
+3.0
+3.6
+4.2
+4.8
+5.4
+6.0
+Re[F " #(i!n) ° F # "(i!n)]/
+p
+2
+1.417
+1.375
+1.333
+1.25
+1.167
+1.083
+1.0
+0.917
+0.833
+0.75
+0.667
+°1.0 °0.5
+0.0
+0.5
+1.0
+!
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+D¥°pairing ° Dnormal
+°0.2°0.1 0.0
+0.1
+0.2
+!n
+0.0
+0.6
+1.2
+1.8
+2.4
+3.0
+3.6
+4.2
+4.8
+5.4
+6.0
+Im[F # #(i!n) ° F " "(i!n)]/
+p
+2
+FIG. 10. (a) The difference between the DOSs of the η-pairing
+and normal states in the cubic lattice model. The values of
+the DOS are shifted by 0.2 for each magnetic field, and the
+gray dotted lines are the zero axes for each magnetic field.
+We also show the Matsubara frequency dependence of (b) the
+imaginary part of [F↓↓(iωn) − F↑↑(iωn)] /
+√
+2 and (c) the real
+part of [F↑↓(iωn) − F↓↑(iωn)] /
+√
+2 for each magnetic field. The
+values of the pair amplitudes are shifted by 0.6.
+characteristic for the η-pairing.
+ACKNOWLEDGEMENT
+This work was supported by KAKENHI Grants No.
+18H01176, No. 19H01842, and No. 21K03459.
+Appendix A: Self-consistent equations in mean-field
+theory
+We derive self-consistent equations for the general in-
+teracting Hamiltonian. Let us begin with the Hamilto-
+nian
+H =
+�
+12
+ε12c†
+1c2 +
+�
+1234
+U1234c†
+1c†
+2c4c3
+(A1)
+where site-spin indices are written as 1 = (i1, σ1). The
+mean-field Hamiltonian is introduced as
+HMF =
+�
+12
+�
+E12c†
+1c2 + ∆12c†
+1c†
+2 + ∆∗
+12c2c1
+�
+.
+(A2)
+We assume ⟨H ⟩ = ⟨HMF⟩ where the statistical average
+is taken with HMF. Then the self-consistent equation is
+obtained as
+E12 = ∂⟨H ⟩
+∂⟨c†
+1c2⟩
+= ε12 +
+�
+34
+(U1324 + U3142 − U1342 − U3124)⟨c†
+3c4⟩
+(A3)
+∆12 = ∂⟨H ⟩
+∂⟨c†
+1c†
+2⟩
+=
+�
+34
+U1234⟨c4c3⟩
+(A4)
+
+11
+where the Wick’s theorem is used for the derivation. Al-
+though the variational principle for the free energy also
+gives the same equation, the above formalism gives a sim-
+ple procedure to derive the self-consistent equations.
+Appendix B: Attractive Hubbard model on Cubic
+lattice
+We analyze the η-pairing on the cubic lattice, whose
+DOS does not have a van Hove singularity near zero en-
+ergy. Here we choose the parameter U = −1.375 and
+the electron concentration is half-filled. As a result, the
+DOS for the η-pairing around zero energy for each mag-
+netic filed on the cubic lattice is smaller than the DOS of
+the normal state as shown in Fig. 10(a). For reference,
+we also show in Figs.
+10(b) and (c) the pair amplitude
+similar to Fig. 4(b) in the main text. In addition, the
+odd-frequency pair amplitude increases when DOS near
+zero energy is enhanced as seen from Figs. 10(a) and (b).
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4dFAT4oBgHgl3EQflx3j/content/tmp_files/2301.08620v1.pdf.txt

@@ -0,0 +1,1060 @@
+Adjoint-based Identification of Sound Sources for
+Sound Reinforcement and Source Localization
+Mathias Lemke and Lewin Stein
+Institut f¨ur Str¨omungsmechanik und Technische Akustik,
+Technische Universit¨at Berlin, Germany
+[email protected]
+Abstract. The identification of sound sources is a common problem in
+acoustics. Different parameters are sought, among these are signal and
+position of the sources. We present an adjoint-based approach for sound
+source identification, which employs computational aeroacoustic tech-
+niques. Two different applications are presented as a proof-of-concept:
+optimization of a sound reinforcement setup and the localization of (mov-
+ing) sound sources.
+Keywords: Computational Aeroacoustics, Adjoint Equations, Source
+Identification, Sound Reinforcement, Source Localization
+1
+Introduction
+A common issue in acoustics is the identification of fixed or moving sound
+sources. In general, several parameters have to be determined; among these are
+the source signal and the position of the sources. This general problem occurs
+in many applications, from environmental to industrial acoustics.
+In this contribution, we discuss an adjoint-based approach for sound source
+identification. The time-domain method is based on the (adjoint) Euler equa-
+tions, which are solved by means of computational aeroacoustic techniques (CAA).
+The approach allows considering complex base flows, such as non-homogeneous
+base flow, thermal stratification as well as complex geometries.
+Adjoint-based methods have been used in the field of fluid mechanics for
+decades. They have proven to be an effective approach for the analysis of flow
+configurations and determining optimal model parameters in various applications
+[7]. Adjoint-based techniques are used to optimize flow configurations by means
+of geometry modifications [9] or for active flow control applications [1]. They are
+applied for the analysis and optimization of reactive flow configurations [13,12]
+and data assimilation applications [23,14,8]. Furthermore, they are employed in
+the field of aeroacoustics [4,20] and sound reinforcement applications [15,21].
+Here, we restrict ourselves to two applications from the areas of sound re-
+inforcement and sound source localization with generic setups as a proof-of-
+concept.
+In the context of sound reinforcement, line arrays are used for the synthesis of
+sound fields. The identification of the geometric arrangement and the electronic
+arXiv:2301.08620v1  [cs.SD]  20 Jan 2023
+
+2
+Mathias Lemke et al.
+drive of the loudspeaker cabinets to optimally (re-)produce a sound field is an
+ill-posed, inverse problem. Typically frequency domain approaches are employed
+[3,22].
+For the localization of moving and non-moving sound sources, usually, micro-
+phone array methods like beam-forming are used. Depending on the specific task,
+different algorithms, working in the time domain or in the frequency domain,
+are applied. See [16] for a recent overview.
+The manuscript is organized as follows: In Sec. 2, the adjoint approach is in-
+troduced, and the adjoint Euler equations are derived. After a short description
+of the numerical implementation in Sec. 3, the derived framework is employed for
+an application in the context of sound reinforcement in Sec. 4. The applicability
+of the approach for localization of sound sources is discussed in Sec. 5.
+2
+Adjoint Approach
+2.1
+General Adjoint Equations
+Adjoint equations can be derived in different ways, e.g., the continuous or the
+discrete approach. Despite different discretizations, the approaches are consistent
+and applicable, see [7] for a discussion. In addition, automatic differentiation
+techniques are used to create adjoint codes from existing simulation programs.
+Recently, a mode-based approach to derive adjoint operators was presented [19]
+as an enhancement of a direct operator construction method [12].
+Here, the adjoint equations are introduced in a discrete manner. A matrix-
+vector notation is used, in which the vector space is the full solution in space
+and time. The section is based on [7,11].
+In general, the adjoint equations arise by a scalar-valued objective function
+J, which is defined by the user and encodes the target of the analysis, e.g., an
+optimization. It is given by the scalar product between a weight vector g and a
+system state vector q
+J = gTq.
+(1)
+The system state q is the solution of the governing system
+Aq = s
+(2)
+with A as governing operator and s as right-hand side forcing. In order to opti-
+mize J by means of s in terms of a brute-force approach, the governing equation
+has to be solved for all possible s.
+Instead, to reduce the computational effort, the adjoint equation can be used
+ATq∗ = g,
+(3)
+with the adjoint variable q∗.
+With
+J = gTq =
+�
+ATq∗�T q = q∗TAq = q∗Ts
+(4)
+
+Adjoint Sound
+3
+a formulation is found, which enables the computation of the objective J without
+solving the governing system for every possible s. With the solution of the adjoint
+equation, the objective can be calculated by a scalar product. Thus, the adjoint
+approach enables efficient computation of gradients for J with respect to s.
+2.2
+Adjoint Euler equations for Acoustic Applications
+The section is based on [11,21]. The objective function J is defined in space and
+time with dΩ = dxidt in the whole computational domain:
+J = 1
+2
+�� �
+q − qtarget�2 dΩ.
+(5)
+The variable q contains the full state q = [ϱ, uj, p] of the system governed by the
+Euler equations. Therein, ϱ denotes the density, uj the velocity in the direction
+xj, and p the pressure.
+For the following aeroacoustic analyses the evaluation of the objective func-
+tion is restricted to the pressure, resulting in
+J = 1
+2
+�� �
+p − ptarget�2 σ dΩ.
+(6)
+The additional weight σ(xi, t) defines where and when the objective is evalu-
+ated. In general, the objective function has to be supplemented by a regular-
+ization term, which is omitted here for the sake of clarity. The target ptarget is
+application-specific. For optimization tasks, as presented in Sec. 4, it is defined
+corresponding to a desired sound field, e.g., optimal listening experience for the
+auditorium of an open-air concert. For the source localization application pre-
+sented in Sec. 5, the target pressure is defined by microphone measurements.
+The microphone positions are included by means of the weight function σ. In
+both cases, a minimum of J is desired.
+This minimum is to be achieved under the constraint that the Euler equa-
+tions
+∂t
+�
+�
+ϱ
+ϱuj
+p
+γ−1
+�
+� + ∂xi
+�
+�
+ϱui
+ϱuiuj + pδij
+uipγ
+γ−1
+�
+� − ui∂xi
+�
+�
+0
+0
+p
+�
+� =
+�
+�
+0
+0
+sp
+�
+� ,
+with γ as heat capacity ratio, are fulfilled. The summation convention applies.
+For details on the formulation, in particular, the reformulation of the energy
+equation in terms of pressure, see [13].
+To ease the derivation, the above system of partial differential equations is
+abbreviated by
+E(q) = s.
+(7)
+The terms s = [0, 0, sp] on the right side of the Euler equations character-
+ize monopole sound sources, which allow controlling the system state, respec-
+tively, the solution of the equations. In general, also mass and momentum source
+
+4
+Mathias Lemke et al.
+terms could be considered. The overall goal is to obtain a solution of the Euler
+equations, which reduces the objective (6) by adapting s. An optimization of s
+corresponds to an optimization of the loudspeakers’ output signals.
+To use the adjoint approach for optimizing s, the objective function (6) and
+the governing system (7) have to be linearized. This results in
+δJ =
+�� �
+q − qtarget�
+σ
+�
+��
+�
+=g
+δpdΩ,
+(8)
+and
+Elinδq = δs.
+(9)
+The weight g = (q − qtarget)σ encodes the difference between the current numer-
+ical solution and the target field. Here, it is evaluated only in terms of pressure,
+as discussed above. Combining the linearized system and the objective in a La-
+grangian manner leads to
+δJ = gTδq − q∗T (Elinδq − δs)
+�
+��
+�
+=0
+(10)
+= q∗Tδs + δqT �
+g − ET
+linq∗�
+.
+Please note, the spatial and temporal integrals are not shown for the sake of
+simplicity.
+The desired adjoint equation E∗ = ET
+lin results from demanding
+g − ET
+linq∗ = 0,
+(11)
+with q∗ = [ϱ∗, u∗
+j, p∗] as adjoint state variable.
+For a detailed derivation of the adjoint Euler equations see [11]. They are
+given by
+∂tq∗ = ˜A
+�
+−(Bi)T∂xiq∗ − ∂xi(Ci)Tq∗ + ˜Ci∂xic − g
+�
+(12)
+with ˜A =
+�
+AT�−1 and ˜Ci as resorting
+q∗
+αδCi
+αβ∂xicβ = q∗
+αδqκ
+∂Ci
+αβ
+∂qκ
+∂xicβ
+(13)
+abbreviated as δqκ ˜Ci
+κβ∂xicβ. The matrices A, Bi and Ci are given in the ap-
+pendix.
+Finally, the change of the objective function is given by
+δJ = q∗Tδs.
+(14)
+Thus, the solution of the adjoint equation can be interpreted as gradient of J
+with respect to the source terms s
+∇sJ = q∗.
+(15)
+Initial and boundary conditions of the adjoint Euler equations as well as the
+derivation of the adjoint compressible Navier-Stokes equations are discussed in
+[11].
+
+Adjoint Sound
+5
+sources
+initial guess
+s0=0
+solution
+Euler equations
+N(q, sn) 
+target
+qtarget
+solution adjoint
+Euler equations
+N*(q,q*,Δ q)
+gradient
+q*
+sources
+update sn+1
+Δ q = q ­ qtarget
+optimal
+s
+source
+positions
+p
+convergence
+loop 1
+Fig. 1. Iterative procedure for the determination of an optimal s. Computationally
+intensive steps are marked in gray. The first gradient provides information on (optimal)
+source positions, see Sec. 5 for a detailed discussion.
+2.3
+Iterative Process
+The adjoint-based gradient is employed in an iterative manner. First, the Euler
+equations (7) are solved forward in time, usually with s0 = 0. Subsequently,
+the adjoint equations (12) are calculated backward in time, deploying the direct
+solution and g. Based on the adjoint solution, the gradient ∇sJ is determined
+and used to update the source distribution sn by means of a steepest gradient
+approach:
+sn+1 = sn + α∇sJ,
+(16)
+with α denoting an appropriate step size and n the iteration number. The gradi-
+ent is calculated for the whole computing region and the entire simulation time.
+For the determination of sound sources with a known position, the gradient is
+evaluated only there. The procedure is repeated, using the current sn, until a
+suitable convergence criterion is reached. Typically, for acoustic problems, con-
+vergence is reached within or less 20 loops.
+The identification of global optima is not ensured as the proposed technique
+optimizes to local extrema only. The computational costs of the approach are
+independent of the number of sources and their arrangement. However, they
+depend on the size and resolution of the computational domain in space and
+time, defined by the considered frequency range. The computational problem is
+fully parallelizable.
+2.4
+Source Localization
+In particular, when s0 = 0 holds, the first adjoint solution contains information
+on the position of the sources. By the pointwise summation of the absolute
+adjoint sensitivities p∗ in the spatial domain over all computed time steps
+¯p =
+tn=end
+�
+tn=0
+|p∗|,
+(17)
+
+6
+Mathias Lemke et al.
+the positions featuring maximum impact on the objective function can be iden-
+tified by means of maxima of ¯p. These correspond to the most likely (monopole)
+source locations. Thus, the adjoint solution allows the localization of sound
+sources, see Sec. 5. A subsequent iterative adaptation of the sources can be
+interpreted as adjoint-based monopole synthesis.
+3
+Adjoint CAA framework
+The set of governing equations (7) is implemented by means of a new MPI-
+parallelized Fortran program. The discretization is realized by a finite difference
+time domain approach (FDTD). For the spatial derivatives, a compact scheme
+of 6th order is employed [10]. The corresponding linear system of equations is
+solved by BLAS routines using an LU-decomposition. For the time-wise inte-
+gration, the standard explicit Runge-Kutta-scheme of fourth-order is used.
+To ensure stability, a compact filter is employed [5]. Boundaries are treated by
+characteristic boundary conditions [18]. The MPI implementation is realized by
+collective communication via all2all v. The parallelization strategy is found to
+be efficient for the governing equations (7), see Fig. 2, and comparable to other
+implementations using collective communication, e.g. [17].
+Thus, the code is prepared to handle large scale problems, e.g., open-air festi-
+val sites in the context of sound reinforcement applications or source localization
+for vehicle aeroacoustics in wind tunnels. However, the examples presented in
+the following are computed using a single workstation or a few cluster nodes.
+Fig. 2. (Left) Strong scaling behaviour. The overall number of grid points is kept
+constant while increasing the number of MPI processes. Nearly linear scaling is found.
+(Right) Weak scaling behaviour. The number of grid points on each process is kept
+constant while increasing the number of MPI processes. An admissible reduction of the
+parallelization efficiency is found.
+The adjoint equations are solved using the same discretization. A detailed
+discussion on the adjoint initial- and characteristic boundary conditions can be
+found in [11].
+
+8
+speedup
+6
+4
+caa
+2
+--ideal
+400
+1200
+2000
+2800
+3600
+MPl processesefficiency
+0.9
+0.8
+caa
+.--ideal
+0.7
+40
+80
+160
+320
+640
+1280 2560
+MPl processesAdjoint Sound
+7
+4
+Application I: Sound Reinforcement
+This section presents a test case regarding the optimization of sound reinforce-
+ment setups. The overall goal is to identify optimal drives (amplitude and phase)
+for given loudspeakers in order to synthesise a desired sound field. The loudspeak-
+ers are approximated by means of monopole sources, which is feasible for low
+frequencies.
+The spatial domain under consideration is 1.6 × 1.6 × 1.6 m3. The domain
+is resolved by 197 × 197 × 99 equidistantly distributed points. The time step,
+and by this, the sampling rate, is given by 48 kHz, corresponding to a CFL-
+condition smaller than 1. The computational time span considered is 31.25 ms.
+The reference values for density and pressure correspond to a speed of sound
+of 343 m/s. All boundaries are treated as non-reflecting. In addition, a sponge
+layer is applied at all boundaries.
+For the test case reference signals for five sources, located in a curved ar-
+rangement in the center x1-x2 plane, are predefined. The signals are charac-
+terised by different amplitudes and phase delays resulting in a steered sound
+field, see Fig. 3 (left). In order to investigate the frequency band 1-3 kHz, a
+corresponding logarithmic sine-sweep is specified as the reference signal. Using
+this setup, a reference sound field is computed by a Complex Directivity Point
+Source (CDPS) algorithm [2]. The resulting reference sound field serves as the
+target for the adjoint-based framework, with the aim to identify the reference
+signals (amplitudes and phases) based on the reference target sound field only.
+After 15 iterative loops of the adjoint framework, the objective function is
+reduced to nearly 3% with respect to the initial solution with s = 0, see Fig. 3
+(right). The general features of the target reference sound field are captured, see
+Fig. 4. A detailed spectral analysis of the occurring deviations at two selected
+microphone positions, presented in Fig. 4, show amplitude deviations less than
+1 dB within the confidence interval from 1.3 to 2.7 kHz. The normalized phase
+derivations, with respect to 2π, are in the limits of -0.07 to 0.07.
+A discussion on how to derive optimal electronic drives from the adjoint-
+based signals s is given in [21]. Therein, the capability of the approach to consider
+complex base flows by means of wind and temperature stratification is shown.
+5
+Application II: Source Localization
+In this section, the localization of fixed and moving sound sources is shown. Two
+generic setups serve as a proof of concept. For the first setup with four stationary
+sound sources and the second setup with a moving source, it is shown that the
+adjoint-based approach is able to identify the sources and track their path in
+case of moving.
+In both cases, the measurements are provided by a reference computation
+with predefined sound sources. Synthetic microphone signals are extracted from
+this reference solution. A spatially discrete planar array with 64 microphones is
+used. The general setup is based on the array benchmark test case B7 provided
+
+8
+Mathias Lemke et al.
+Fig. 3. (Left) Sound reinforcement setup including a selected time step of the CDPS-
+based reference sound field shown at the center x1-x2 plane of the computational do-
+main. The five monopole speakers in a curved arrangement are denoted by (*). Different
+driving functions (in amplitude and phase) for the speaker result in a steered sound
+field. The area/volume marked by the dashed line corresponds to the spatial weight σ
+in the objective function. Please note, the employed CDPS technique for computing
+the reference sound field does not provide reliable solutions near the source positions;
+therefore, p′
+ref is discontinuous for x1 = [0.32, 0.62] m. (Right) Progress of the objective
+function with a logarithmic y-axis. Convergence is reached. The objective is reduced
+by nearly two orders of magnitude with respect to the initial guess s = 0.
+Fig. 4. Reference target (left) and resulting optimized (right) sound field at t = 15.63
+ms for the center x1-x2 plane. The general features of the reference field are (re-)
+captured. The influence of the employed sponge layer in the adjoint-based sound field
+is visible. The dashed line encodes the spatial weight σ within the objective function.
+The marked positions correspond to synthetic microphone positions x1,2 = [1.1, 1.1]
+and x1,2 = [0.8, 0.8] which are used for spectral analysis, see text for details.
+
+1.5
+0.8
+米‘
+speaker
+0.6
+0.4
+a
+m
+0.
+8
+0
+d-
+0.5
+0.P
+-0.4
+0.5
+1
+1.5
+X, /m1
+0.5
+0.25
+r/
+0.1
+0.05
+0.02
+5
+10
+15
+iteration1.5
+0.5
+1
+a
+P
+0
++
+ref
+p
+0.5
+-0.5
+0.5
+1
+1.5
+X, / m1.5
+0.5
+1
+a
+P
+0
++
+opt
+2
+p
+0.5
+-0.5
+0.5
+1
+1.5Adjoint Sound
+9
+Fig. 5. (Left) Normalized amplitude difference between resulting optimized and refer-
+ence target sound field at selected microphone positions, see Fig. 4. (Right) Normalized
+phase difference between resulting optimized and reference target sound field at the
+selected microphone positions.
+by the Brandenburg university of technology, see [6]. Modifications are discussed
+below. An example in which experimental data are used is shown in [11].
+The spatial domain under consideration is 1.7 × 1.7 × 1.25 m3. The domain
+is resolved by 240 × 240 × 176 equidistantly distributed points. The time step,
+and by this, the sampling rate of the microphone measurements, is given by
+53.33 kHz, corresponding to a CFL-condition smaller than 1. In both cases, no
+base flow is considered. The reference values for density and pressure correspond
+to a speed of sound of 343 m/s. The spiral-like microphone array is located at
+x3 = 0 m and centered in the corresponding plane. The spatial distribution of
+the microphones is described in more detail in [6]. All boundaries are treated as
+non-reflecting. In addition, a sponge layer is applied at all boundaries.
+5.1
+Four sources
+As in the array benchmark test case B7 four monopole sources are located in
+the x1-x2-plane at x3 = 0.75 m, see Fig. 6 (left). For the reference computation,
+the original benchmark source signals are replaced by incoherent random sig-
+nals, frequency-band limited between 750 and 2500 Hz, see Fig. 6 (right). The
+computational time span is 14.06 ms.
+Using a corresponding reference forcing s = �
+i si a simulation of the Euler
+equations (7) is carried out. From the results, discrete microphone signals are
+extracted, see Fig. 7 (left), which are the result of the superposition of all sources
+and the associated signals.
+The 64 signals are encoded in the objective function J (6) using the spatial
+weight σ. To avoid an unstable discrete forcing of the adjoint equations, σ is
+chosen as Gauss-distribution with a half-width of 2∆x for each microphone
+position. After determining the solution of the direct equations with an initial
+guess for s = 0, here, constant environmental conditions for all time steps, the
+
+mic 1
+B
+0.4
+mic 2
+ta
+0.2
+p
+opt
+0
+-0.2
+-0.4
+-0.6
+1500
+2000
+2500
+f /Hz0.1
+mic 1
+mic 2
+2π
+0.05
+tar
+0
+opt
+-0.05
+-0.1
+1500
+2000
+2500
+f / Hz10
+Mathias Lemke et al.
+Fig. 6. (Left) Acoustic setup for source localization of four sources (*) by 64 micro-
+phones (o) located in the planes x3 = 0.75 m respectively x3 = 0 m. (Right) Normalized
+signals si of the four reference sources, shown for the whole computational time.
+adjoint equations are solved backwards in time. From the resulting gradient, the
+source positions can be derived, as discussed before. That way, the reference
+source positions are identified, see Fig. 7 (right).
+Fig. 7. (Left) Captured pressure signal at the center microphone in the array. The
+initial silence results from the distance between the sources and the array. (Right) Re-
+sulting pointwise summation of the absolute adjoint sensitivities p∗ (17). The reference
+source positions (∗) are recovered.
+Please note, the analysis is based on the first adjoint-based gradient only.
+The required computational time for the analysis is less than 15 min on a 16
+core workstation. Iterative optimization of s might improve the results.
+5.2
+Single moving source
+Again, the aforementioned test case B7 from [6] serves as a base for the following
+test setup. The planar microphone array is located in the same plane (x3 = 0) but
+
+米
+米
+0.5
+米
+米
+8
+00
+0
+8
+0
+0
+0
+0
+00
+00
+00
+00
+0
+0
+0
+0
+0
+00
+0
+0.
+00
+00
+0
+0
+0
+00
+8
+0.5
+0.5
+0
+0
+-0.5
+-0.5
+×2 /m
+X, / mS
+(normalized)
+0.5
+S
+2
+S
+1
+3
+.
+S
+0
+4
+..
+: i
+11
+-0.5
+11
+S
+II
+.....
+11
+二
+-1
+2
+6
+10
+14
+t/ mscenter mic
+0.1
+a
+P
+8
+d-(
+p
+-0.1
+2
+6
+10
+14
+t/msref. sources
+0.5
+0.8
+(normalized)
+0.6
+米
+米
+0
+*
+米
+0.4
+p
+-0.5
+0.2
+-0.5
+0
+0.5
+/ mAdjoint Sound
+11
+scaled by a factor of 0.8, resulting in smaller distances between the microphones.
+The incoherent sources are replaced by a single source with a harmonic 2 kHz
+reference signal. The source is moving in the x1-x2-plane, see Fig. 8 (left). The
+movement is described by an acceleration and deceleration, taking place along
+the x1 axis. It starts at the beginning of the computational time and ends with
+the simulation after 8.44 ms. The highest speed of the movement is reached
+midway.
+Again a reference solution provides synthetic microphone signals, which are
+encoded in the objective function. Using constant environmental conditions as
+solution of the direct equations (s0 = 0), the adjoint equations are solved. Eval-
+uation of the adjoint sensitivity p∗ over time at the reference source position
+provides information of the reference signal, see Fig. 8 (right). The phase of
+the reference signal is determined with very good agreement. The amplitude
+shows deviations at the beginning and end of the simulation. The influence of
+the directional characteristic of the used microphone array is presumed.
+Fig. 8. (Left) Acoustic setup for source localization of a single moving source (*) by
+means of 64 microphones (o) located in the planes x3 = 0.75 m, respectively x3 = 0
+m. The movement of the source is visualized by it waypoints, chosen with a constant
+time interval. (Right) Normalized adjoint-based sensitivity p∗ at the reference source
+positions over time in comparison to the reference forcing. See text for a detailed
+discussion.
+Besides, the identification of the source signal also its position might be
+tracked. In Fig. 9 the adjoint-based sensitivity p∗ is shown for the plane x3 = 0.75
+m for different time steps. Occurring maxima give rise to the actual sound source
+position, besides its signal.
+Again, the analysis is based on the first adjoint-based gradient only. The
+required computational time for the analysis is less than 10 min on 8 cluster
+nodes with 8 cores each.
+
+m
+0.5
+0
+0
+CD
+0
+000
+00
+0
+00
+0
+8
+0
+00
+0.5
+0.5
+0
+0
+-0.5
+-0.5
+m
+m
+25 二 1
+I 
+0.5
+(normalized)
+?
+P-0.5
+adjoint-based
+4
+--- reference
+-
+1!
+2
+4
+6
+8
+t/ms12
+Mathias Lemke et al.
+Fig. 9. Normalized adjoint-based sensitivity p∗ at the plane x3 = 0.75 for different
+time steps. The reference source location is marked by (*) in a white circle. In the
+inset, the normalized reference signal is shown.
+
+t= 3.88125 / ms
+0.5
+0.5
+(normalized)
+m
+0
+0
++
+p
+-0.5
+-0.5
+/
+米
+3.5
+4
+4.5
+t / ms
+-0.5
+0
+0.5
+, / mt= 3.99375/ ms
+0.5
+0.5
+(normalized)
+m
+0
+0
+/
++
+米
+p
+-0.5
+-0.5
+3.5
+4
+4.5
+t / ms
+-0.5
+0
+0.5
+X, / mt= 4.12500 / ms
+0.5
+0.5
+(normalized)
+m
+0
+0
+/
+*
+米
+p
+-0.5
+-0.5
+3.5
+4
+4.5
+t / ms
+-0.5
+0
+0.5
+_ / mt= 4.25625 / ms
+0.5
+0.5
+(normalized)
+m
+0
+0
+/
+p
+-0.5
+-0.5
+3.6
+44.4
+4.8
+t / ms
+-0.5
+0
+0.5
+_ /mAdjoint Sound
+13
+6
+Summary
+An adjoint-based framework for the identification of sound sources is presented.
+It is shown that the approach is able to determine (optimal) source signals and
+to track moving sources.
+By design, the time-domain approach allows the consideration of base flows,
+such as velocity profiles and temperature stratification, and complex geometries,
+which will be the focus of the upcoming work. The first results that take into
+account a complex base flow in the context of sound reinforcement are shown in
+[21].
+Acknowledgments
+The authors gratefully acknowledge financial support by the Deutsche Forschungs-
+gemeinschaft (DFG) within the project LE 3888/2-1.
+We thank Florian Straube (Audio Communication Group, TU Berlin) for
+defining the target sound field for the sound reinforcement test case.
+References
+1. A. Carnarius, F. Thiele, E. ¨Ozkaya, A. Nemili, and N. Gauger. Optimal control of
+unsteady flows using a discrete and a continuous adjoint approach. In D. H¨omberg
+and F. Tr¨oltzsch, editors, System Modeling and Optimization, volume 391 of IFIP
+Advances in Information and Communication Technology, pages 318–327. Springer
+Berlin Heidelberg, 2013.
+2. S. Feistel. Modeling the radiation of modern sound reinforcement systems in high
+resolution, volume 19. Logos Verlag Berlin GmbH, 2014.
+3. S. Feistel, M. Sempf, K. K¨ohler, and H. Schmalle. Adapting loudspeaker array
+radiation to the venue using numerical optimization of FIR filters. In Proc. of the
+135th Audio Eng. Soc. Conv., New York, number #8937, 2013.
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+navier-stokes equations. AIAA Journal, 38:2103–2112, Nov. 2000.
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+seen 12. Dec. 2019.
+7. M. Giles and N. Pierce. An introduction to the adjoint approach to design. Flow,
+Turbulence and Combustion, 65:393–415, 2000.
+8. J. Gray, M. Lemke, J. Reiss, C. Paschereit, J. Sesterhenn, and J. Moeck. A compact
+shock-focusing geometry for detonation initiation: Experiments and adjoint-based
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+9. A. Jameson. Optimum aerodynamic design using cfd and control theory. AIAA
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+10. S. K. Lele. Compact finite difference schemes with spectral-like resolution. Journal
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+11. M. Lemke. Adjoint based data assimilation in compressible flows with application to
+pressure determination from PIV data. PhD thesis, Technische Universit¨at Berlin,
+2015.
+
+14
+Mathias Lemke et al.
+12. M. Lemke, L. Cai, J. Reiss, H. Pitsch, and J. Sesterhenn. Adjoint-based sensitiv-
+ity analysis of quantities of interest of complex combustion models. Combustion
+Theory and Modelling, 23(1):180–196, 2019.
+13. M. Lemke, J. Reiss, and J. Sesterhenn.
+Adjoint based optimisation of reactive
+compressible flows. Combustion and Flame, 161(10):2552 – 2564, 2014.
+14. M. Lemke and J. Sesterhenn.
+Adjoint-based pressure determination from PIV
+data in compressible flows — validation and assessment based on synthetic data.
+European Journal of Mechanics - B/Fluids, 58:29 – 38, 2016.
+15. M. Lemke, F. Straube, F. Schultz, J. Sesterhenn, and S. Weinzierl. Adjoint-based
+time domain sound reinforcement. In Audio Engineering Society Conference: 2017
+AES International Conference on Sound Reinforcement – Open Air Venues, Aug
+2017. featured in Ramsey, F. (2017): ’Sound Reinforcement in the Open Air.’ In:
+J. Audio Eng. Soc., vol. 65, no. 12, pp. 1051 - 1055 (December).
+16. R. Merino-Mart´ınez, P. Sijtsma, M. Snellen, T. Ahlefeldt, J. Antoni, C. J. Bahr,
+D. Blacodon, D. Ernst, A. Finez, S. Funke, T. F. Geyer, S. Haxter, G. Herold,
+X. Huang, W. M. Humphreys, Q. Lecl`ere, A. Malgoezar, U. Michel, T. Padois,
+A. Pereira, C. Picard, E. Sarradj, H. Siller, D. G. Simons, and C. Spehr. A review
+of acoustic imaging methods using phased microphone arrays. CEAS Aeronautical
+Journal, 10(1):197–230, Mar 2019.
+17. D. Pekurovsky. P3dfft: A framework for parallel computations of fourier transforms
+in three dimensions. SIAM Journal on Scientific Computing, 34(4):C192–C209,
+2012.
+18. T. Poinsot and S. Lele. Boundary conditions for direct simulations of compressible
+viscous flows. Journal Computational Physics, 101:104–129, 1992.
+19. J. Reiss, M. Lemke, and J. Sesterhenn. Mode-based derivation of adjoint equations
+- a lazy man’s approach. on ArXiv, 2018.
+20. J. Schulze, P. Schmid, and J. Sesterhenn.
+Iterative optimization based on an
+objective functional in frequency-space with application to jet-noise cancellation.
+Journal of Computational Physics, 230(15):6075 – 6098, 2011.
+21. L. Stein, F. Straube, J. Sesterhenn, S. Weinzierl, and M. Lemke. Adjoint-based
+optimization of sound reinforcement including non-uniform flow. The Journal of
+the Acoustical Society of America, 146(3):1774–1785, 2019.
+22. A. Thompson and J. Luzarraga. Drive granularity for straight and curved loud-
+speaker arrays. Proc. of the Inst. of Acoustics, 35(2):210–218, 2013.
+23. Y. Yang, C. Robinson, D. Heitz, and E. M´emin. Enhanced ensemble-based 4dvar
+scheme for data assimilation. Computers & Fluids, 115:201 – 210, 2015.
+A
+Appendix
+A.1
+Adjoint equations
+As stated above, linearization of the governing Euler equations with respect to
+all state variables by q = q0 + δq results in
+∂tAδq + ∂xiBiδq + Ci∂xiδq + δCi∂xic = δs.
+(18)
+Again, the summation convention applies. The corresponding linearization ma-
+trices are
+
+Adjoint Sound
+15
+A =
+�
+�����
+1 0 0 0
+0
+u1 ρ 0 0
+0
+u2 0 ρ 0
+0
+u3 0 0 ρ
+0
+0 0 0 0
+1
+γ−1
+�
+�����
+,
+B1 =
+�
+�����
+u1
+ρ
+0
+0
+0
+u2
+1
+2ρu1
+0
+0
+1
+u1u2 ρu2 ρu1
+0
+0
+u1u3 ρu3
+0 ρu1
+0
+0
+γp
+γ−1
+0
+0
+γu1
+γ−1
+�
+�����
+,
+B2 =
+�
+�����
+u2
+0
+ρ
+0
+0
+u1u2 ρu2 ρu1
+0
+0
+u2
+2
+0 2ρu2
+0
+1
+u2u3
+0
+ρu3 ρu2
+0
+0
+0
+γp
+γ−1
+0
+γu2
+γ−1
+�
+�����
+,
+B3 =
+�
+�����
+u3
+0
+0
+ρ
+0
+u1u3 ρu3
+0
+ρu1
+0
+u2u3
+0 ρu3 ρu2
+0
+u2
+3
+0
+0 2ρu3
+1
+0
+0
+0
+γp
+γ−1
+γu3
+γ−1
+�
+�����
+,
+Ci =
+�
+�����
+0 0 0 0
+0
+0 0 0 0
+0
+0 0 0 0
+0
+0 0 0 0
+0
+0 0 0 0 −ui
+�
+�����
+,
+δCi =
+�
+�����
+0 0 0 0
+0
+0 0 0 0
+0
+0 0 0 0
+0
+0 0 0 0
+0
+0 0 0 0 −δui
+�
+�����
+.
+The full adjoint Navier-Stokes equations, in particular, the friction terms,
+are derived and discussed in [11]. The two-dimensional adjoint Euler equations
+can be found in [15].
+status: draft for review
+last modified: January 23, 2023 by (ML)
+

4tE4T4oBgHgl3EQfbgyP/content/tmp_files/2301.05074v1.pdf.txt

@@ -0,0 +1,351 @@
+Identification of light leptons and pions in the electromagnetic calorimeter of Belle II
+Anja Novosela,b, Abtin Narimani Charanc, Luka ˇSanteljb,a, Torben Ferberd, Peter Kriˇzanb,a, Boˇstjan Golobe,a
+aJoˇzef Stefan Institute, Ljubljana, Slovenia
+bFaculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
+cDeutsches Elektronen-Synchrotron (DESY), Hamburg, Germany
+dKarlsruhe Institute of Technology (KIT) , Karlsruhe, Germany
+eUniversity of Nova Gorica, Nova Gorica, Slovenia
+Abstract
+The paper discusses new method for electron/pion and muon/pion separation in the Belle II detector at transverse momenta below
+0.7 GeV/c, which is essential for efficient measurements of semi-leptonic decays of B mesons with tau lepton in the final state. The
+method is based on the analysis of patterns in the electromagnetic calorimeter by using a Convolutional Neural Network (CNN).
+Keywords: Electromagnetic calorimeter, Particle identification, Convolutional Neural Network
+1. Introduction
+Searches for New Physics at the intensity frontier are based
+on very precise measurements of rare processes within the Stan-
+dard Model. Of particular interest, because of persistent hints of
+Lepton Flavour Universality (LFU) violation, are semi-leptonic
+decays of B mesons, e.g. decays mediated by the b → cτ+ντ
+transitions with a tau lepton in the final state and decays in-
+volving b → sµ+µ− and b → se+e− transitions. In decays with
+tau lepton in the final state, the tau lepton must be reconstructed
+from its long-lived decay products, for example from the decays
+τ− → µ−¯νµντ or τ− → e−¯νeντ. In the Belle II experiment [1, 2],
+the momentum spectrum of light leptons from tau decays is
+rather soft, a sizable fraction being below 0.7 GeV/c. One of
+the crucial steps in the analysis of these decays is identifying
+low momenta light leptons (e or µ) from hadronic background
+(mostly π). The simplest baseline feature for separating elec-
+trons from other charged particles (muons and pions) is E/p,
+the ratio between the energy measured in the electromagnetic
+calorimeter and the reconstructed momentum of topologically
+matched charged track. This variable provides an excellent sep-
+aration for particles with p > 1 GeV/c, but due to increased en-
+ergy losses from bremsstrahlung for low momentum electrons,
+the separation is less distinct [3]. Muons are identified in the
+KL and muon system. However, its efficiency is very poor for
+low momentum muons that are out of acceptance of the ded-
+icated sub-detector. Other sub-detectors designed for particle
+identification, the time of propagation detector and the aerogel
+ring-imaging Cherenkov detector, are not able to provide effi-
+cient µ/π separation in this momentum range because at low
+momenta multiple scattering in the material of the detector as
+well as the material in front of it blurs the pattern considerably.
+Our main goal is to improve the identification of low momen-
+tum leptons using the information of energy deposition in the
+electromagnetic calorimeter in a form of images. As a classifier
+we are using a Convolutional Neural Network (CNN), a power-
+ful machine learning technique designed for working with two-
+dimensional images. Using CNN on the images allows us to ac-
+cess the information on the shape of the energy deposition with-
+out depending on cluster reconstruction or track-cluster match-
+ing.
+In what follows, we will describe the electromagnetic
+calorimeter of Belle II, discuss the analysis of simulated pion,
+muon and electron patterns in the electromagnetic calorimeter,
+and present the results.
+2. Electromagnetic calorimeter of Belle II
+The Belle II detector is a large-solid-angle magnetic spec-
+trometer designed to reconstruct the products of collisions pro-
+duced by the SuperKEKB collider. The detector consists of
+several sub-detectors arranged around the interaction point in
+cylindrical geometry: the innermost Vertex Detector (VXD)
+used for reconstructing decay vertices, a combination of the
+Pixel Detector (PXD) and Silicon Vertex Detector (SVD); the
+Central Drift Chamber (CDC) is the main tracking system; the
+Time of Propagation (TOP) detector in the barrel region and
+the Aerogel Ring-Imaging Cherenkov detector (ARICH) in the
+forward endcap region are used for hadron identification; the
+Electromagnetic Calorimeter (ECL) is used to measure the en-
+ergy of photons and electrons and the outermost K-Long and
+Muon (KLM) detector detects muons and neutral K0
+L mesons
+[1].
+The sub-detector relevant for this work is the ECL, more
+specifically its central barrel region barrel region which con-
+sists of 6624 CsI(Tl) scintillation crystals, covering the po-
+lar angle region 32.2◦ < θ < 128.7◦ with respect to the
+beam axis. A solenoid surrounding the calorimeter generates
+a uniform 1.5 T magnetic field filling its inner volume [2].
+We are mainly interested in the transverse momentum range
+0.28 < pT < 0.7 GeV/c, where the minimal pT threshold en-
+sures the tracks are within the ECL barrel region acceptance.
+Preprint submitted to Nucl. Instr. Meth. A
+January 13, 2023
+arXiv:2301.05074v1  [hep-ex]  12 Jan 2023
+
+Currently, two methods for the particle identification in the ECL
+are available. The first method relies exclusively on the ratio
+of the energy deposited by a charged particle in the ECL and
+the reconstructed momentum of topologically matched charged
+track, E/p. While for electrons this variable enables powerful
+discrimination, as electrons completely deposit their energy in
+the ECL, the µ/π separation is strongly limited, especially for
+low-momentum particles with a broader E/p distribution as can
+be seen on Fig. 1. The second method uses Boosted Decision
+Trees (BDT) with several expert-engineered observables char-
+acterising the shower shape in the ECL [4].
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+E/p [c]
+0
+2
+4
+6
+8
+Events (normalised / (0.02 c))
+Belle II Simulation, ECL barrel, 0.28 
+ pT < 0.7 GeV/c
+e
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+E/p [c]
+0
+1
+2
+3
+4
+5
+Events (normalised / (0.02 c))
+Belle II Simulation, ECL barrel, 0.28 
+ pT < 0.7 GeV/c
+Figure 1: Distribution of E/p for simulated single particle candidates: e
+(green), µ (red) and π (blue) for 0.28 ≤ pT < 0.7 GeV/c in the ECL barrel
+region.
+3. Analysis of the patterns in the electromagnetic calorime-
+ter
+Our proposed method to improve the identification of low-
+momentum leptons is to exploit the specific patterns in the spa-
+tial distribution of energy deposition in the ECL crystals us-
+ing a Convolutional Neural Network (CNN)1. The images are
+consistent with the 11 x 11 neighbouring crystals around the
+entry point of the extrapolated track into the ECL, where each
+pixel corresponds to an individual ECL crystal and pixel inten-
+sity to the energy deposited by charged particle in the crystal.
+Examples of the obtained images are shown on Fig. 2. While
+electrons generate electromagnetic showers depositing the ma-
+jority of their energy in the ECL, the dominant interaction in
+CsI(Tl) for muons and pions in the aforementioned transverse-
+momentum range is ionization. Besides, pions can strongly in-
+teract with nuclei producing less localized images compared to
+muons [5].
+Energy [GeV]
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Belle II Simulation, ECL barrel, 0.28 
+ pT <  0.7 GeV/c
+Figure 2: Examples of simulated energy depositions and the average over 80000
+images for e (left), µ (middle) and π (right).
+For each binary classification we generated 1.5 × 106 events
+using the Belle II Analysis Software Framework [6], where the
+1CNN is built using TensorFlow software available from tensorflow.org.
+data set consists of the same number of signal (e or µ) and back-
+ground (π) events with uniformly distributed transverse mo-
+menta, polar angle and azimuthal angle. The two data sets were
+split on the train-validation-test set as 70 − 10 − 20% and we
+use the same CNN architecture for e/π and µ/π case. As an
+input to the convolutional layers we use 11 x 11 images. Before
+fully connected layers we add the information about pT and θID,
+where the later represents an integer number corresponding to
+the location of the ECL crystal and is in the network imple-
+mented as an embedding. To perform a binary classification,
+we have 1 neuron in the output layer with a sigmoid activation
+function that outputs the signal probability that the image was
+produced by a lepton.
+4. Performance
+To validate the performance of a binary classifier we use
+a Receiver Operating Characteristic (ROC) curve by plotting
+true positive rate (µ or e efficiency) against the false positive
+rate (π mis-ID rate). As the reference for the existing ECL
+information, we use the log-likelihood difference, a powerful
+discriminator between the competing hypotheses, defined as
+∆LLECL = log LECL
+e,µ
+− log LECL
+π
+based only on E/p [3] and
+BDT ECL using the shower-shape information from the ECL,
+thoroughly described in [4]. The ROC curves obtained by these
+three methods are shown on Fig. 3 for e/π and on Fig. 4 for µ/π
+classification.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ mis-ID rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+e efficiency
+Belle II Simulation, ECL barrel, 0.28 
+ pT < 0.5 GeV/c
+LLECL (AUC: 89.34)
+BDT ECL (AUC: 94.12)
+CNN (AUC: 99.35)
+0.0
+0.1
+0.6
+0.7
+0.8
+0.9
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ mis-ID rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+e efficiency
+Belle II Simulation, ECL barrel, 0.5 
+ pT < 0.7 GeV/c
+LLECL (AUC: 98.58)
+BDT ECL (AUC: 99.25)
+CNN (AUC: 99.86)
+0.0
+0.1
+0.6
+0.7
+0.8
+0.9
+1.0
+Figure 3: The performance of three different classifiers for e/π based on only
+ECL information: ∆LLECL, BDT ECL, and ∆LLCNN.
+2
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ mis-ID rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ efficiency
+Belle II Simulation, ECL barrel, 0.28 
+ pT < 0.5 GeV/c
+LLECL (AUC: 69.02)
+BDT ECL (AUC: 86.50)
+CNN (AUC: 93.56)
+0.0
+0.1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ mis-ID rate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ efficiency
+Belle II Simulation, ECL barrel, 0.5 
+ pT < 0.7 GeV/c
+LLECL (AUC: 69.65)
+BDT ECL (AUC: 79.89)
+CNN (AUC: 84.94)
+0.0
+0.1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 4: The performance of three different classifiers for µ/π based on only
+ECL information: ∆LLECL, BDT ECL, and ∆LLCNN.
+Looking at the shapes of ROC curves and the Area Under the
+Curve (AUC) values, it is evident that the CNN outperforms
+the existing classifiers, ∆LLECL and BDT ECL for both e/π and
+µ/π. The performance of the CNN drops with increasing mo-
+mentum as the path in the ECL gets shorter and the specific
+patterns in the images become less evident.
+5. Summary and outlook
+We can conclude there is more information in the ECL that is
+currently used for particle identification. We saw that the sep-
+aration between low-momentum light leptons and pions can be
+improved using a CNN on the ECL images. The newly pro-
+posed method looks very promising and worthwhile to be fur-
+ther developed. A comparison of the method presented in this
+article to a novel BDT-based analysis is a subject of a forthcom-
+ing publication [7].
+6. Acknowledgements
+We thank Anˇze Zupanc for his support with ideas and ad-
+vice in the early stages of the project. This work was supported
+by the following funding sources: European Research Coun-
+cil, Horizon 2020 ERC-Advanced Grant No. 884719; BMBF,
+DFG, HGF (Germany); Slovenian Research Agency research
+grants No. J1-9124, J1-4358 and P1-0135 (Slovenia).
+References
+[1] T. Abe et al., KEK Report 2010-1 (2010)
+[2] I. Adachi et al., Nucl. Instrum. Meth. A 907 (2018)
+[3] E. Kou et al., PTEP, Volume 2019, Issue 12, 123C01 (2019)
+[4] M. Milesi, J. Tan, P. Urquijo, EPJ Web of Conferences 245, 06023 (2020)
+[5] S. Longo, J. M. Roney et al., Nucl. Instrum. Meth. A 982 (2020)
+[6] T. Kuhr, C. Pulvermacher, M. Ritter et al., Comput Softw Big Sci 3, 1
+(2019)
+[7] M. Milesi et al., in preparation for Nucl. Instrum. Meth. A
+3
+

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+page_content=' Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Slovenia  bFaculty of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' University of Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
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+page_content=' Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Germany  dKarlsruhe Institute of Technology (KIT) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Karlsruhe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Germany  eUniversity of Nova Gorica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Nova Gorica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Slovenia  Abstract  The paper discusses new method for electron/pion and muon/pion separation in the Belle II detector at transverse momenta below  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='7 GeV/c, which is essential for efficient measurements of semi-leptonic decays of B mesons with tau lepton in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' The  method is based on the analysis of patterns in the electromagnetic calorimeter by using a Convolutional Neural Network (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  Keywords: Electromagnetic calorimeter, Particle identification, Convolutional Neural Network  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Introduction  Searches for New Physics at the intensity frontier are based  on very precise measurements of rare processes within the Stan-  dard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Of particular interest, because of persistent hints of  Lepton Flavour Universality (LFU) violation, are semi-leptonic  decays of B mesons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' decays mediated by the b → cτ+ντ  transitions with a tau lepton in the final state and decays in-  volving b → sµ+µ− and b → se+e− transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' In decays with  tau lepton in the final state, the tau lepton must be reconstructed  from its long-lived decay products, for example from the decays  τ− → µ−¯νµντ or τ− → e−¯νeντ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' In the Belle II experiment [1, 2],  the momentum spectrum of light leptons from tau decays is  rather soft, a sizable fraction being below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='7 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' One of  the crucial steps in the analysis of these decays is identifying  low momenta light leptons (e or µ) from hadronic background  (mostly π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' The simplest baseline feature for separating elec-  trons from other charged particles (muons and pions) is E/p,  the ratio between the energy measured in the electromagnetic  calorimeter and the reconstructed momentum of topologically  matched charged track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' This variable provides an excellent sep-  aration for particles with p > 1 GeV/c, but due to increased en-  ergy losses from bremsstrahlung for low momentum electrons,  the separation is less distinct [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Muons are identified in the  KL and muon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' However, its efficiency is very poor for  low momentum muons that are out of acceptance of the ded-  icated sub-detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Other sub-detectors designed for particle  identification, the time of propagation detector and the aerogel  ring-imaging Cherenkov detector, are not able to provide effi-  cient µ/π separation in this momentum range because at low  momenta multiple scattering in the material of the detector as  well as the material in front of it blurs the pattern considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  Our main goal is to improve the identification of low momen-  tum leptons using the information of energy deposition in the  electromagnetic calorimeter in a form of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' As a classifier  we are using a Convolutional Neural Network (CNN), a power-  ful machine learning technique designed for working with two-  dimensional images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Using CNN on the images allows us to ac-  cess the information on the shape of the energy deposition with-  out depending on cluster reconstruction or track-cluster match-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  In what follows, we will describe the electromagnetic  calorimeter of Belle II, discuss the analysis of simulated pion,  muon and electron patterns in the electromagnetic calorimeter,  and present the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Electromagnetic calorimeter of Belle II  The Belle II detector is a large-solid-angle magnetic spec-  trometer designed to reconstruct the products of collisions pro-  duced by the SuperKEKB collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' The detector consists of  several sub-detectors arranged around the interaction point in  cylindrical geometry: the innermost Vertex Detector (VXD)  used for reconstructing decay vertices, a combination of the  Pixel Detector (PXD) and Silicon Vertex Detector (SVD);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' the  Central Drift Chamber (CDC) is the main tracking system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' the  Time of Propagation (TOP) detector in the barrel region and  the Aerogel Ring-Imaging Cherenkov detector (ARICH) in the  forward endcap region are used for hadron identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' the  Electromagnetic Calorimeter (ECL) is used to measure the en-  ergy of photons and electrons and the outermost K-Long and  Muon (KLM) detector detects muons and neutral K0  L mesons  [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  The sub-detector relevant for this work is the ECL, more  specifically its central barrel region barrel region which con-  sists of 6624 CsI(Tl) scintillation crystals, covering the po-  lar angle region 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='2◦ < θ < 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='7◦ with respect to the  beam axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' A solenoid surrounding the calorimeter generates  a uniform 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='5 T magnetic field filling its inner volume [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  We are mainly interested in the transverse momentum range  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='28 < pT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='7 GeV/c, where the minimal pT threshold en-  sures the tracks are within the ECL barrel region acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  Preprint submitted to Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' A  January 13, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='05074v1  [hep-ex]  12 Jan 2023  Currently, two methods for the particle identification in the ECL  are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' The first method relies exclusively on the ratio  of the energy deposited by a charged particle in the ECL and  the reconstructed momentum of topologically matched charged  track, E/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' While for electrons this variable enables powerful  discrimination, as electrons completely deposit their energy in  the ECL, the µ/π separation is strongly limited, especially for  low-momentum particles with a broader E/p distribution as can  be seen on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' The second method uses Boosted Decision  Trees (BDT) with several expert-engineered observables char-  acterising the shower shape in the ECL [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='2  E/p [c]  0  2  4  6  8  Events (normalised / (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='02 c))  Belle II Simulation, ECL barrel, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='28  pT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='7 GeV/c  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='2  E/p [c]  0  1  2  3  4  5  Events (normalised / (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='02 c))  Belle II Simulation, ECL barrel, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='28  pT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='7 GeV/c  Figure 1: Distribution of E/p for simulated single particle candidates: e  (green), µ (red) and π (blue) for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='28 ≤ pT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='7 GeV/c in the ECL barrel  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Analysis of the patterns in the electromagnetic calorime-  ter  Our proposed method to improve the identification of low-  momentum leptons is to exploit the specific patterns in the spa-  tial distribution of energy deposition in the ECL crystals us-  ing a Convolutional Neural Network (CNN)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' The images are  consistent with the 11 x 11 neighbouring crystals around the  entry point of the extrapolated track into the ECL, where each  pixel corresponds to an individual ECL crystal and pixel inten-  sity to the energy deposited by charged particle in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  Examples of the obtained images are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' While  electrons generate electromagnetic showers depositing the ma-  jority of their energy in the ECL, the dominant interaction in  CsI(Tl) for muons and pions in the aforementioned transverse-  momentum range is ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Besides, pions can strongly in-  teract with nuclei producing less localized images compared to  muons [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  Energy [GeV]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='10  Belle II Simulation, ECL barrel, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='28  pT <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='7 GeV/c  Figure 2: Examples of simulated energy depositions and the average over 80000  images for e (left), µ (middle) and π (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  For each binary classification we generated 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='5 × 106 events  using the Belle II Analysis Software Framework [6], where the  1CNN is built using TensorFlow software available from tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  data set consists of the same number of signal (e or µ) and back-  ground (π) events with uniformly distributed transverse mo-  menta, polar angle and azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' The two data sets were  split on the train-validation-test set as 70 − 10 − 20% and we  use the same CNN architecture for e/π and µ/π case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' As an  input to the convolutional layers we use 11 x 11 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Before  fully connected layers we add the information about pT and θID,  where the later represents an integer number corresponding to  the location of the ECL crystal and is in the network imple-  mented as an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' To perform a binary classification,  we have 1 neuron in the output layer with a sigmoid activation  function that outputs the signal probability that the image was  produced by a lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Performance  To validate the performance of a binary classifier we use  a Receiver Operating Characteristic (ROC) curve by plotting  true positive rate (µ or e efficiency) against the false positive  rate (π mis-ID rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' As the reference for the existing ECL  information, we use the log-likelihood difference, a powerful  discriminator between the competing hypotheses, defined as  ∆LLECL = log LECL  e,µ  − log LECL  π  based only on E/p [3] and  BDT ECL using the shower-shape information from the ECL,  thoroughly described in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' The ROC curves obtained by these  three methods are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' 3 for e/π and on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
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+page_content='89)  CNN (AUC: 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
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+page_content='0  Figure 4: The performance of three different classifiers for µ/π based on only  ECL information: ∆LLECL, BDT ECL, and ∆LLCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  Looking at the shapes of ROC curves and the Area Under the  Curve (AUC) values, it is evident that the CNN outperforms  the existing classifiers, ∆LLECL and BDT ECL for both e/π and  µ/π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' The performance of the CNN drops with increasing mo-  mentum as the path in the ECL gets shorter and the specific  patterns in the images become less evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Summary and outlook  We can conclude there is more information in the ECL that is  currently used for particle identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' We saw that the sep-  aration between low-momentum light leptons and pions can be  improved using a CNN on the ECL images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' The newly pro-  posed method looks very promising and worthwhile to be fur-  ther developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' A comparison of the method presented in this  article to a novel BDT-based analysis is a subject of a forthcom-  ing publication [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Acknowledgements  We thank Anˇze Zupanc for his support with ideas and ad-  vice in the early stages of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' This work was supported  by the following funding sources: European Research Coun-  cil, Horizon 2020 ERC-Advanced Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' 884719;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' BMBF,  DFG, HGF (Germany);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Slovenian Research Agency research  grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' J1-9124, J1-4358 and P1-0135 (Slovenia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content='  References  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
+page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQfbgyP/content/2301.05074v1.pdf'}
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+MNRAS 000, 1–14 (2021)
+Preprint 4 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Multi-wavelength study of TeV blazar 1ES 1218+304 using
+gamma-ray, X-ray and optical observations
+Rishank Diwan,1⋆ Raj Prince,2 Aditi Agarwal,3 Debanjan Bose,4† Pratik Majumdar,5
+Aykut Özdönmez,6 Sunil Chandra,7,8 Rukaiya Khatoon,8 Ergün Ege,9
+1Laboratory for Space Research, The University of Hong Kong, 405B Cyberport 4, 100 Cyberport Road, Cyberport, Hong Kong
+2Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
+3 Raman Research Institute, C. V. Raman Avenue, Sadashivanagar, Bengaluru - 560080, India
+4 S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata-700106
+5 Saha Institute of Nuclear Physics, a CI of Homi Bhabha National Institute, Kolkata 700064, West Bengal, India
+6 Ataturk University, Faculty of Science, Department of Astronomy and Space Science, 25240, Yakutiye, Erzurum
+7 South African Astronomical Observatory, Observatory Road, Observatory, Cape Town 7925, South Africa
+8 Center for Space Research, North-West University, Potchefstroom, 2520, South Africa
+9 Istanbul University, Faculty of Science, Department of Astronomy and Space Sciences, 34116, Beyazıt, Istanbul, Turkey
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We report the multi-wavelength study for a high-synchrotron-peaked BL Lac 1ES 1218+304 using near-simultaneous
+data obtained during the period from January 1, 2018, to May 31, 2021 (MJD 58119-59365) from various instruments
+including Fermi-LAT, Swift-XRT, AstroSat, and optical from Swift-UVOT & TUBITAK observatory in Turkey. The
+source was reported to be flaring in TeV γ-ray during 2019 but no significant variation in Fermi-LAT is observed. A
+minute scale variability is seen in SXT light curve suggesting a compact emission region for their variability. However,
+Hour’s scale variability is observed in the γ-ray light curve. A "softer-when-brighter" trend is observed in γ-ray and an
+opposite trend is seen in X-ray suggesting both emissions are produced via two different processes as expected from an
+HBL source. We have chosen the two epochs in January 2019 to study and compare their physical parameters. A joint
+fit of SXT and LAXPC provides a great constraint on the synchrotron peak roughly estimated to be ∼2.68×1017 Hz.
+A clear shift in the synchrotron peak is observed from 1017−18 to 1020 Hz revealing its extreme nature or behaving like
+an EHBL-type source. The optical observation provides color-index variation as "blue-when-brighter". The broadband
+SED is fitted with a single-zone SSC model and their parameters are discussed in the context of a TeV blazar and
+possible mechanism behind the broadband emission.
+Key words:
+galaxies: active – galaxies: jets – gamma-rays: galaxies – radiation mechanisms: non-thermal – BL
+Lacertae objects: individual: 1ES 1218+304
+1 INTRODUCTION
+Active galactic nuclei (AGN) host a supermassive black hole
+(SMBH) at the center which accretes matter from the sur-
+rounding. The matters are in Keplerian orbit and fall into
+the SMBH via an accretion disk. The mechanism proposed
+in Blandford & Znajek (1977) suggests that the magnetic
+field lines from the accretion disk get twisted and collimated
+due to the high spin of SMBH and eject the matter through
+a bipolar jet perpendicular to the accretion disk plane. Later,
+the AGNs were classified based on how they are viewed
+commonly known as the AGN unification scheme (Urry &
+Padovani 1995). Blazars are a subclass of active galactic nu-
+clei that have their relativistic jet pointed to the observer.
+⋆ E-mail: [email protected]
+† E-mail: [email protected]
+They are characterized by rapid variability from hours to
+days’ timescales across all wavelengths, high polarization, and
+superluminal jet speeds. Blazars can be further subdivided
+into two classes: flat spectrum radio quasars (FSRQs) and
+BL Lacertae (BL Lac) objects. The broad-band continuum
+spectra of blazars are dominated by non-thermal emission.
+The spectral energy distribution of blazars is characterized
+by a double hump structure: the first hump is generally at-
+tributed to the synchrotron radiation in the radio to X-ray
+bands whereas there is intense debate about the origin of
+the second hump. The commonly accepted emission mech-
+anism is via inverse Compton scattering of the low-energy
+photons by high-energy electrons in the system from GeV
+to TeV energies. There are alternative scenarios proposed
+by several authors which involve hadronic interactions pro-
+ducing neutral pions. These pions decay to generate photons
+in the GeV-TeV energies (Mannheim 1993; Aharonian 2000;
+© 2021 The Authors
+arXiv:2301.00991v1  [astro-ph.HE]  3 Jan 2023
+
+2
+R. Diwan et al.
+Böttcher et al. 2013). The BL Lac-type sources are further
+subdivided into three main classes depending on the position
+of their low-energy peak. If the synchrotron peak is observed
+at < 1014Hz, those BL Lacs are called low-frequency peaked
+BL Lacs (LBLs). If the synchrotron peak is observed be-
+tween 1014Hz and 1015Hz, then they are called intermediate-
+frequency peaked BL Lacs (IBLs). Finally, BL Lacs with syn-
+chrotron peak ≥ 1015Hz is called high-frequency peaked BL
+Lacs (HBLs). There is also a newly defined class of ultra-
+high-frequency peaked BL Lacs (UHBLs) with the spectral
+peak of the second bump (high energy peak) in the SED lo-
+cated at an energy of 1 TeV or above. These blazars are also
+known as "extreme blazars" or EHBLs. (Abdo et al. 2010).
+Multiwavelength observation of blazars is a very important
+tool for investigating the various properties of the blazars and
+the jet. For example, the shortest variability timescale allows
+one to put strong constraints on the size of the emission re-
+gion of the blazar. The location of the emission region along
+the jet axis is another challenging problem in blazar physics.
+Many studies have been done in the past to locate the emis-
+sion region, in some cases, it has been found that the emission
+happens very close to the SMBH within the broad-line region
+(BLR) (Prince 2020; Prince et al. 2021). However, in some
+studies, it has been proposed to be at higher distances be-
+yond the broad-line region (Cao & Wang 2013; Nalewajko
+et al. 2014; Barat et al. 2022). The break or curvature in the
+γ-ray spectrum above 10-20 GeV suggests the emission region
+within the BLR as the BLR is opaque to high energy pho-
+tons above 10 GeV ( Liu & Bai 2006). The cross-correlation
+studies among the various wavebands are another way to lo-
+cate the emission region along the jet axis. In many studies,
+it has been reported that simultaneous broadband emissions
+generally have a co-spatial origin. However, in some cases, a
+significant time lag has been reported strongly suggesting the
+different locations for the different emissions (Prince 2019).
+In the first case scenario, one zone emission model is favored
+to explain the broadband SED, and in the later case, the
+multi-zone emission model is preferred (Prince et al. 2019).
+The production of high-energy γ-rays in blazar suggests an
+acceleration of charged particles to very high energy and
+many models have been proposed to explain the acceleration.
+The most accepted mechanisms are the diffusive shock accel-
+eration (Schlickeiser 1989a,b) and the magnetic re-connection
+(Shukla & Mannheim 2020). In many studies shock accelera-
+tion has been favored which also demands the emission region
+close to the SMBH within the BLR because the shocks are
+produced and are strong at the base of the jet. On the other
+hand, the magnetic reconnection happens due to external per-
+turbation and hence demands the jet to be less collimated i.e.
+the emission region is farther from the base.
+In this paper, we report on a multiwavelength study of the
+TeV blazar 1ES1218+304 to understand the broadband prop-
+erties of the source. It is located at a redshift, z = 0.182 with
+R.A. = 12 21 26.3 (hh mm ss), Dec = +30 11 29 (dd mm ss).
+It has been observed in TeV energy with VERITAS (Fortin
+2008, Acciari et al. 2009) and MAGIC (Albert et al. 2006,
+Lombardi et al. 2011) and are part of TeV Catalog1.
+The paper is arranged in the following way. We discuss the
+multiwavelength observations and the data analysis proce-
+1 http://tevcat.uchicago.edu/
+dures from different instruments used in this study in Section
+2. In section 3, we have discussed the results from Astrosat
+alone and the broadband light curves and spectral energy dis-
+tributions at length. In Section 4 we summarise and discussed
+the important findings in the context of blazar physics and
+eventually conclude our work in Section 5.
+2 MULTIWAVELENGTH OBSERVATIONS,
+DATA ANALYSIS AND DATA REDUCTION
+The following section describes the data analysis technique
+used to generate a multi-waveband light curve. In the sub-
+sections, we provide a description of the data analysis tech-
+nique of γ-ray data collected from Fermi-Lat. X-ray, and
+UV-optical data were collected from Swift-XRT and Swift-
+UVOT. Also, soft X-ray and hard X-ray data were collected
+from AstroSat-SXT and AstroSat-LAXPC, respectively and
+Optical Data from TUBITAK National Observatory.
+2.1 Fermi-LAT γ-ray Observatory
+Large Area Telescope (LAT) is a gamma-ray telescope placed
+on Fermi gamma-ray space observatory2 which was launched
+in 2008. It has a working energy range of 20 MeV to 1
+TeV with a field of view of 2.4 Sr (Atwood et al. 2009).
+The orbital period of the telescope is around ∼ 96 mins
+in each hemisphere and covers the entire sky in total ∼ 3
+hr. Blazar 1ES 1218+304 is continuously being monitored
+since 2008. In this study, we have analyzed the data from
+1st January 2018 - 31st May 2021 when the source was
+reported to be flaring in gamma-ray (January 2019). The
+analysis was performed using Fermipy v0.17.43(Wood et al.
+2021) and the standard Fermi tools software (Fermitools
+v1.2.23)4 between 0.3-300 GeV. A 15◦ circular region was
+chosen around the source to extract the photon events with
+evclass=128 and evtype=3 and the time intervals were re-
+stricted using ‘(DATA_QUAL>0)&&(LAT_CONFIG==1)’
+as recommended by the Fermi-LAT team in the fermitools
+documentation. The source model file was generated using
+the Fermi 4FGL catalog (Abdollahi et al. 2020) and the back-
+ground gamma-ray emission was taken care of by using the
+gll_iem_V07.fits file along with the isotropic background
+emission by using the iso_P8R3_SOURCE_V2_v1.txt file. In
+addition, the zenith angle cut was chosen as 90◦ to reduce the
+contamination from the Earth limb’s γ-ray. The source and
+background were modeled by the binned Likelihood method.
+Initially, the spectral parameters of all the sources were kept
+free to optimize the γ-ray emission from them. Eventually,
+we generated the γ-ray light curves for 7, 15, and 30 days
+of binning for our scientific purpose. To extract lightcurve
+and perform spectral fitting normalization of the sources only
+within 2◦ of ROI were kept free, and the rest of the param-
+eters and other source models were frozen, except that of
+Source of Interest, in this case, blazar 1ES 1218+304 and a
+high flux source 4FGL J1217.9+3007, with an offset of 0.753◦
+from 1ES 1218+304, which constitutes to 10 parameters for
+2 https://fermi.gsfc.nasa.gov/
+3 Fermipy webpage
+4 Fermtools Github page
+MNRAS 000, 1–14 (2021)
+
+Multi-wavelength study of 1ES 1218+304
+3
+likelihood analysis. PowerLaw model was used for the source
+as given below:
+dN(E)
+dE
+= No ×
+� E
+Eo
+�−α
+(1)
+where Eo and No are the scale factor and the prefactor, re-
+spectively provided in the 4FGL catalog and α is the spectral
+index.
+2.2 AstroSat
+On January 03, 2019 MAGIC reported a gamma-ray activ-
+ity and detection of very high energy γ ray from blazar 1ES
+1218+304 (Mirzoyan 2019). Later, VERITAS also detected a
+γ-ray flare from this source (Mukherjee & VERITAS Collab-
+oration 2019). Following these two events, we proposed a tar-
+get of opportunity proposal in India’s first space-based multi-
+wavelength observatory, AstroSat5. Observations were car-
+ried out from 17th to 20th January with a soft-Xray telescope
+(SXT) and large area X-ray proportional counter (LAXPC).
+2.2.1 SXT
+The SXT working energy range is 0.3-7.0 keV and the ob-
+servation was performed with photon counting mode (PC).
+The level-1 data was downloaded from the webpage and fur-
+ther reduction was performed with the latest SXT pipeline,
+sxtpipeline1.4b (Release Date: 2019-01-04). It produces
+the cleaned level-2 data products which were used for fur-
+ther analysis (Singh et al. 2016, Singh et al. 2017). The ob-
+servations were done in various orbits and therefore it was
+merged together with the help of SXTEVTMERGERTOOL. The
+X-ray light curve is extracted using XSELECT with a circular
+region of 16′ centered on the source. The energy selection
+of 0.3-7.0 keV was applied in XSELECT itself using the chan-
+nel filtering through pha_cutoff filter. The source spectrum
+was extracted for 0.3-7.0 keV energy range and the back-
+ground spectrum file was used provided by the AstroSat
+SkyBkg_comb_EL3p5_Cl_Rd16p0_v01.pha. The spectrum was
+grouped in GRPPHA in order to have good photon statistics in
+each bin. The ancillary response file (arf) was generated using
+sxtARFModule and the RMF file (sxt_pc_mat_g0to12.rmf)
+was provided by the SXT-POC (Payload Operation Cen-
+ter) team. Eventually, the X-ray spectra from 0.3-7.0 KeV
+with proper background and response files were loaded in
+XSPEC and fitted with the simple absorbed power-law and
+log-parabola spectral models with the correction of ISM ab-
+sorption model at NH = 1.91×1020 cm−2 (HI4PI Collabora-
+tion et al. 2016).
+2.2.2 LAXPC
+LAXPC works in the hard X-ray energy range from 3.0-80.0
+keV (Yadav et al. 2016) consisting of three identical detec-
+tors namely LAXPC10, LAXPC20, and LAXPC30. Unfor-
+tunately, LAXPC 10 was operating at a lower gain during
+the time of observation period. Also, the LAXPC30 detec-
+tor has a gain instability issue caused by substantial gas
+5 https://www.isro.gov.in/AstroSat.html
+leakage. Therefore, we used only LAXPC20 for the analy-
+sis, and the corresponding results are presented here. The
+Level-1 data were processed using the LaxpcSoft package
+available in AstroSat Science Support Cell (ASSC)6. We
+generated the Level-2 combined event file using the com-
+mand laxpc_make_event. During the data processing, a
+good time interval was applied to exclude the time inter-
+vals corresponding to the Earth occultation periods, SAA
+passage, and standard elevation angle screening criteria
+by using the laxpc_make_stdgti tool. Finally, the tools
+laxpc_make_spectra and laxpc_make_lightcurve were used
+to produce the spectra and lightcurve of the source, using the
+gti file. We restricted the spectra to the energy range of
+4-20 keV since the background dominates the spectra above
+this energy. In the spectral analysis, a 3% systematic un-
+certainty was added to the data. The obtained lightcurve is
+not background subtracted, therefore we estimated the back-
+ground following the faint source routine (Misra et al. 2021).
+However, due to insignificant variations observed in the ex-
+tracted lightcurve from LAXPC20, we did not use them in
+our study.
+2.3 The Neil Gehrels Swift Observatory
+Simultaneous to AstroSat, blazar 1ES 1218+304 was also ob-
+served in X-ray with Swift-XRT and in optical-UV by Swift-
+UVOT telescopes7. It provides a unique opportunity to have
+simultaneous broadband light curves and spectrum which is
+important to decipher the cause behind the flare and the
+broadband emission.
+2.3.1 XRT
+X-ray telescope (XRT) works in an energy range between 0.3-
+10.0 keV. Multiple observations were done during this period
+with an average of 2ks exposure. We have analyzed the data
+following the standard Swift xrtpipeline and the details can
+be found on Swift webpage8. The cleaned event files were pro-
+duced and a circular region of 10” was chosen for the source
+and background around the source and away from the source.
+Tool XSELECT was used to extract the source light curve and
+the spectrum. The spectrum was binned by using the tool
+GRPPHA to have a sufficient number of counts in each bin. A
+proper ancillary response file (ARF) and the redistribution
+matrix files (RMF) were used to model the X-ray spectra in
+XSPEC. A simple unabsorbed power law was used to fit the X-
+ray 0.3-10.0 keV spectra and extract the X-ray flux. The soft
+X-ray (below 1 keV) is prone to go through interstellar ab-
+sorption in Milky-way and hence a correction is applied with
+NH = 1.91×1020 cm−2 (HI4PI Collaboration et al. 2016).
+2.3.2 UVOT
+Having an ultraviolet-optical telescope has the advantage of
+getting simultaneous observations to X-ray. UVOT has six
+filters namely U, B, and V in optical and W1, M2, and W2
+in the ultraviolet band. The image files were opened in DS9
+6 http://astrosat-ssc.iucaa.in
+7 https://swift.gsfc.nasa.gov/
+8 https://www.swift.ac.uk/analysis/xrt/
+MNRAS 000, 1–14 (2021)
+
+4
+R. Diwan et al.
+Table 1. Best fit spectral parameters of 1ES 1218+304 from SXT observations of 17-20 January 2019. X-ray flux is presented in the unit
+(erg cm−2 s−1). The spectrum is fitted with both the power-law and log-parabola models. In the last row, we show the joint fit of the
+SXT and LAXPC spectrum. We also added a 3% systematic in the fit as suggested by the AstroSat team. The parameters are compared
+for free and fixed NH (HI4PI Collaboration et al. 2016) values. The overall fit provide better fit with free NH.
+Model
+Parameters
+Value
+Power-law
+Fixed nH
+Free nH
+TBabs
+NH(1022cm−2)
+0.0191
+0.057±0.005
+Index
+Γ
+1.95±0.01
+2.11±0.02
+Flux
+F0.3−10.0 keV
+(1.427 ± 0.004) × 10−10
+(1.474 ± 0.006) × 10−10
+χ2/dof
+777/434
+595.75/433
+Logparabola
+TBabs
+NH(1022cm−2)
+0.0191
+0.075±0.014
+Index
+α
+1.90±0.02
+2.21±0.08
+β
+0.28±0.04
+0.15±0.11
+Flux
+F0.3−10.0 keV
+(1.300 ± 0.009) × 10−10
+(1.585 ± 0.037) × 10−10
+χ2/dof
+642.28/433
+590.55/432
+Logparabola
+joint fit
+SXT + LAXPC
+TBabs
+NH(1022cm−2)
+0.0191
+0.042±0.010
+Index
+α
+1.85±0.02
+1.98±0.06
+β
+0.33±0.03
+0.22±0.06
+Norm
+0.0262±0.0002
+0.0281 ± 0.0009
+Constant factor
+-
+0.96±0.04
+0.96±0.04
+χ2/dof
+601.16/402
+587.72/401
+software and the source and background region of 5" and 10"
+were selected around the source and away from the source,
+respectively. The task UVOTSOURCE has been used to get the
+magnitudes which were later corrected for galactic reddening,
+E(B-V)=0.0176 (Schlafly & Finkbeiner 2011) and converted
+into the fluxes using zero points and the conversion factor
+(Giommi et al. 2006).
+2.4 Optical
+The optical observations of our source were performed in the
+Johnson BVRI bands using the three ground-based facilities
+in Turkey, namely, 0.6m RC robotic (T60) and the 1.0m RC
+(T100) telescopes at TUBITAK National Observatory, and
+0.5m RC telescope at Ataturk University in Turkey. Techni-
+cal details of these telescopes are explained in Agarwal et al.
+(2022). The standard data reduction of all CCD frames, i.e.
+the bias subtraction, twilight flat-fielding, and cosmic-ray re-
+moval, was done as mentioned in (Agarwal et al. 2019a).
+2.5 Archival
+We have used the archival optical data from ASAS-SN (All-
+Sky Automated Survey for Supernovae) (Shappee et al. 2014;
+Kochanek et al. 2017).We have also used long-term high flux
+observation in UV/Optical range from NASA/IPAC Extra-
+galactic Database (NED)9 for providing the reference points
+in our SED analysis. We have also extracted the NuSTAR
+SED data points from (Sahakyan 2020) and plotted them
+alongside our SED analysis.
+9 https://ned.ipac.caltech.edu/
+3 RESULTS
+In this section, we provide the main results of our work using
+the above broadband observations. We have explained various
+characteristics of broadband light curves and spectral energy
+distributions.
+3.1 Astrosat results
+Astrosat observations in SXT and LAXPC were done dur-
+ing 17-20 January 2019 after two weeks of TeV detection.
+We have produced the SXT light curve and the spectrum
+as shown in Figure 1 and Figure 2 for 0.3-7.0 keV energy
+band. The source appears to be variable on a short-time
+scale and the corresponding fractional variability and vari-
+ability time is estimated in section 3.2. A spectrum is ex-
+tracted in the energy range of 0.3-7 keV and fitted with the
+power law and log-parabola models. The best-fit parame-
+ters are presented in Table 1. We started with a power-law
+with fixed hydrogen column density, NH = 0.0191×1020 cm−2
+and ended up getting χ2/dof = 777/434 with photon spec-
+tral index, Γ = 1.95±0.01 and 0.3-7 keV flux, F0.3−7keV =
+(14.27±0.04)×10−11 ergs/cm2/s. Next, we keep NH as a free
+parameter and the best fit value is estimated as 0.057±0.005
+in units of 1020 cm−2. The χ2/dof has improved to 595.75/433
+and the spectral index was found to be 2.11±0.02 with almost
+the same 0.3-7 keV flux. We repeat the same procedure with
+the log parabola model and with both the cases of fixed and
+free NH and it gives a better fit than the power law. With
+the free NH parameter we achieved a better fit with χ2/dof
+= 590.55/432 compared to the power-law case. The best-fit
+spectral index is 2.21±0.08 a bit softer than the power-law
+index. The details about the other parameters are provided
+in Table 1.
+MNRAS 000, 1–14 (2021)
+
+Multi-wavelength study of 1ES 1218+304
+5
+0
+20000
+40000
+60000
+80000
+100000 120000
+Time(s)
+1.6
+1.7
+1.8
+1.9
+2.0
+2.1
+2.2
+2.3
+Counts/sec
+SXT 0.3-7.0 keV
+Figure 1. AstroSat-SXT light curve for energy 0.3-7.0 keV. The
+bin size is taken as 856 sec.
+We could not get a good light curve in LAXPC but ex-
+tracted the spectrum from 4-20 keV. The SXT and LAXPC
+spectra are jointly fitted with Power law and Log-parabola
+models. In the case of the Power-law, we get the χ2/dof
+= 948.46/403 and 623.25/402 for fixed and free NH val-
+ues. In both cases, the reduced-χ2 is much higher than
+the case of Log-parabola (Table 1) and hence not pur-
+sued further. For the joint fit, we used the total model as
+constant*tbabs*logpar. The constant factor is fixed at 1.0
+for data group 1 and kept as a free parameter for data group
+2. The best fit value for the constant factor is 0.96±0.04 for
+both fixed and free NH. The overall reduced-χ2 is improved
+when the NH is free and it is estimated as 4.2±1.0 (×1020
+cm−2), almost two times higher than the fixed NH value. Fig-
+ure 3 shows the best fit plot with a log-parabola model. We
+found that the spectral index, α, and the curvature parame-
+ter, β are a bit different during fixed and free NH. The math-
+ematical representation of the log-parabolic model is given
+as,
+F(E) = K(E/E1)(−α+βlog(E/E1))ph cm−2 s−1 keV,
+(2)
+where K is the normalization and the E1 is the reference
+energy fixed at 1 keV. Using the best-fit parameters of the
+log-parabola model we can estimate the location of the syn-
+chrotron peak, which is given as Ep = E1 10(2−α)/2β keV.
+For α=1.98 and β=0.22, the Ep is estimated as 1.11 keV or
+2.68×1017 Hz. The peak of the synchrotron emission is mostly
+constrained by the X-ray as shown in Figure 3 which peaks
+at ∼ 2.68×1017 Hz.
+3.2 Broadband Light curves
+We have collected the γ-ray data between 2018 to 2021. The
+source was found to be in a flaring state in γ-ray during Jan
+2019. Simultaneous observation in Swift-XRT and UVOT
+also confirms the flaring behavior in X-ray as well as in
+optical-UV. On 02 January 2019 source was reported to be
+flaring in very high energy gamma-ray by MAGIC (Mirzoyan
+2019) which was followed by VERITAS (Mukherjee &
+VERITAS Collaboration 2019) and observation was done on
+4, 5, and 6 January 2019 show high flux state above 100 GeV
+and the corresponding period is marked by light pink color
+in Figure 4. We identify this period as Flare A. In X-ray
+10−3
+0.01
+0.1
+1
+normalized counts s−1 keV−1
+1
+0.5
+2
+5
+0.5
+1
+1.5
+2
+2.5
+ratio
+Energy (keV)
+Figure 2. The 0.3 - 7.0 keV energy spectrum of 1ES 1218+304
+fitted with Logparabola spectral model with free galactic absorp-
+tion. The SXT data were taken during the period 17-20 January
+2019.
+10−10
+2×10−11
+5×10−11
+ν Fν (ergs cm−2 s−1)
+1017
+1018
+2×1017
+5×1017
+2×1018
+1
+1.5
+2
+ratio
+Energy (Hz)
+Figure 3. The joint SXT (red) and LAXPC (blue) spectra are
+modeled together. The SXT energy range is taken as 0.3 - 7.0
+keV and LAXPC is taken from 3.0-20.0 keV. The joint spectra are
+fitted with a log parabola spectral model. Both spectra were taken
+simultaneously during the period of 17-20 January 2019.
+and optical source was reported to be historically bright
+with flux around ∼ 2×10−10 erg cm−2 s−1 in X-ray and
+with R band flux 2.35±0.05 mJy (Ramazani et al. 2019).
+We also proposed this source in India’s first space mission,
+AstroSat for broadband observation. Our observation was
+done between 17-20 January 2019. This period is marked
+as a vertical green line in Figure 4 and identified as Flare
+B. The first two panels of Figure 4 represent the long-term
+γ-ray (GeV) light curve and corresponding photon spectral
+index. The source is not very bright in Fermi-LAT but a
+clear variability in the flux is seen. Panel 3 & 4 represent the
+long-term Swift-XRT light curve and corresponding photon
+spectral index. A clear X-ray brightening during Jan 2019 is
+observed. During this period, we do not have many optical
+observations (panel 5), and hence it’s difficult to comment
+on the flux level. However, in UV (W1, M2, W2) bands
+(panel 6) high flux state is observed corresponding to TeV
+and X-ray activity. In panel 7, we show the archival optical
+data from ASAS-SN, and no short time scale variability
+MNRAS 000, 1–14 (2021)
+
+6
+R. Diwan et al.
+is seen. We also have optical data from the ground-based
+observatory (panel 5) which covers the last part of the light
+curve showing a nice variation from a high flux state to a low
+flux state, suggesting a long-term variation in optical bands.
+3.3 Variability Study
+In general, blazar shows significant variability during the flar-
+ing period. The properties of these flares can depend on var-
+ious factors like particle injection, particle acceleration, and
+energy dissipation in the jets of the blazars. To study this in-
+trinsic property we calculate the Fractional Variability Am-
+plitude (Fvar) from the multi-wavelength light curve of the
+source. The relation given in (Vaughan et al. 2003) is used to
+determine the fractional variability (Fvar)
+Fvar =
+�
+S2 − E2
+F 2
+(3)
+err(Fvar) =
+�
+�
+�
+�
+��
+1
+2N
+E2
+F 2Fvar
+�2
++
+��
+E2
+N
+1
+F
+�2
+(4)
+where S2 is the variance of the light curve, F is the aver-
+age flux, E2 is the mean of the squared error in the flux
+measurements and N is the number of flux points in a light
+curve. We have estimated the Fvar for all the light curves
+and the corresponding values are tabulated in Table 2. We
+found that the source is more variable in UV followed by X-
+ray and gamma-ray. We also plot the Fvar with respect to
+the corresponding frequency in Figure 5. A similar behavior
+is also seen for another TeV blazar 1ES 1727+502 for one of
+the states (Prince et al. 2022). In past studies, it has also
+been argued that the variability pattern resembles the shape
+of the broadband SED seen in blazar if the source is observed
+from radio to very high energy gamma-ray. One of the best
+examples is Mrk 421 which is also a TeV source, where the
+variability pattern during its two flaring states resembles the
+blazar SED (Aleksić et al. 2015a,b). A long-term study, using
+10 yrs data, is done on 1ES 1218+304 by Singh et al. (2019)
+using the multi-wavelength data from radio to γ-ray and the
+Fvar estimated on long-term period is different from what we
+have found in our study. Singh et al. (2019) have found that
+source is more variable in radio at 15 GHz followed by X-ray
+and then optical-UV and γ-ray.
+The timescale of variability is yet another important pa-
+rameter that sets the bound on the size of the emission re-
+gion. Doubling/Halving timescales are calculated for all time
+bins from MJD 58119 to 59365 for the 7-day binned γ-ray
+light curve. The formula used is:
+F(t2) = F(t1) × 2(t2−t1)/Td
+(5)
+Here F(t1) and F(t2) are the fluxes measured at time t1
+and t2, respectively. Td is the flux doubling/halving time
+scale. The fastest doubling/halving time (Tf) in γ-ray was
+found to be 0.396 days. The value for tvar can be given by
+tvar = ln(2)×Tf which is 0.275 days or 6.6 hours. The hour’s
+scale variability is very common in blazar suggesting a com-
+pact emitting region close to the central supermassive black
+hole.
+Waveband
+Fvar
+err(Fvar)
+Fermi γ-ray
+0.2601
+0.0964
+AstroSat-SXT X-ray
+0.0421
+0.0058
+Swift X-ray
+0.5074
+0.01513
+W1
+0.9448
+0.0006
+W2
+0.6805
+0.0005
+M2
+0.9448
+0.0007
+U
+0.0242
+3.3185E-05
+V
+0.0147
+0.0002
+B
+0.0171
+0.0002
+R
+0.0144
+6.5188E-05
+I
+0.0120
+8.2755E-05
+Table 2.
+Fractional variability amplitude (Fvar) parameter for
+the blazar 1ES 1218+304 from optical to HE γ-rays using observa-
+tions during January 1, 2018 - May 31, 2021 (MJD 58119-59365)
+with different instruments.
+Using the same equation we also calculate the time-scale vari-
+ability for the 856 sec binned AstroSat SXT light curve shown
+in Figure 1. The flux doubling/halving time is estimated as
+Tf = 1848.645 sec and the tvar is 1281.29 sec (1.2 ksec) or
+21.35 minutes. A similar flux variability time of 1.1 ksec is
+also estimated for Mrk 421 in SXT light curve by Chatter-
+jee et al. (2021). Considering the fact that 1ES 1218+304
+is a high synchrotron peaked blazar the X-ray will explain
+the synchrotron emission. As argued by many authors that
+the variability time can be associated with the characteristic
+time scale in the system. Here, we consider that the X-ray
+variability timescale can be linked with the radiation cooling
+time scale due to synchrotron only. Under this assumption
+the cooling time can be the fast X-ray variability time and
+can be defined as (Rybicki & Lightman 1979),
+tcool ≃ 7.74 × 108 (1 + z)
+δ
+B−2γ−1 sec.
+(6)
+Where, B is the strength of the magnetic field in Gauss and
+tcool is the synchrotron cooling timescale in seconds. Follow-
+ing Rybicki & Lightman (1979), We can also derive the char-
+acteristic frequency of the electron population responsible for
+the synchrotron emission at the peak frequency,
+νch,e = 4.2 × 106
+δ
+(1 + z)Bγ2 Hz.
+(7)
+Using the above two equations, we eliminate the γ since it
+changes with different states and derives a single equation
+given as,
+B3δ ≃ 2.5(1 + z)(νch,e/1018)−1τ −2
+d .
+(8)
+Using the above equation we derive the magnetic field
+strength for Doppler factor, δ, =30 and variability time scale
+of 1.2 ksec and it is found to be 0.1 G. The strength of the
+magnetic field derived from the broadband SED modeling is
+a factor lower than this estimated value. This discrepancy
+could be because of the many assumptions made in deriving
+the eqn (7) or due to the degeneracy in the SED modeling.
+3.4 Flux-Index Correlation
+We computed flux-index correlation for the γ-ray and X-ray
+data to study index hardening/softening. The flux vs index
+plot is shown in Figure 6 with γ-ray on the upper panel and
+MNRAS 000, 1–14 (2021)
+
+Multi-wavelength study of 1ES 1218+304
+7
+0
+1
+2
+3
+4
+5
+Flux0.3
+300 GeV
+1.0
+1.5
+2.0
+2.5
+Index
+0.5
+1.0
+1.5
+2.0
+Flux0.3
+10 KeV
+1.5
+2.0
+2.5
+3.0
+Index
+15.0
+15.5
+16.0
+16.5
+17.0
+Optical (mag)
+U
+B
+V
+R
+I
+15.5
+16.0
+16.5
+17.0
+17.5
+UV (mag)
+W1
+M2
+W2
+58200
+58400
+58600
+58800
+59000
+59200
+MJD
+14
+15
+16
+17
+Optical (mag)
+ASAS-SN
+Figure 4. Multi-wavelength light curve of 1ES 1218+304 from January 2018 to May 2021. 7-day binned γ-ray flux are presented in units
+of 10−8 ph cm−2 s−1, and X-ray fluxes are in units of 10−10 erg cm−2 s−1. The vertical red line represents the Flare period from 5-7
+January 2019 and the vertical green line represents the Flare period from 15-20 January 2019. This period also includes the data from
+AstroSat for the period 17-20 January 2019. We identify these periods as Flare A and Flare B.
+X-ray on the lower panel. In the case of γ-ray, we have taken
+data points with TS≥16. We also observe a positive corre-
+lation between the flux and index, with Pearson correlation
+coefficient, R = 0.644 and p-value ≈ 0. The trend follows
+the linear function with slope = 0.212. In contrast to the
+above plot, X-ray data shows an inverse trend i.e; a negative
+correlation between flux and index, with Pearson correlation
+coefficient, R = -0.748 and p-value ≈ 0. It can also be fit-
+ted by a linear function with a slope = -0.423. This plot
+shows two contrasting trends, we can see the ’harder-when-
+brighter’ trend in the X-ray energy range and the ’softer-
+when-brighter’ trend in the γ-ray energy range. A similar
+trend is also observed for one of the TeV blazar 1ES 1727+502
+(Prince et al. 2022). One of the possible explanations for hav-
+ing different trends in X-ray and gamma-ray is that they are
+produced via two different processes. For BL Lac-type sources
+such as 1ES 1218+304, it is well-known that the X-rays are
+produced by the synchrotron process and γ-rays are produced
+via the inverse-Compton process. A long-term study done by
+Singh et al. (2019) also found a mild harder-when-brighter
+MNRAS 000, 1–14 (2021)
+
+8
+R. Diwan et al.
+1016
+1018
+1020
+1022
+1024
+1026
+Frequency (Hz)
+0.0
+0.2
+0.4
+0.6
+0.8
+Fractional Variability amplitude
+Gamma-ray
+SWIFT X-ray
+Optical
+SWIFT-UV
+AstroSat SXT
+Figure 5. Fractional variability for various wavebands is plotted
+with respect to their frequency.
+0
+1
+2
+3
+4
+5
+6
+7
+Photon Flux (10
+8 ph cm
+2 sec
+1)
+1.0
+1.5
+2.0
+2.5
+3.0
+Index
+-ray
+r= 0.644, p-value= 1.04×10
+11
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Flux_(0.3-10 KeV) (10
+10 erg cm
+2 sec
+1)
+1.6
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+3.0
+Index
+X-ray
+r= -0.748, p-value= 1.139×10
+5
+Figure 6. Scatter plot for the correlation between flux and index of
+the blazar 1ES 1218+304. The top plot represents the 7-day binned
+Fermi-Lat data. The slope is positive and the Person correlation
+coefficient is 0.644. The bottom plot represents Swift-XRT data
+for Flux (0.3-10 KeV) vs Photon Index. The slope is negative and
+the Pearson correlation coefficient is -0.748, it follows an inverse
+trend as the γ-ray data. The orange line is a linear fit for reference.
+trend in X-rays using almost 10 yrs of data. The average
+spectral index is estimated as 1.99±0.16 which is consistent
+with our estimated value as ∼2.0. These results are also con-
+sistent with the long-term study done by Wierzcholska &
+Wagner (2016) where they found the average photon spec-
+tral index as ∼2.0±0.01 for different values of galactic ab-
+sorption taken from different models. A recent study done by
+Sahakyan (2020) estimated the average photon spectral index
+≥2 for the period considering from 2008 to 2020. The spec-
+tra can be even harder during the bright state as 1.60±0.05
+which is consistent with our result (see Figure 6).
+103
+104
+105
+Energy (MeV)
+10
+6
+10
+5
+10
+4
+E2 dN
+dE [MeV cm
+2 s
+1]
+Likelihood Fit
+5-7 Jan
+15-20 Jan
+Total Time Period
+Figure 7. The γ-ray SED extracted for both the period and fit-
+ted with power-law using the Likelihood fit method. The fitting
+parameters are discussed in the corresponding Section 3.5.
+3.5 Fermi-LAT γ-ray spectral fitting
+The process for data extraction and fitting is provided in
+subsection 2.1. We have used the fermipy to extract the γ-
+ray SED for the two periods (5-7 and 15-20 January 2019).
+The SEDs are then fitted with a simple power law spectral
+model. We noticed that the spectra are very hard and still
+increasing with energy suggesting the involvement of high-
+energy particles in their production. The fitted parameters
+are given in Table 3 and the spectral index for period A
+(Γ=1.55±0.23) and B (Γ=1.54±0.19) are much harder than
+the average power law index, (Γ=1.75±0.03) for the total pe-
+riod. The harder spectra suggest that the IC peak is even at
+higher energy which is clearly seen in broadband SED model-
+ing. A study by Costamante et al. (2018) also shows a harder
+gamma-ray spectrum for many TeV blazar. A harder gamma-
+ray spectrum is also seen in another TeV extreme blazar. In-
+cluding the TeV data in broadband SED Aguilar-Ruiz et al.
+(2022) modeled the SED for six such sources with a two-
+zone emission model. Few new EHBL types sources are also
+discovered with the MAGIC telescope and the Fermi-LAT
+gamma-ray spectra were found to be very hard for all the
+sources suggesting an extreme location of the second SED
+peak above 100 GeV energy range (Acciari et al. 2020). A
+long-term gamma-ray spectral index was also estimated for
+1ES 1218+304 by Singh et al. (2019) and they found it to
+be harder with 1.67±0.05, similar to our estimated value. Sa-
+hakyan (2020) also estimated the γ-ray spectra averaged over
+∼11.7 years which found to be 1.71±0.02 mostly consistent
+with above discussed results. These values are also consistent
+with the long-term average photon spectral index reported in
+the recent 4FGL catalog.
+3.6 Color-Magnitude Variations
+The color-magnitude relation helps us understand the differ-
+ent variability scenarios of the blazar. Fluctuations in optical
+flux are often followed by spectral changes. Therefore study-
+ing the color-magnitude (CM) relationship can further shed
+light on the dominant emission mechanisms in the blazar.
+To obtain a better understanding of the CM relation for our
+source, we fit a linear plot (CI = m V +c) between the color
+MNRAS 000, 1–14 (2021)
+
+Multi-wavelength study of 1ES 1218+304
+9
+Parameter
+Flare A
+Flare B
+Whole Time Period
+Units
+Spectral Index (α)
+-1.547 ± 0.230
+-1.540 ± 0.191
+-1.745 ± 0.030
+-
+Flux (F0.3−300GeV )
+3.306
+3.063
+1.310
+10−8× photon(s) cm−2 s−1
+Prefactor (N0)
+9.538 ± 3.633
+8.902 ± 2.796
+2.966 ± 0.122
+10−13× photon(s) cm−2 s−1 MeV−1
+TS
+43.497
+48.297
+2913.496
+-
+Table 3. Best fit spectral parameters of 1ES 1218+304 from Fermi-Lat observations using equation 1 for two flaring periods 58488-58490
+MJD (Flare A), 58498-58503 MJD (Flare B) and whole time period MJD 58119-59365.
+15.6
+15.8
+16.0
+16.2
+16.4
+16.6
+(B+V)/2
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Color Indices
+B-V + 1.3
+B-I
+R-I + 0.2
+V-R
+Figure 8. Colour magnitude plot for 1ES 1218+304. The various
+color indices are plotted against (B+V)/2.
+indices (CI) and (B+V)/2 magnitude. We then estimate the
+fit values, i.e., slope (m), constant (c), along with the corre-
+lation coefficient (r) and the respective null hypothesis prob-
+ability (p) using two methods, Pearson and Spearman, as
+listed in Table 4. The generated CM plots are shown in Fig-
+ure 8. Offsets of 1.3 and 0.2 are used for (B-V) and (R-I).
+A positive slope with p < 0.05 implies a bluer-when-brighter
+(BWB) trend or a redder-when-fainter trend (Agarwal et al.
+2021) while a negative slope indicates a redder-when-brighter
+trend (RWB). As evident from Table 4, a significant BWB is
+dominant during our observation period for all possible color
+indices, namely; (B-V), (B-I), (R-I), and (V-R). Blazars, in
+general, display BWB from their quasi-simultaneous optical
+observations (Ghosh et al. 2000; Agarwal et al. 2015; Gupta
+et al. 2016a).
+The BWB trend can be attributed to the process of elec-
+tron acceleration to higher energies at the shock front, fol-
+lowed by losing energy by radiative cooling while propagat-
+ing away (Kirk et al. 1998). On the other hand, the opposite
+trend of redder when brighter is observed more commonly
+in FSRQs due to the contribution of bluer thermal emission
+from the accretion disc (Villata et al. 2006). In addition to
+BWB and RWB trends, other optical studies have revealed
+cycle or loop-like trends (Agarwal et al. 2021), a mixed trend
+where BWB is dominant during higher state while RWB dur-
+ing the fainter state, or a stable-when-brighter (SWB) which
+is no significant color-magnitude correlation in the data at
+any timescale (Gupta et al. 2016b; Isler et al. 2017; Negi
+et al. 2022; Agarwal et al. 2022). However, due to the lack
+of simultaneous observations for a larger sample of blazars,
+color-magnitude trends are still a topic of debate.
+3.7 Broadband SED modeling
+The broadband SED modeling in blazar is used to un-
+derstand the simultaneous multi-wavelength emission from
+the source along with the possible physical mechanism re-
+sponsible for broadband flaring event. Simultaneous multi-
+wavelength SEDs were generated for two time periods, which
+overlapped with proposed flaring periods. The model fit-
+ting was done using a publicly available code JetSet10 (Tra-
+macere et al. 2009, 2011, 2020; Massaro, E. et al. 2006).
+Broadband emission of BL Lac sources like 1ES 1218+304 is
+better explained by the one-zone Synchrotron-Self Compton
+(SSC) model. Leptonic models assume that relativistic lep-
+tons (mostly electrons and positrons) interact with the mag-
+netic field in the emission region and produce synchrotron
+photons in the frequency region of radio to soft-X-ray or the
+first hump of the SED. The emission in the frequency region
+of X-ray to γ-ray or the second hump of the SED is pro-
+duced by inverse Compton (IC) scattering of a photon popu-
+lation further classified into synchrotron-self Compton (SSC)
+or external Compton (EC) categories based on the source
+of the seed photons. In the case of SSC models (Ghisellini
+1993; Maraschi et al. 1992) relativistic electrons up-scatter
+the same synchrotron photons which they have produced in
+the magnetic field. The model assumes a spherically sym-
+metric blob of radius (R) in the emission region, surrounded
+by relativistic particles accelerated by the magnetic field (B).
+The blob makes an angle θ with the observer and moves along
+the jet with the bulk Lorentz factor Γ, affecting emission re-
+gion by the beaming factor δ = 1/Γ(1 − β cos θ). The blob
+is filled with a relativistic population of electrons following
+an empirical lepton distribution relation and the power law
+with an exponential cut-off (PLEC) distribution of particles
+is assumed:
+Ne(γ) = N0γ−αexp(−γ/γcut)
+(9)
+where γcut is the highest energy cut-off in the electron spec-
+trum. We see that the optical/UV measurements are higher
+than the non-thermal emission from the jet predicted by
+the SSC model. We also see high flux points in UV/optical
+range from the long-term observation of 1ES 1218+304, from
+NASA/IPAC Extragalactic Database (NED)11. These obser-
+vations suggest that the stellar emission from the host galaxy
+of the source is dominant at optical/UV frequencies. In order
+to accurately account for this emission due to the host galaxy,
+we have added the host galaxy component during modeling
+the SED using JetSet. Modeling of blazar 1ES 1218+304 is
+based on the SSC model in reference to equation 9. Results
+10 https://jetset.readthedocs.io/en/latest/
+11 https://ned.ipac.caltech.edu/
+MNRAS 000, 1–14 (2021)
+
+10
+R. Diwan et al.
+Colour
+In-
+dices
+Slope
+Intercept
+Pearson
+Coeffi-
+cient
+Pearson
+P-value
+Spearman
+Coeffi-
+cient
+Spearman
+P-value
+(B-V)
+0.216
+±
+0.024
+−3.152
+±
+0.390
+0.752
+7.88E-
+13
+0.774
+6.33E-
+14
+(B-I)
+0.446
+±
+0.031
+−6.002
+±
+0.506
+0.893
+1.15E-
+19
+0.928
+6.06E-
+24
+(R-I)
+0.156
+±
+0.019
+−1.982
+±
+0.317
+0.550
+1.67E-
+05
+0.734
+2.65E-
+10
+(V-R)
+0.085
+±
+0.018
+−1.070
+±
+0.292
+0.745
+1.52E-
+10
+0.787
+2.77E-
+12
+Table 4. Colour magnitude fitting and correlations coefficient.
+2
+0
+2
+4
+6
+8
+10
+12
+14
+log(E) (eV)
+12
+14
+16
+18
+20
+22
+24
+26
+28
+log( )  (Hz)
+14
+13
+12
+11
+10
+9
+8
+log( F  )  (erg cm
+2  s
+1)
+  -Sync
+  -SSC
+host_galaxy
+Total SED
+FERMI
+SWIFT UVOT
+SWIFT XRAY
+archived
+Nustar
+Figure 9. Broadband SED Modelling for 5-7 January 2019 (Flare
+A). Optical data are fitted with the host galaxy template available
+in JetSet. Archival NuSTAR data are also plotted in cyan color
+which does not match with the current state X-ray spectral shape.
+Due to the hard X-ray spectral index, the synchrotron peak is
+shifted to higher energy (∼1020 Hz) compared to the synchrotron
+peak location (1017−18 Hz) during 15-20 January as constrained
+by AstroSat observation in Figure 3 and also visible in Figure 10.
+for the SSC model are shown in Figure 9 and Figure 10 for
+Flare A and Flare B. The model parameters are given in table
+5.
+3.7.1 The constraint on Doppler factor
+We can calculate the minimum value of the Doppler factor
+using the detection of high-energy photons from the source.
+This calculation assumes the optical depth, τγγ(Eh), of the
+highest energy photon, Eh, to γγ interaction is 1. The formula
+for calculating the minimum value of the Doppler factor is
+δmin =
+�σtd2
+l (1 + z)2fϵEh
+4tvarmec4
+�1/6
+(10)
+where σt is the Thomson scattering cross-section for the elec-
+tron (6.65 × 10−25cm2), dl is the luminosity distance of the
+source, fϵ is the X-ray flux in 0.3-10 KeV energy range, Eh
+is the highest energy photon, tvar is the observed variability
+time. For 1ES 1218+304, z=0.182, dl is 924 Mpc and tvar is
+0.275 days. Using the value of highest energy photon Eh =
+162.822 GeV for Flare A and 278.132 GeV for Flare B, and
+fϵ = 1.94 × 10−10 for Flare A and 1.55 × 10−10 for Flare B,
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+log(E) (eV)
+12
+14
+16
+18
+20
+22
+24
+26
+28
+log( )  (Hz)
+14
+13
+12
+11
+10
+9
+log( F  )  (erg cm
+2  s
+1)
+  -Sync
+  -SSC
+host_galaxy
+Total SED
+FERMI
+SWIFT UVOT
+archived
+Nustar
+AstroSat-SXT
+SWIFT XRAY
+Figure 10. The plot is the same as Figure 8 but for 15-20 January
+2019 (Flare B). Here also the archival NuSTAR spectrum does
+not match the current state X-ray spectral shape which suggests
+that the NuSTAR spectrum was taken in low-flux states. Here the
+synchrotron peak is decided by both the XRT and SXT spectra
+plotted on top of each other which peaks at roughly ∼2.68×1017
+Hz as estimated in section 3.1 using AstroSat data.
+we get the δmin value to be 13.725 for Flare A and 14.455 for
+Flare B.
+3.7.2 The size of emission region
+The information on the size and location of the emission re-
+gion is very important for performing the SED modeling. The
+variability time scale estimated from the γ-ray light curve is
+used to estimate the size of the emission region. The radius
+R can be estimated by using the equation,
+R = cδmintvar/(1 + z),
+(11)
+where R is estimated to be 8.27 − 8.71 × 1015cm, using the
+δmin calculated in the previous section, and tvar is calculated
+in section 3.3. During SED modeling we have used the values
+1.06 × 1016 cm for Flare A and 1.40 × 1016 cm for Flare B.
+The location of the emission region along the jet axis from
+the supermassive black hole can also be estimated from the
+variability time assuming a spherical emission region by using
+the expression d ∼ 2cΓ2tvar/(1+z). Using the Lorentz factor,
+Γ = δmin and tvar = 0.275 days and z = 0.182, the location is
+estimated to be, d ∼ 2×1017 cm. To optimize the broadband
+SED modeling, we have fixed the location of the emission
+region to 1.0 × 1017 cm along the jet axis.
+MNRAS 000, 1–14 (2021)
+
+Multi-wavelength study of 1ES 1218+304
+11
+Sr. No.
+Model Parameters
+Unit
+Flare A
+Flare B
+5-7 Jan
+15-20 Jan
+1.
+γmin
+-
+88.342
+5.9990
+2.
+γmax
+-
+6.3346 × 107
+6.2115 × 107
+3.
+γcut
+-
+2.8153 × 107
+6.2216 × 105
+4.
+RH
+1017cm
+1.0
+1.0
+5.
+R
+1016cm
+1.0658
+1.4
+6.
+α
+-
+1.482500
+1.530156
+7.
+N
+cm−3
+85.34312
+37.58231
+8.
+B
+G
+2.7378 × 10−3
+1.3035 × 10−2
+9.
+z
+-
+0.182
+0.182
+10.
+δ
+-
+15.97827
+30.30340
+11.
+Ue
+erg cm−3
+3.470401
+4.179746 × 10−2
+12.
+UB
+erg cm−3
+2.982449 × 10−7
+6.760266 × 10−6
+13.
+Pe
+erg s−1
+9.460334 × 1045
+7.081457 × 1044
+14.
+PB
+erg s−1
+8.130172 × 1038
+1.145346 × 1041
+15.
+Pjet
+erg s−1
+1.060629 × 1046
+7.370064 × 1044
+16.
+Reduced Chi-Squared
+-
+1.079990
+2.707362
+Host Galaxy
+17.
+nuFnu_p_host
+erg cm−2 s−1
+-10.373
+-10.373
+18.
+nu_scale
+Hz
+0.496
+0.493
+Table 5. [1-3] Minimum, maximum and cut Lorentz factor of injected electron spectrum [4] The position of the region [5] The size of
+emission region [6] Spectral Index [7] Particle density [8] Magnetic field [9] Red Shift [10] Doppler factor [11] Electron energy density [12]
+Magnetic field energy density [13] Jet power in electrons [14] Jet power in magnetic field [15] Total jet power
+3.7.3 Jet Power
+We have estimated the power carried by individual compo-
+nents (leptons, protons, and magnetic fields) and the total
+jet power. The total power of the jet was estimated using
+Pjet = πR2Γ2c(U ′
+e + U ′
+p + U ′
+B)
+(12)
+Here Γ is the bulk Lorentz factor. U ′
+e, U ′
+p, U ′
+B are the energy
+densities of electrons-positrons, cold protons and the mag-
+netic field respectively in the co-moving jet’s frame (primed
+quantities are in the co-moving jet frame while unprimed
+quantities are in the observer frame). The power in leptons
+is given by
+Pe = 3Γ2c
+4R
+� Emin
+Emax
+EQ(E)dE
+(13)
+where Q(E) is the injected particle spectrum. The integration
+limits, Emin and Emax are calculated by multiplying the min-
+imum and maximum Lorentz factor (γmin and γmax) of the
+electrons with the rest-mass energy of the electron respec-
+tively. The power in the magnetic field is calculated using
+PB = R2Γ2cB2
+8
+(14)
+where B is the magnetic field strength obtained from the
+SED modeling. The energy densities for electron-positron and
+magnetic field for both Flare events were returned by our
+model. The energy density for cold proton was not estimated
+as it was too small. We calculated Pe, PB which are the power
+carried by the leptons and the magnetic field respectively. The
+total power Pjet ≈ Pe + PB along with the power of the in-
+dividual components has been mentioned in Table 5. The jet
+is dominated by the lepton’s power and its value decreases
+for the second flare period. The luminosities have been com-
+puted for a pure electron/positron jet since the proton con-
+tent is not well known, and can be considered as the lower
+limit. The absolute jet power Ljet ≃ 1×1046ergs−1 for Flare
+A and is below the Eddington luminosity for a 5.6 × 108M⊙
+black hole mass (LEdd = 7.3 × 1046ergs−1) estimated from
+the properties of the host galaxy in the optical band (Rüger
+et al. 2010). For Flare B, Ljet ≃ 7.37 × 1044ergs−1 is signifi-
+cantly below the LEdd.
+3.7.4 Broadband emission during flaring states
+We choose two flaring periods during the month of January
+2019, MJD 58488-58490 (5-7 January 2019, Fig 9) and MJD
+58498-58503 (15-20 January 2019, Fig 10) were modeled with
+a one-zone leptonic scenario. The modeled parameters are
+mentioned in Table 5. The model parameters inferred from
+this fitting suggest that Flare A had more activity compared
+to Flare B. Although the γmax and α are almost the same for
+both the flares inferring that there was very little variability
+in VHE γ-ray band, we see from Table 5 that γmin, γcut have
+significantly higher values for Flare A compared to Flare B,
+which may be due to the flaring seen in the X-ray band. The
+magnetic field (B) for Flare A (2.73×10−3) is also less than
+that of Flare B (1.30×10−2). During the fitting of SED, we
+kept RH and δ as free parameters. We find that the value of
+RH is close to the value we calculate using equation 11. We
+also calculate the minimum doppler factor δmin between the
+range (13.725-14.455), but during the SED modeling, we find
+that for Flare A δ = 15.98 and for Flare B it is much higher δ
+= 30.30 then the calculated value. It suggests that variation
+in δ could be one of the reasons for different flux states.
+During these flares, the optical-UV emission is dominated
+MNRAS 000, 1–14 (2021)
+
+12
+R. Diwan et al.
+by thermal emission from the host galaxy and hence has been
+modeled using the host galaxy model using JetSet. It is also
+seen that the X-ray data is better explained by synchrotron
+radiation of electrons. The SSC component of SED model-
+ing dominates above 1020 Hz (∼ 1 MeV) and it is useful in
+describing the data up to the VHE γ-ray band.
+4 SUMMARY AND DISCUSSIONS
+In our work, we present the multi-wavelength study of HBL
+blazar 1ES 1218+304 from 1st January 2018 to 31st March
+2021 (58119-59365), which also include the high flux event in
+VHE γ-rays detected by both MAGIC and VERITAS obser-
+vatories during January 2019. This high flux rate was also
+seen in Swift-XRT and UVOT instruments. Hence we di-
+vided our SED analysis into two flaring periods 5-7 Jan-
+uary 2019 and 15-20 January 2019 for simultaneous multi-
+wavelength observation of 1ES 1218+304. The fastest vari-
+ability timescale was found to be 0.275 days from analyzing
+the γ-ray light curve, constraining the size of the emission
+region to 8.27 − 8.71 × 1015 cm, which came out to be higher
+than previous modeling results (Rüger et al. 2010, Sahakyan
+2020, Singh et al. 2019) but comparable to SED modeled
+results in our case, see Table 5. The location of the emis-
+sion region is estimated to be d ∼ 2 × 1017cm was similar
+to that used for SED modeling. The highest energy photon
+detected was 278.132 GeV which arrived during Flare B. We
+can also see the ’harder-when-brighter’ trend in the X-ray en-
+ergy range and the ’softer-when-brighter’ trend in the γ-ray
+energy range.
+The broadband SED modeling of the source was repro-
+duced by a leptonic simple one-zone SSC model with the
+electron energy distribution described by a Power-law with
+an exponential cut-off (PLEC) function. Parameters like the
+magnetic field, injected electron spectrum, and minimum and
+maximum energy of injected electrons have been optimized
+to get a good fit to the SEDs data points. So this study sug-
+gests that a single-zone model can also be good enough to
+explain the multi-waveband emissions from 1ES 1218+304.
+The optical and UV emissions from the source are found to
+be dominated by the stellar thermal emissions from the host
+galaxy and can be modeled using the JetSet code by a simple
+blackbody approximation (Rüger et al. 2010).
+Costamante et al. (2018) argued that the broadband SED
+modeling in hard-TeV blazar can be explained by the one-
+zone SSC model at the expense of extreme electron ener-
+gies with very low radiative efficiency. The maximum elec-
+tron Lorentz factor estimated in their modeling for all the six
+sources is orders of 107 which is consistent with our results
+for 1ES 1218+304. The other modeling parameters such as
+the size of the emission region, magnetic field strength, and
+the magnetization parameters (UB/Ue) are very similar to
+our SED modeling result for 1ES 1218+304. In our case, the
+UB/Ue = 10−4 - 10−6 and in Costamante et al. (2018) it order
+of 10−2 - 10−5. Similar results were also obtained by Kauf-
+mann et al. (2011) where they model the broadband SED of
+extreme TeV source 1ES 0229+200. The magnetic field and
+the magnetization parameter (10−5) are consistent with our
+results for 1ES 1218+304. But their model requires a narrow
+electron energy distribution with γmin ∼ 105 to γmax ∼ 107
+rather than the broad energy range obtained in our study,
+Costamante et al. (2018), and Acciari et al. (2020).
+Acciari et al. (2020) have observed ten new TeV sources
+with MAGIC from 2010 to 2017 for a total period of 262
+hours and the simultaneous X-ray observations confirm that
+out of 10, 8 sources are of extreme nature. Their γ-SED
+was found to be very hard between 1.4 to 1.9. Blazar 1ES
+1218+304 is also an extreme TeV blazar and in our study, the
+gamma-ray SED is found to be 1.5 consistent with the above
+TeV sources. They have modeled all the sources with a sin-
+gle zone conical-jet SSC model. Additionally, they also used
+the proton-synchrotron and a leptonic scenario with a struc-
+tured jet. They also argue that all the model provides a good
+fit to the broadband SED but the individual parameters in
+each model differ substantially. Comparing their SSC model
+results to our SSC modeling the maximum electron energy is
+consistent. The electron spectral index in our case is harder
+than their results and also the magnetic field in our case is
+much smaller. The estimated Lorentz factor is more or less
+consistent with the Γ used for all the sources in their study.
+In their recent work Aguilar-Ruiz et al. (2022) have modeled
+the six well-known extreme BL Lac sources with a lepto-
+hadronic two-zone emission model to explain the broadband
+SED. In another study, Zech & Lemoine (2021) have shown
+that the broadband SED of extreme BL Lac sources can be
+explained by considering the co-acceleration of electrons and
+protons on internal or recollimation shocks inside the rela-
+tivistic jet. Sahakyan (2020) has modeled the average state
+of 1ES 1218+304 with one-zone SSC model. The parameter
+estimated in their study is mostly consistent with ours. How-
+ever, our study focuses on the smaller period including two
+flaring events. During the flaring event (15-20 Jan) the mag-
+netic field and the magnetization parameters are estimated
+as 1.30×10−2 Gauss and ∼10−4 which is comparable to the
+value for the same parameters estimated by modeling the av-
+erage state of the source in Sahakyan (2020). However, the
+Doppler factor required in Sahakyan (2020) is much higher
+than the Doppler factor needed to fit the flaring state in our
+case. Singh et al. (2019) also modeled the average broadband
+SED collected for almost 10 years with a one-zone SSC model.
+The required γmin, γmax and Doppler factor are consistent
+with our result but the size of the emission region is one order
+of magnitude smaller than ours, and also the magnetic field
+estimated in their model is much higher than what we found.
+The difference in some of the parameters could be because
+they modeled the average SED and in our case, we are more
+focused on a short period of time. The optical-UV SED is
+mostly off to the general trend of broadband SED of blazar
+and hence in both cases is fitted with a host-galaxy contri-
+bution. Singh et al. (2019) used a specific model to fit the
+host-galaxy and estimated the black hole mass of the source,
+however, in JetSet we can not include a specific model, and
+hence host-galaxy is fitted as a free parameter.
+The above discussion suggests that the known extreme BL
+Lac sources are very less in number and need careful attention
+and more broadband study to exactly quantify their nature
+and the physical emission mechanism.
+MNRAS 000, 1–14 (2021)
+
+Multi-wavelength study of 1ES 1218+304
+13
+5 CONCLUSIONS
+In this work, we present the long-term study of the blazar
+1ES 1218+304 using 3.5 years of near-simultaneous multi-
+wavelength data from Fermi-LAT, SWIFT-XRT, SWIFT-
+UVOT, AstroSat, and TUBITAK observations taken between
+January 1, 2018, and March 31, 2021. This study explores the
+broadband temporal and spectral behavior of the source dur-
+ing flaring states. The main results of our study are provided
+below:
+• During the month of January 2019, VHE γ-rays detected
+by both MAGIC and VERITAS observatory. This high flux
+state was also seen in Fermi, Swift-XRT, and UVOT instru-
+ments. The fractional variability estimated across the wave-
+bands suggests that UV is more variable followed by X-ray,
+γ-ray, and optical.
+• The fast flux variability in γ-ray is calculated to be
+0.275 days, the size of the emission region is estimated to
+be ∼8×1015 cm, and the emission region is located at a dis-
+tance of ∼ 2 × 1017 cm. A "harder-when-brighter" trend was
+seen in X-ray whereas a "softer-when-brighter" trend was in
+γ-ray. The γ-ray emission from 1ES 1218+304 can also be
+described by a power law with a spectral index of ∼ 1.745.
+• The Astrosat SXT light curve reveals a minute scale of
+variability of the order of 20 minutes and the X-ray spectrum
+is well fitted with both power-law and the log parabola mod-
+els. However, the LP provides a better fit. A joint fit with the
+LAXPC spectrum provides a great constrain on the location
+of synchrotron peak roughly around 2.68×1017Hz.
+• As seen in many other TeV blazars, a shift in syn-
+chrotron peak is observed from one state to another state
+from ∼1017−18 Hz to ∼1020 suggesting an extreme nature of
+the source.
+• The broadband SED modeling of the source is repro-
+duced by a one-zone leptonic SSC model with the electron
+energy distribution described by a Power-law with an expo-
+nential cut-off (PLEC) function. We also find that the Opti-
+cal/UV emissions from the source are dominated by the stel-
+lar thermal emissions from the host galaxy which are modeled
+by a simple blackbody approximation (Rüger et al. 2010) us-
+ing JetSet. The JetSet code uses an approximation of the host
+galaxy model to help fit the SED modeling. We need more
+precise and dedicated observation in the UV/Optical band
+for a better understanding of the host galaxy.
+• 1ES 1218+304 is also an important source for obser-
+vations within the upcoming high-energy ground-based tele-
+scopes like CTA (Cherenkov Telescope Array)12 observatory
+to establish the link beyond the GeV energy range, in the
+realm of TeV γ-ray emission and MeV-GeV emission mea-
+sured from the Fermi-LAT and its extreme blazar behavior.
+ACKNOWLEDGEMENTS
+D. Bose acknowledges the support of Ramanujan Fellowship-
+SB/S2/RJN-038/2017. R. Prince is grateful for the support of
+the Polish Funding Agency National Science Centre, project
+2017/26/A/ST9/-00756 (MAESTRO 9) and MNiSW grant
+DIR/WK/2018/12. This work made use of Fermi telescope
+12 https://www.cta-observatory.org
+data and the Fermitool package obtained through the Fermi
+Science Support Center (FSSC) provided by NASA. This
+work also made use of publicly available packages JetSet, Fer-
+mipy, and PSRESP. This publication uses the data from the
+AstroSat mission of the Indian Space Research Organisation
+(ISRO), archived at the Indian Space Science Data Centre
+(ISSDC). This work has used the data from the Soft X-ray
+Telescope (SXT) developed at TIFR, Mumbai, and the SXT
+POC at TIFR is thanked for verifying and releasing the data
+via the ISSDC data archive and providing the necessary soft-
+ware tools. We thank the LAXPC Payload Operation Center
+(POC) at TIFR, Mumbai for providing the necessary soft-
+ware tools. We have also made use of the software provided
+by the High Energy Astrophysics Science Archive Research
+Center (HEASARC), which is a service of the Astrophysics
+Science Division at NASA/GSFC.
+DATA AVAILABILITY
+For this work, we have used data from the Fermi-LAT, Swift-
+XRT, Swift-UVOT, and AstroSat which are available in the
+public domain. We have also used optical data collected by
+the TUBITAK telescope. This optical data was given to us
+on request. Details are given in Section 2.
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+MNRAS 000, 1–14 (2021)
+

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@@ -0,0 +1,1431 @@
+arXiv:2301.08624v1  [math.AP]  20 Jan 2023
+ALMOST MINIMIZERS TO A TRANSMISSION PROBLEM
+FOR (p, q)-LAPLACIAN
+SUNGHAN KIM AND HENRIK SHAHGHOLIAN
+Abstract. This paper concerns almost minimizers of the functional
+J(v, Ω) =
+ˆ
+Ω
+�
+|Dv+|p + |Dv−|q�
+dx,
+where 1 < p ̸= q < ∞ and Ω is a bounded domain of Rn, n ≥ 1. We
+prove the universal H¨older regularity of local (1 + ε)-minimizers, when
+ε is universally small. Moreover, we prove almost Lipschitz regularity
+of the local (1 + ε)-minimizers, when |p − q| ≪ 1 and ε ≪ 1.
+Contents
+1.
+Introduction
+1
+2.
+Technical Tools
+4
+3.
+H¨older regularity
+10
+4.
+Almost Lipschitz regularity
+14
+Declarations
+18
+References
+18
+1. Introduction
+In this paper, we study regularity properties of almost minimizers to the
+functional
+(1.1)
+J(u, Ω) ≡ Jp,q(u, Ω) :=
+ˆ
+Ω
+(|Du+|p + |Du−|q) dx,
+where Ω ⊂ Rn is a bounded domain and 1 < p, q < ∞.
+Our primary
+goal is to prove a universal H¨older estimate for the almost minimizers. We
+shall also study various scenarios, on the relation between p and q, to see
+if the regularity can be improved. In particular, we aim at proving almost
+Lipschitz regularity provided that p and q are close to each other.
+The notion of local K-minimizers is given as follows.
+H. Shahgholian was supported in part by Swedish Research Council. This project
+was finalized during the program Geometric aspects of nonlinear PDE at Institute Mittag
+Leffler, Stockholm.
+1
+
+2
+SUNGHAN KIM AND HENRIK SHAHGHOLIAN
+Definition 1.1 (Local K-minimizers). Let K ≥ 1 be a constant. We shall
+call u ∈ W 1,p∧q
+loc
+(Ω) a local K-minimizer of the functional J, if for any cube
+Q ⊂ Ω, J(u, Q) < ∞, and
+(1.2)
+J(u, Q) ≤ KJ(v, Q),
+for any v ∈ u + W 1,p∧q
+0
+(Q) such that J(v, Q) < ∞.
+In the course of this paper, we shall be interested in the case K = 1 + ε,
+for some small ε > 0. We remark that our analysis does not change, as
+one replaces cubes with balls in the above definition. However, it is worth
+mentioning that the notion with cubes is in general not equivalent to hat
+with balls, unless K = 1, and local K-minimizers with cubes are known to
+be less restrictive; see [Giu03, Example 6.5].
+In the framework of standard functionals (i.e., those without break across
+some level set), the universal H¨older regularity is established for quasi-
+minimzers (those with K > 1 any, and Q in (1.2) replaced with spt(u − v)),
+as the essential arguments for the proof of the H¨older regularity for exact
+minimizers remain unchanged upon the extension; see [Giu03]. In contrast,
+thanks to the particular break across the zero-level set in Jp,q, many impor-
+tant steps in the proof of [CKS21, Theorem 1.2] for the H¨older regularity of
+exact minimizers to our functional Jp,q are destroyed when applied to quasi-
+minimzers. Still, we were able to extend the argument to (1+ε)-minimizers,
+when ε is universally small.
+Theorem 1.2. There are constants ε > 0 and σ ∈ (0, 1), depending only on
+n, p+, and p−, such that if u ∈ W 1,p+∧p−(Q2) is a local (1 + ε)-minimizer
+of Jp+,p−, then u± ∈ C0,σ±
+loc (Q1) with σ+ = σ, σ− = 1 − (1 − σ)p−
+p+ , and
+[u±]C0,σ±(Q1) ≤ c
+�ˆ
+Q2
+((u+)p+ + (u−)p−) dx
+� 1
+p± ,
+where c depends only on n, p+, and p−.
+We remark that the above theorem also shows the exact relation between
+the H¨older exponents for each phase; this was not contained in the authors
+earlier collaboration [CKS21, Theorem 1.2] with M. Colombo. Our proof
+involves a careful extension of the main ingredients for [CKS21, Theorem
+1.2] to local (1 + ε)-minimizers, and a compactness argument.
+A key feature of local (1 + ε)-minimizers, ε ≥ 0, for the functional Jp,q
+is that the positive and negative phase scales differently from each other.
+Namely if u is a local (1+ε)-minimizer in Q2, then one needs ∥u+∥X compa-
+rable with ∥u−∥q/p
+X , with X = Lp(Q1) or L∞(Q1). As for the case of the local
+minimizers, i.e., ε = 0, the comparability was proved by a Harnack inequal-
+ity argument [CKS21, Lemma 3.7, Corollary 3.8], which played an essential
+role in the proof of their universal H¨older regularity [CKS21, Theorem 1.2].
+The main difference, which also amounts to the challenges here, for the
+case of local (1+ε)-minimizers, ε > 0, is the lack of such a Harnack inequality
+
+3
+argument. More fundamentally, local (1 + ε)-minimizers do not possess the
+subsolution properties as opposed to local minimizers (see [CKS21, Lemma
+3.4]). One of the consequences is that the basic estimates for one phase,
+such as the Cacciopoli inequality (Lemma 2.2) and the comparison lemma
+(Lemma 3.1) for local (1+ε)-minimizers, involve an additional ε-factor of the
+other phase. Hence, our main task here is to effectively control the additional
+ε-term, which amounts to some technical difficulties. It is worthwhile to
+mention that the absence of the Harnack inequality argument is overcome
+by a careful compactness argument, by which both phases, although scaled
+differently, survive at the limit. The latter part is new, to the best of the
+authors’ knowledge, and can be applied to a wider range of problems.
+Our second result is about the almost Lipschitz regularity for local (1+ε)-
+minimizers for the functional Jp,q, when |p − q| ≪ 1 and ε ≪ 1.
+Theorem 1.3. Let 1 < p+ < ∞ and σ ∈ (0, 1) be given.
+Then there
+exist ε, δ > 0, depending only on n, p+ and σ, such that for any p− ∈
+(p+−δ, p+ +δ) and any local (1+ε)-minimizer u ∈ W 1,p+∧p−(Q2) of Jp+,p−,
+one has u± ∈ C0,σ±(Q1), with σ+ = σ, σ− = 1 − (1 − σ)p−
+p+ , and
+[u]C0,σ±(Q1) ≤ c
+�ˆ
+Q2
+((u+)p± + (u−)p−) dx
+� 1
+p± ,
+where c depends only on n, p+ and σ.
+A similar statement is proved in [AT15] for uniformly elliptic function-
+als when governing conductivity matrices are close with each other; [AT15]
+however considers local minimizers (i.e., ε = 0) only. Our problem is philo-
+sophically the same, as the limit case is clean, thus possess better regularity.
+On the technical level, our argument is needs slight more care than that
+of [AT15, Theorem 7.1], as the proof for the growth of the functional Jp,q
+changes as (p, q) varies. Moreover, one needs to make sure that the argument
+works well regardless of the relation between p (or q) and the dimension n.
+These are all rigorously treated in Sect. 4.
+Recently, free boundaries for almost minimizers are investigated in various
+settings, see e.g., [DET19], [DS20], and [DJS22] to mention a few. There
+is a possibility of extending the approach with viscosity solutions employed
+in [DS20], but it is beyond the scope of this paper. It would be already
+interesting to extend the result for the clean case, p = q.
+In [CKS21], the authors analyze the free boundary of local minimizers for
+Jp,q, using the measure ∆pu+, which is nonnegative and supported on the
+free boundary, ∂{u > 0}(=∂{u < 0}). This is mainly due to the subsolu-
+tion property of u+, which is no longer valid for almost minimizers. The
+same issue appears in the case of the two-phase Alt-Caffarelli functional
+(see [DET19, Section 4]), which is resolved by the NTA property of the free
+boundary and a clever use of barriers. The NTA property was obtained
+there by the use of the ACF monotonicity formula, which is absent in our
+
+4
+SUNGHAN KIM AND HENRIK SHAHGHOLIAN
+regime. The construction of the barriers and the comparison with the al-
+most minimizers require some regularity of the free boundary, which in the
+case of [DET19] was the NTA property. However, in our problem, none of
+these seems to be analogously carried out. For this reason, we leave out the
+analysis of the free boundary for our almost minimizers to the interested
+reader.
+The paper is organized as follows. In Section 2, we collect some technical
+tools to prepare the proof of Theorem 1.2. In Section 3, we prove Theorem
+1.2. In Section 4, we prove Theorem 1.3.
+We follow the standard notation and terminology. In particular, n denotes
+the dimension of the underlying space, and there is no restriction other than
+n ≥ 1. By Qr(x0), we denote the cube centered at x0 with side-length r,
+i.e., Qr(x0) := {x ∈ Rn : |xi − x0i| < r, 1 ≤ i ≤ n}. For simplicity, we set
+Qr := Qr(0). Given a set A ⊂ Rn, by |A| we denote the Lebesgue measure
+of A. The function spaces C0,σ and W 1,p are standard H¨older and Sobolev
+spaces, and C0,σ
+loc , W 1,p
+loc are their local versions.
+2. Technical Tools
+In this section, we shall present and verify some technical tools, most of
+which generalize those appeared in [CKS21, Sect. 4–5]. The main goal of
+this section is to prove the following proposition, which roughly tells us that
+negative values cannot penetrate the interior if a local (1 + ε)-minimizer
+attains large positive values in most of the domain.
+Let us remark that
+this proposition corresponds to [CKS21, Proposition 5.2] for the case of
+minimizers.
+The main difference here is that (1 + ε)-minimizers do not
+possess in general the subsolution properties. Here we exploit the techniques
+to circumvent this issue. Unless stated otherwise, the constant c throughout
+this section is a positive constant that may differ at each occurrence, and
+will depend at most on n, p, and q. Moreover, the parameter ε will be a
+small constant, whose smallness is determined solely by n, p, and q.
+Proposition 2.1. There exist ε > 0 and µ > 0, depending only on n, p, and
+q, such that if u ∈ W 1,p∧q(Q1) is a local (1 + ε)-minimizer of the functional
+J, satisfyingˆ
+Q1
+((u+)p + (u−)q) dx ≤ 1,
+|{u ≤ 1/2} ∩ Q1| ≤ ε,
+then u > 0 a.e. in Qµ.
+The proof for this proposition will be postponed to the end of this section.
+Let us begin with the Cacciopoli-type inequality.
+Lemma 2.2. Let u ∈ W 1,p∧q(Q2) be a local (1 + ε)-minimizer of the func-
+tional J. There exists ¯ε ∈ (0, 1), depending only on n, p, and q, such that if
+ε ≤ ¯ε, then
+(2.1)
+ˆ
+Q1
+|Du+|p dx ≤ c
+ˆ
+Q2
+((u+)p + ε(u−)q) dx,
+
+5
+where c depends only on n, p, and q.
+Proof. Fix r, R with 1 < r < R < 2, and choose any s, t with r < s < t < R.
+Let η ∈ C1
+c (Qt) be a cutoff function such that η ≡ 1 in Qs, |Dη| ≤
+2c
+t−s in
+Qt, and spt(η) ⊂ Q(t+s)/2. Set w := (1 − η)u+ − u− ∈ W 1,p∧q(Qt). Since
+w+ = (1 − η)u+, w− = u−, and spt(u − w) ⊂ spt(η) ⊂ Q(t+s)/2, we derive
+from the (1 + ε)-minimizerslity of u for Jp,q in Qt that
+ˆ
+Qr
+|Du+|p dx ≤ (1 + ε)
+ˆ
+Qt
+|D((1 − η)u+)|p dx + ε
+ˆ
+Qt
+|Du−|q dx.
+Applying H¨older’s inequality and Young’s inequality, and then using spt(η) ⊂
+Q(t+s)/2 and |Dη| ≤ c/(t − s), we deduce that
+ˆ
+Qs
+|Du+|p dx ≤ c
+ˆ
+Qt
+� (u+)p
+(t − s)p + ε|Du−|q
+�
+dx + cε
+ˆ
+Qt
+|Du+|p dx.
+Since this part is by now standard, we omit the details. Note that the last
+display holds for all s, t, r < s < t < R. Hence, choosing ε small enough such
+that cε < 1
+2, we can employ the standard iteration lemma [Giu03, Lemma
+6.1] to derive that
+(2.2)
+ˆ
+Qr
+|Du+|p dx ≤ c
+ˆ
+QR
+� (u+)p
+(R − r)p + ε|Du−|q
+�
+dx.
+Now replace QR in the right-hand side with Q(R+r)/2, and then apply
+the same argument above to (−u) with Qr replaced with Q(R+r)/2; note
+that (−u) is a local (1 + ε)-minimizer of Jq,p in place of Jp,q. Then we may
+proceed as follows,
+ˆ
+Qr
+|Du+|p dx ≤ c
+ˆ
+Q(R+r)/2
+� (u+)p
+(R − r)p + ε|Du−|q
+�
+dx
+≤ c
+ˆ
+QR
+� (u+)p
+(R − r)p + cε (u−)q
+(R − r)q
+�
+dx + c2ε2
+ˆ
+QR
+|Du+|p dx.
+Recall that r, R were any numbers between 1 and 2. Hence, taking ε smaller
+if necessary such that c2ε2 < 1
+2, we can make use of the iteration lemma
+once again to arrive at (2.1).
+□
+Remark 2.3. In what follows, we shall always assume that ε < ¯ε, with ¯ε
+as in Lemma 2.2.
+Let us remark that the above Cacciopoli inequality is too weak to bring
+forth a local L∞-estimate. Besides, local quasi-minimizers are not neces-
+sarily bounded, even for functionals under standard growth condition (of
+course, only if p ≤ n). Nevertheless, with the aid of the Cacciopoli inequal-
+ity above, we shall observe that the blowup rate of local (1 + ε)-minimizers
+can be made arbitrarily small, for small ε, in case p ≤ n.
+
+6
+SUNGHAN KIM AND HENRIK SHAHGHOLIAN
+Lemma 2.4. Let u ∈ W 1,p∧q(Q1) be a local (1 + ε)-minimizer of the func-
+tional J. Suppose that
+∥u+∥Lp(Q1) ≤ 1,
+sup
+r∈(0,1)
+∥u−∥Lq(Qr)
+r1− p
+q ∥u+∥
+p
+q
+Lp(Qr)
+≤ κ,
+for some constant κ > 0. Then for any δ > 0, there exists a positive constant
+εκ,δ, depending only on n, p, q, κ and δ, such that if ε ≤ εκ,δ, then
+sup
+r∈(0,1)
+1
+rn−δp
+ˆ
+Qr
+(u+)p dx ≤ cκ,δ,
+where cκ,δ depends only on n, p, q, Λ, δ and κ.
+Proof. We remark that the conclusion is trivial for p > n, due to the Sobolev
+embedding theorem. Henceforth, we shall assume that 1 < p ≤ n.
+Let κ and δ be arbitrary positive constants, and suppose the conclusion
+of the lemma is false. Then for each j = 1, 2, · · · , one can find some positive
+constant εj ց 0, and a local (1 + εj)-minimizer uj ∈ W 1,p∧q(Q1) of the
+functional J, such that
+∥u+
+j ∥Lp(Q1) ≤ 1,
+sup
+r∈(0,1)
+∥u−
+j ∥Lq(Qr)
+r1− p
+q ∥u+
+j ∥
+p
+q
+Lp(Qr)
+≤ κ,
+but
+Sj =
+sup
+rj≤r≤1
+1
+rn−δp
+ˆ
+Qr
+(u+
+j )p dx → ∞,
+for some constant rj ∈ (0, 1). In order to have Sj → ∞ to be compatible
+with ∥u+
+j ∥Lp(Q1) = 1, we must have rj → 0.
+Consider an auxiliary function vj : Qr−1
+j
+→ R, defined by
+vj(y) =
+u+
+j (rjy)
+r
+− n
+p
+j
+∥u+
+j ∥Lp(Qrj )
+−
+u−
+j (rjy)
+r
+1− p
+q − n
+q
+j
+∥u+
+j ∥
+p
+q
+Lp(Qrj )
+.
+One easily verifies that vj ∈ W 1,p∧q(Qr−1
+j ) is a local (1 + εj)-minimizer of
+the functional J, and
+(2.3)
+sup
+1≤R≤r−1
+j
+1
+Rn−δp
+ˆ
+QR
+(v+
+j )p dy = 1,
+where the supremum is attained at R = 1, and
+(2.4)
+sup
+1≤R≤r−1
+j
+1
+Rn+q−(1+δ)p
+ˆ
+QR
+(v−
+j )q dy ≤ κq.
+Due to Lemma 2.2, along with (2.3) and (2.4),
+(2.5)
+ˆ
+QR
+(|Dv+
+j |p + |Dv−
+j |q) dx ≤ cRn−(1+δ)p,
+
+7
+where c depends only on n, p and q, whenever 2Rrj ≤ 1. By the Sobolev em-
+bedding theory, there exists a function v ∈ W 1,p∧q
+loc
+(Rn) with v+ ∈ W 1,p
+loc (Rn)
+and v− ∈ W 1,q
+loc (Rn) such that v+
+j → v+ and v−
+j → v−
+j weakly in W 1,p
+loc (Rn)
+and respectively W 1,q
+loc (Rn), after extracting a subsequence if necessary; we
+shall denote this subsequence by vj, for brevity. The weak convergence im-
+plies that v ∈ W 1,p∧q(BR) is a minimizer of the functional J. Since v+
+j → v+
+strongly in Lp(BR) and v−
+j → v− strongly in Lq(BR), letting j → ∞ in (2.3)
+yields that
+(2.6)
+sup
+R≥1
+1
+Rn−δp
+ˆ
+QR
+(v+)p dy = 1.
+However, since v is a minimizer of the functional J, by [CKS21, Lemma
+3.4], v+ is a weak p-subsolution. As a result, the local L∞-estimates [Giu03,
+Theorem 7.3] applies to v+, which along with (2.6) yields
+∥v+∥L∞(QR) ≤ c
+Rδ .
+Hence, letting R → ∞ yields that v+ = 0 a.e. in Rn. This yields a contra-
+diction against (2.6).
+□
+We also have a growth estimate for the p-th Dirichlet energy of the positive
+phase. The idea is the same as in [CKS21, Lemma 4.5], which is based on
+some approximation by positive p-harmonic functions of the positive phase
+of local quasi-minimizers, in terms of the size of the negative phase.
+Lemma 2.5. Let u ∈ W 1,p∧q
+loc
+(Q2) be a local (1 + ε)-minimizer of the func-
+tional J, and v ∈ u+ + W 1,p
+0 (Q1) be the p-harmonic function. Then
+0 ≤
+ˆ
+Q1
+(|Du+|p − |Dv|p) dx ≤ c
+ˆ
+Q2
+((u−)q + ε|Du+|p) dx,
+and
+ˆ
+Qr
+|Du+|p dx ≤ c
+ˆ
+Q1
+((rn + ε)|Du+|p + (u−)q) dx,
+∀r ∈ (0, 1),
+where c depends only on n, p and q.
+Proof. The proof is essentially the same as that of [CKS21, Lemma 4.5].
+The additional term ε
+´
+Q2 |Du+|p dx appears due to the different Cacciopoli
+inequality; more exactly, we use (2.2) with u replaced with −u. We shall
+not repeat this argument here.
+□
+The following lemma corresponds to [CKS21, Lemma 4.8]. The key in-
+gredient of the proof there is the Poincar´e inequality, and Lemma 2.5, which
+corresponds to [CKS21, Lemma 4.5]. As noted above, Lemma 2.5 differs
+from [CKS21, Lemma 4.5] by the additional term, ε
+´
+Q2 |Du+|p dx. How-
+ever, this does not make any difference in the proof of the lemma below.
+Thus, we shall skip the proof.
+
+8
+SUNGHAN KIM AND HENRIK SHAHGHOLIAN
+Lemma 2.6 (Essentially due to [CKS21, Lemma 4.8]). Let u ∈ W 1,p∧q(Q4)
+be a local 2-minimizer for the functional J, satisfying
+ˆ
+Q4
+(u+)p dx = 1,
+ˆ
+Q4
+((u−)q + |Du+|p) dx ≤ ε,
+for some ε > 0. Then
+|{u ≤ 1/2} ∩ Q1| ≤ cε,
+where c depends only on n, p and q.
+Let us prove Proposition 2.1 with additional assumptions that ∥u−∥Lq(Q1)
+and ∥Du+∥Lp(Q!) are sufficiently small. The proof follows the idea of that
+of [CKS21, Lemma 5.5], with some modifications addressing the lack of
+subsolution properties of each phase.
+Lemma 2.7. There exists ε > 0, depending only on n, p and q, such that if
+u ∈ W 1,p∧q(Q4) is a local (1 + ε)-minimizer of the functional J, satisfying
+ˆ
+Q4
+(u+)p dx = 1,
+ˆ
+Q4
+((u−)q + |Du+|p) dx ≤ ε,
+then u > 0 a.e. in Q1.
+Proof. Let us consider the case q < n first. Following the proof of [CKS21,
+Lemma 4.3], we obtain that for σ ∈ (0, 1),
+(2.7)
+ˆ
+Qr
+�(u−)q
+rq
++ |Du+|p
+�
+dx ≤ cεr−(1−σ)p
+ˆ
+Qr
+(u+)p dx,
+∀r ∈ (0, 1),
+where c depends only on n, p, q and σ. The proof is essentially the same,
+as Lemma 2.5 and 2.6 replace [CKS21, Lemma 3.5–3.7], which are the key
+ingredients of the proof there; moreover Lemma 2.2 replaces the usual Cac-
+ciopoli inequality for weak q-subsolutions. These lemmas have additional
+ε-term, which arise from the (1 + ε)-local minimizerslity of u, but this does
+not contribute any major difference from the proof for [CKS21, Lemma 4.3].
+Hence, we shall omit the details.
+We observe that due to (2.7) (as well as the assumption
+´
+Q4(u+)p dx = 1),
+the hypothesis of Lemma 2.4 is satisfied (with κ = 1 > εrσp). Thus, choosing
+ε ≤ εδ with εδ as in Lemma 2.4 with δ < σ, we deduce
+(2.8)
+ˆ
+Qr
+(u+)p dx ≤ cr−δp,
+∀r ∈ (0, 1).
+Inserting (2.8) into (2.7) yields that
+(2.9)
+ˆ
+Qr
+|Du+|p dx ≤ cεr−(1−(σ−δ))p,
+∀r ∈ (0, 1);
+now c depends only on n, p, q, σ and δ. Let us remark that this step does
+not appear for the case of minimizers [CKS21, Lemma 4.3] because for the
+latter case we can use the subsolution property [CKS21, Lemma 3.4] for u+
+to obtain its local boundedness.
+
+9
+The growth estimate in (2.9) is obtained by choosing ε sufficiently small.
+Taking ε even smaller if necessary, we may repeat the above argument
+around any point z ∈ Q1, and obtain
+ˆ
+Qr(z)
+|Du+|p dx ≤ cεr−(1−(σ−δ))p,
+∀r ∈ (0, 1), ∀z ∈ Q1,
+possibly with a larger constant c. Therefore, by Morrey’s lemma, we deduce
+that u+ ∈ C0,σ−δ(Q1) and
+(2.10)
+[u+]C0,σ−δ(Q1) ≤ cε
+1
+p .
+Finally, by Lemma 2.6, |{u ≤ 1
+2} ∩ Q1| ≤ cε. Hence, with cε ≤ 2−2n−1, we
+have |{u > 1
+2} ∩ Q1| > 0, which now implies via (2.10) that
+inf
+Q1 u+ ≥ 1
+2 − cε
+1
+p > 0,
+provided that we choose ε even smaller. Note that the smallness condition
+for ε at this stage can be determined solely by n, p and q, by for instance
+selecting σ = 1
+2 and δ = 1
+4. This finishes the proof for the case q < n.
+The case for q ≥ n can be treated similarly, following the proof of [CKS21,
+Lemma 4.3]; we omit the details.
+□
+We are ready to prove Proposition 2.1.
+Proof of Proposition 2.1. Let ¯ε be as in Lemma 2.7, and suppose that cε ≤ ¯ε.
+Using |{u ≤ 1
+2} ∩ Q1| ≤ ε, we may follow the proof of [CKS21, Proposition
+4.2] to find a constant ρ, depending only on n, p and q, such that
+(2.11)
+ˆ
+Q4ρ
+�(u−)q
+ρq
++ |Du+|p
+�
+dx ≤ cερq−p
+ˆ
+Q4ρ
+(u+)p dx.
+Therefore, defining uρ : Q4 → R by
+uρ(x) =
+u+(ρx)
+(4ρ)− n
+p ∥u+∥Lp(Q4ρ)
+−
+u−(ρx)
+4− n
+q ρ1− p
+q − n
+q ∥u+��
+p
+q
+Lp(Q4ρ)
+,
+we see that uρ ∈ W 1,p∧q(Q4) is a local (1 + ε)-minimizer of the functional
+J, such that
+ˆ
+Q4
+(u+
+ρ )p dx = 1,
+ˆ
+Q4
+((u−
+ρ )q + |Du+
+ρ |p) dx ≤ cε.
+Since cε ≤ ¯ε, with ¯ε as in Lemma 2.6, we obtain
+uρ > 0
+a.e. in Q1.
+Rescaling back, we obtain that u > 0 a.e. in Q4ρ as desired.
+□
+
+10
+SUNGHAN KIM AND HENRIK SHAHGHOLIAN
+3. H¨older regularity
+In this section, we study the universal H¨older regularity of local (1 + ε)-
+minimizers for the functional Jp,q, and prove our first main result, Theorem
+1.2. Let us begin with a lemma that tells us how each phase of local mini-
+mizers for the functional Jp,q should scale relatively to one another.
+Lemma 3.1. Let u ∈ W 1,p∧q(Q1) be a local minimizer of the functional J,
+such that ∥u+∥Lp(Q1) = 1 and u(0) = 0. If ∥u+∥Lp(Q1/2) ≥ β for some β > 0,
+then ∥u−∥Lq(Q1) ≥ cβ, for some positive constant cβ depending only on n,
+p, q and β.
+Proof. Let β be any constant, with 0 < β < 1. Assume by way of contradic-
+tion that there exists a minimizer uj ∈ W 1,p∧q(Q1) of the functional J, such
+that ∥u+
+j ∥Lp(Q1) = 1, ∥u+
+j ∥Lp(Q1/2) ≥ β, uj(0) = 0 but ∥u−
+j ∥Lq(Q1) ≤ 1
+j . By
+[CKS21, Theorem 1.1], uj ∈ C0,σ(Q1/2) and ∥u+
+j ∥C0,σ(Q1/2) ≤ c∥u+
+j ∥Lp(Q1) ≤
+c, and similarly, ∥u−
+j ∥C0,σ(Q1/2) ≤ c
+j, where both c and σ depend only on n, p
+and q. This together with the Cacciopoli inequality (Lemma 2.2 with ε = 0)
+implies that u+
+j → u0 weakly in W 1,p(Q1/2) and uniformly in Q1/2, while
+u−
+j → 0 weakly in W 1,q(Q1/2) and uniformly in Q1/2, for some nonnegative
+function u0 ∈ W 1,p(Q1/2). The uniform convergence along with uj(0) = 0
+implies that u0(0) = 0. In addition, passing to the limit in ∥u+
+j ∥Lp(Q1/2) ≥ β
+ensures that ∥u0∥Lp(Q1/2) ≥ β. However, the weak convergence of the gradi-
+ent of uj implies that u0 is also a minimizer of the functional J. As u0 ≥ 0
+in Q1/2, u0 is a p-harmonic function, but then it violates the minimizer
+principle, as ∥u0∥Lp(Q1/2) ≥ β > 0.
+□
+Lemma 3.2. Let u ∈ W 1,p∧q(Q1) be a local minimizer of the functional J,
+such that
+∥u+∥Lp(Q1) ≤ 1,
+u(0) = 0,
+sup
+0<r<1
+1
+rn+σ−q
+ˆ
+Qr
+(u−)q dx ≤ c−,
+for some constants c− > 0 and σ− ∈ (0, 1]. Then with σ+ = 1 − (1 − σ−) q
+p,
+sup
+0<r<1
+1
+rn+σ+p
+ˆ
+Qr
+(u+)p dx ≤ c+,
+where c+ depends only on n, p, q, σ− and c−.
+Proof. Let c−, σ− be given, and set σ+ as in the statement. Suppose that
+the conclusion of this lemma is false. Then for each j ∈ N, one can find a
+minimizer uj ∈ W 1,p∧q(Q1) for the functional J, such that
+(3.1)
+ˆ
+Q1
+(u+
+j )p dx ≤ 1,
+uj(0) = 0,
+sup
+0<r<1
+1
+rn+σ−q
+ˆ
+Qr
+(u−
+j )q dx ≤ c−,
+but
+(3.2)
+Sj :=
+sup
+rj
+2 ≤r≤1
+1
+rn+σ+p
+ˆ
+Qr
+(u+
+j )p dx → ∞
+
+11
+where the supremum is achieved at r = 1
+2rj; since ∥u+
+j ∥Lp(Q1) ≤ 1, we must
+have rj → 0. Define
+vj(y) :=
+u+
+j (rjy)
+r
+− n
+p
+j
+∥u+
+j ∥Lp(Qrj )
+−
+u−
+j (rjy)
+r
+1− p
+q − n
+q
+j
+∥u+
+j ∥
+p
+q
+Lp(Qrj )
+.
+Then vj is a local minimizer of the functional J, such that by (3.1) and (3.2),
+∥v+
+j ∥Lp(Q1) = 1, ∥v+
+j ∥Lp(Q1/2) ≥ 2−σ+ and vj(0) = 0. Therefore, Lemma 3.1
+yields that ∥v−
+j ∥Lq(Q1) ≥ cσ+. This implies that
+(3.3)
+1
+rn
+j
+ˆ
+Qrj
+(u−
+j )q dx ≥ cq
+σ+rq−p
+j
+ˆ
+Qrj
+(u+
+j )p dx ≥
+S
+q
+p
+j cq
+σ+
+2σ+q rq−p+σ+p
+j
+.
+Putting (3.1) and (3.3) together, and recalling that σ+ = 1 − (1 − σ−) q
+p,
+cq
+σ− ≥
+S
+q
+p
+j cq
+σ+
+2σ+q ,
+a contradiction to the assumption that Sj → ∞.
+□
+Thanks to the above lemma, we can prove Theorem 1.2 for minimizers of
+the functional J.
+Proof of Theorem 1.2 for minimizers. It suffices to consider the case p > q,
+and
+ˆ
+Q1
+((u+)p + (u+)q) dx = 1.
+By [CKS21, Theorem 1.1], we already know that u ∈ C0,σ(Q1) and that
+[u]C0,σ(Q1) ≤ c, where both c > 0 and σ ∈ (0, 1) depend only on n, p and q.
+Hence, if u(z) = 0 at some z ∈ Q1/2, then
+sup
+0<r< 1
+2
+1
+rn+σq
+ˆ
+Qr(z)
+(u−)q dx ≤ c,
+which along with Lemma 3.2 implies that
+sup
+0<r< 1
+2
+1
+rn+p−q+σq
+ˆ
+Qr(z)
+(u+)p dx ≤ c,
+where the constant c in both displays depends only on n, p and q. Since
+p > q, 1 − (1 − σ) q
+p > σ > 0. Now setting σ− = σ and σ+ = 1 − (1 − σ) q
+p,
+we immediately verify the relation required between σ+ and σ−. Since the
+above growth estimates hold uniformly around all z ∈ {u = 0} ∩ Q1, and
+since ∆pu = 0 in {u > 0}∩Q1 and ∆qu = 0 in {u < 0}∩Q1, one may arrive
+at the conclusion via some standard manipulation. We skip the detail.
+□
+
+12
+SUNGHAN KIM AND HENRIK SHAHGHOLIAN
+Given a measurable function u : Ω → R, define D+(u), D−(u) and Γ(u)
+by the subset of Ω as follows:
+D+(u) = {z ∈ Ω : u > 0 a.e. in some Qr(z) ⊂ Ω},
+D−(u) = D+(−u),
+and
+Γ(u) = Ω \ (D+(u) ∪ D−(u)).
+By definition, both D+(u) and D−(u) are open and hence Γ(u) is closed
+(relative to the topology of Ω). Moreover, z ∈ Γ(u) if and only if |{u ≥
+0} ∩ Qr(z)||{u ≤ 0} ∩ Qr(z)| > 0 for any cube Qr(z) ⊂ Ω.
+With Proposition 2.1 at hand, we shall obtain, as a contraposition along
+with Lemma 3.4 below, that if a local (1 + ε)-minimizer vanishes (in an ap-
+propriate Lebesgue sense) at certain point in the interior, then each phase
+exhibits certain universal H¨older growth. More exactly, we assert the fol-
+lowing.
+Proposition 3.3. There exists a constant ¯σ ∈ (0, 1), depending only on
+n, p, and q, for which the following holds: for each σ ∈ (0, ¯σ), one can
+find a constant εσ ∈ (0, 1), depending only on n, p, q, and σ, such that if
+u ∈ W 1,p∧q(Q1) is a local (1 + εσ)-minimizer of the functional J satisfying
+ˆ
+Q1
+((u+)p + (u−)q) dx ≤ 1,
+0 ∈ Γ(u),
+then with σ+ = σ and σ− = 1 − (1 − σ)p
+q, one has
+sup
+0<r<1
+�
+1
+rσ+p
+ˆ
+Qr
+(u+)p dx +
+1
+rσ−q
+ˆ
+Qr
+(u−)q dx
+�
+≤ cσ,
+where cσ depends only on n, p, q, and σ.
+The following lemma will play a key role (together with Proposition 2.1).
+Lemma 3.4. There exists a constant ¯σ ∈ (0, 1), depending only on n, p,
+and q, for which the following holds: for each σ ∈ (0, ¯σ) and each τ ∈ (0, 1
+2],
+one can find εσ,τ ∈ (0, 1), depending only on n, p, q, σ, and τ, such that if
+u ∈ W 1,p∧q(Q1) is a local (1+εσ+,τ)-minimizer of the functional J satisfying,
+(3.4)
+ˆ
+Q1
+((u+)p + (u−)q) dx = 1,
+|E+(u, Qr)|
+|Qr|
+∧ |E−(u, Qr)|
+|Qr|
+≥ τ,
+for some r ∈ (0, 1), where
+E+(u, Qr) =
+�
+u ≤ 1
+2rΛ(u, Qr)
+1
+p
+�
+,
+E−(u, Qr) =
+�
+u ≥ −1
+2rΛ(u, Qr)
+1
+q
+�
+,
+Λ(u, Qr) =
+ˆ
+Qr
+�(u+)p
+rp
++ (u−)q
+rq
+�
+dx,
+
+13
+then with σ+ = σ and σ− = 1 − (1 − σ)p
+q, one has
+sup
+r≤ρ≤1
+�
+1
+ρσ+p
+ˆ
+Qρ
+(u+)p dx +
+1
+ρσ−q
+ˆ
+Qρ
+(u−)q dx
+�
+≤ cσ,τ,
+where cσ,τ depends on the same parameters that determine εσ,τ.
+Proof. Let ¯σ be determined later, and fix τ ∈ (0, 1
+2], σ ∈ (0, ¯σ), and set σ±
+as in the stastement. Suppose by way of contradiction that for each j ∈ N,
+we can find some positive constant εj → 0, some local (1 + εj)-minimizer
+uj ∈ W 1,p∧q(Q1) of the functional J, and a radius rj ∈ (0, 1) such that
+(3.5)
+ˆ
+Q1
+((u+
+j )p + (u−
+j )q) dx ≤ 1,
+|E+(uj, Qrj)|
+|Qrj|
+∧ |E−(uj, Qrj)|
+|Qrj|
+≥ τ, ,
+but
+(3.6)
+Sj =
+sup
+rj≤r≤1
+�
+1
+rσ+p
+ˆ
+Qr
+(u+
+j )p dx +
+1
+rσ−q
+ˆ
+Qr
+(u−
+j )q dx
+�
+→ ∞,
+with the supremum achieved at level r = rj. In order for (3.6) to be com-
+patible with the first equality in (3.5), we must have rj → 0.
+Define vj : Qr−1
+j
+→ R by
+vj(y) =
+u+
+j (rjy)
+S
+1
+p
+j rσ+
+j
+−
+u−
+j (rjy)
+S
+1
+q
+j rσ−
+j
+.
+By the way that it is rescaled, vj is a local (1+εj)-minimizer of the functional
+J in Q1/rj. Moreover, by (3.5) along with the relation Sjrσ+p
+j
+= Λjrp
+j and
+Sjrσ−q
+j
+= Λjrq
+j, where Λj = Λ(uj, Qrj),
+(3.7)
+����
+�
+|vj| ≤ 1
+2
+�
+∩ Q1
+���� ≥ τ,
+and by (3.6),
+(3.8)
+sup
+1≤R≤r−1
+j
+�
+1
+Rσ+p
+ˆ
+QR
+(v+
+j )p dy +
+1
+Rσ−q
+ˆ
+QR
+(v−
+j )q dy
+�
+= 1,
+where the supremum is achieved at R = 1.
+Thanks to (3.7) and (3.8), one can argue analogously in the proof of
+Lemma 2.4 to obtain a minimizer v ∈ W 1,p∧q
+loc
+(Rn) of the functional J, with
+v+ ∈ W 1,p
+loc (Rn) and v− ∈ W 1,q
+loc (Rn) such that
+(3.9)
+����
+�
+|v| ≤ 1
+2
+�
+∩ Q1
+���� ≥ τ,
+and
+(3.10)
+sup
+R≥1
+�
+1
+Rσ+p
+ˆ
+QR
+(v+)p dy +
+1
+Rσ−q
+ˆ
+QR
+(v−)q dy
+�
+= 1,
+
+14
+SUNGHAN KIM AND HENRIK SHAHGHOLIAN
+where the supremum is achieved at R = 1. In particular, the latter obser-
+vation indicates that v is nontrivial.
+At this point, we choose ¯σ as the positive exponent for Theorem 1.2 for
+minimizers; let us remind the readers that the statement for minimizers is
+proved right after the proof of Lemma 3.2. Set ¯σ+ := ¯σ and ¯σ− := 1 − (1 −
+¯σ) q
+p. As v is a local minimizer of the functional J in Q2R, v+ ∈ C0,¯σ+(QR),
+v− ∈ C0,¯σ−(QR) and then by (3.10), we derive that
+[v+]p
+C0,¯σ+(QR) + [v−]q
+C0,¯σ−(QR) ≤
+c
+R(¯σ+−σ+)p +
+c
+R(¯σ−−σ−)q ,
+for any R > 1. As σ+ = σ < ¯σ+ and σ− = 1 − (1 − σ) q
+p < ¯σ−, sending
+R → ∞ implies that both v+ and v− must be constant. Then by (3.9),
+|v| ≤ 1
+2 everywhere in Q1, whence
+´
+Q1((v+)p + (v+)q) dx ≤ 2−p + 2−q < 1, a
+contradiction to the observation that the supremum in (3.10) is attained at
+R = 1.
+□
+We are ready to prove Proposition 3.3
+Proof of Proposition 3.3. As 0 ∈ Γ(u), there are three cases to consider: (i)
+|{u > 0} ∩ Qρ||{u < 0} ∩ Qρ| > 0 for all ρ ∈ (0, 1), (ii) u ≥ 0 a.e. in Qρ
+for some small ρ > 0, and (iii) u ≤ 0 a.e. in Qρ for some small ρ > 0. The
+last two cases are symmetric, and in those cases u becomes a local (1 + ε)-
+minimizer for the functional Jp,p, or Jq,q depending on its sign. Thus, the
+growth estimate follows easily, once we establish the estimate for the first
+case. We leave out this part as an exercise for the reader.
+Henceforth, let us assume that the first case holds. Let (ε, τ, µ) be the
+triple of constants from Proposition 2.1 that are determined solely by n, p
+and q. Fix any r ∈ (0, 1). Since |{u > 0} ∩ Qµr| · |{u < 0} ∩ Qµr| > 0,
+as a contraposition (applied to both u and −u, after suitable rescaling), we
+obtain that
+(3.11)
+|E+(u, Qr)|
+|Qr|
+∧ |E−(u, Qr)|
+|Qr|
+≥ τ,
+with E+(u, Qr) and E−(u, Qr) defined as in Lemma 3.4. As τ being a con-
+stant depending only on n, p and q, the conclusion of this proposition follows
+immediately from Lemma 3.4; this final step introduces another condition
+on the size of ε, which through the dependence of τ would be determined
+again solely by n, p, q, and σ.
+□
+4. Almost Lipschitz regularity
+Here we prove almost Lipschitz regularity of almost minimizers to J =
+Jp,q, when p and q are close. Our proof is based on the compactness argu-
+ment. The basic ingredient is the universal H¨older estimate for local mini-
+mizers of the functional Jp,q, see [CKS21, Theorem 1.3]. Although it is not
+specified in the statement, one can observe from the higher integrability of
+each phase that the H¨older regularity is uniform when p (or q) is close to n.
+
+15
+We record this fact as a lemma below, as the proof of [CKS21, Theorem 1.3]
+makes use of the local boundedness and the Harnack inequality for weak p-
+harmonic functions, and the constants involved in the latter assertions may
+vary as p → n.
+Lemma 4.1. Let u ∈ W 1,p+∧p−(Q2) be a local minimizer of Jp+,p−. There
+exists ¯δ > 0, depending only on n, such that if |n − p±| ≤ ¯δ, then
+[u±]C0,¯σ(Q1) ≤ ¯c∥u±∥Lp±(Q2),
+where ¯σ ∈ (0, 1) and ¯c > 1 depend only on n.
+Proof. Since u is a local minimizer (instead of (1 + ε)-minimizer) of Jp+,p−,
+u± is a weak p±-subsolution in Q2, according to [CKS21, Lemma 3.4]. Hence,
+by [GG82, Corollary 4.2], there exist constants ¯δ > 0 and ¯γ ∈ (0, 1), both
+depending only on n, such that if |p± − n| < ¯δ, then u± ∈ W 1,p±+¯δ(Q1) ⊂
+W 1,n+¯γ¯δ(Q1). Now setting ¯σ := 1 −
+n
+n+¯γ¯δ, it follows from the Sobolev em-
+bedding, the higher integrability and the Cacciopoli inequality for weak
+p±-subsolutions that
+[u±]C0,¯σ(Q1) ≤ c1(n)
+�ˆ
+Q1
+|Du±|n+¯γ¯δ dx
+�
+1
+n+¯γ¯δ
+≤ c1(n)c2(n, p±)
+�ˆ
+Q3/2
+|Du±|p± dx
+� 1
+p±
+≤ c1(n)c2(n, p±)c3(n, p±)
+�ˆ
+Q2
+(u±)p± dx
+� 1
+p± .
+Note that c2(n, p±), and c3(n, p±) are constants from the higher integrabil-
+ity and respectively the Cacciopoli inequality, and these are all uniformly
+bounded by a constant c(n), as p → p±. Hence, our proof is finished.
+□
+Let us first verify the uniform growth of order σ at free boundary points
+for minimizers. We prove it by compactness.
+Lemma 4.2. Let u ∈ W 1,p∧q(Q1) be a local minimizer of Jp,q such that
+(4.1)
+ˆ
+Q1
+((u+)p + (u−)q) dx ≤ 1,
+u(0) = 0.
+Then for any σ ∈ (0, 1), there exists δ > 0, depending only on n, p, and σ,
+such that if |p − q| < δ, then with σ+ = σ and σ− = 1 − (1 − σ)p
+q,
+1
+rn+σ+p
+ˆ
+Qr
+(u+)p dx +
+1
+rn+σ−q
+ˆ
+Qr
+(u−)q dx ≤ c,
+∀r ∈ (0, 1),
+where c > 1 depends only on n, p, and σ.
+Proof. Let σ > 0 and p ∈ (1, ∞) be given. Suppose that the conclusion of
+this lemma does not hold. Then for each j = 1, 2, · · · , there must exist an
+
+16
+SUNGHAN KIM AND HENRIK SHAHGHOLIAN
+exponent qj > 1 with |qj − p| ց 0, a local minimizer uj ∈ W 1,p∧qj(Q1) of
+the functional Jp,qj, and a scale rj ∈ (0, 1), such that
+(4.2)
+ˆ
+Q1
+((u+
+j )p + (u−
+j )qj) dx ≤ 1,
+uj(0) = 0,
+but with σ+ = σ and σj,− = 1 − (1 − σ) p
+qj → σ,
+(4.3)
+Sj :=
+sup
+rj≤r≤1
+�
+1
+rσ+p
+ˆ
+Qr
+dx +
+1
+rσj,−qj
+ˆ
+Qr
+(u−
+j )qj
+�
+dx ր ∞.
+To have the first inequality in (4.2) and (4.3) to be compatible, we must have
+rj ց 0 up to a subsequence. As in the proof of Lemma 3.4, we consider the
+rescaling
+vj(y) :=
+u+
+j (rjy)
+S
+1
+p
+j rσ+
+j
+−
+u−
+j (rjy)
+S
+1
+qj
+j rσj,−
+j
+.
+Then vj is a minimizer of Jp,qj in Q1/rj and that
+(4.4)
+sup
+1≤R≤ 1
+rj
+�
+1
+Rσ+p
+ˆ
+QR
+(v+
+j )p dx +
+1
+Rσj,−qj
+ˆ
+QR
+(v−
+j )qj dx
+�
+= 1.
+Then by [CKS21, Theorem 1.2], we have
+(4.5)
+sup
+j
+∥vj∥C0,¯σ(QR) < ∞,
+where both c > 1 and ¯σ ∈ (0, 1) depend only on n and p; see Lemma
+4.1 for the stability of ¯σ and c for the case p = n. Moreover, by [CKS21,
+Lemma 3.4], v+
+j (and v−
+j ) is a weak p-(resp. qj-)subsolution, so the higher
+integrability [GG82, Theorem 4.1] applies. Utilizing |qj−p| ց 0, there exists
+η > 0, depending only on n and p, such that
+(4.6)
+sup
+j
+ˆ
+QR
+|Dvj|p+η dx < ∞.
+Also observe from (4.2) that
+(4.7)
+vj(0) = 0.
+By (4.5) and (4.6), we can extract a subsequence of {vj}∞
+j=1 along which
+vj → v weakly in W 1,p+η
+loc
+(Rn) and locally uniformly in Rn, for some v ∈
+W 1,p+η
+loc
+∩ C0,σ
+loc (Rn). Let us continue to denote this subsequence by {vj}∞
+j=1.
+The uniform convergence along with (4.7) implies that
+(4.8)
+v(0) = 0.
+We claim that v is a (weak) p-harmonic function in Rn.
+For any large j, we have qj ∈ (p − η, p + η). By (4.6) and the compact
+embedding, vj → w strongly in W 1,p
+loc (Rn). Now fix R ≥ 1, and let ϕ ∈
+W 1,p+η
+0
+(QR) be arbitrary. Then since |qj − p| ց 0, |Dv−
+j |qj → |Dv−|p and
+
+17
+|D(vj+ϕ)−|qj → |D(v+ϕ)|p a.e. in QR. Then by the dominated convergence
+theorem and the minimizerslity of Jp,qj(vj, QR),
+(4.9)
+ˆ
+QR
+|Dv|p dx = lim
+k→∞
+ˆ
+QR
+(|Dv+
+j |p + |Dv−
+j |qj) dx
+≤ lim
+k→∞
+ˆ
+QR
+(|D(vj + ϕ)+|p + |D(vj + ϕ)−|qj) dx
+=
+ˆ
+QR
+|D(v + ϕ)|p dx.
+Thus, v minimizes Jp,p(·, QR) over all variations v+ϕ with ϕ ∈ W 1,p+η
+0
+(QR).
+This suffices to guarantee v to be (weak) p-harmonic in QR, see [Lin19].
+Since R was any number larger than 1, the claim is now verified.
+Now letting k → ∞ in (4.4) and using qj → p, we obtain
+(4.10)
+sup
+R≥1
+1
+Rσp
+ˆ
+QR
+|v|p dx = 1.
+By the interior Lipschitz estimate for p-harmonic functions,
+(4.11)
+[v]C0,1(QR) ≤
+c
+R1−σ ,
+for some c independent of R. Taking R → ∞ in (4.11), we derive that v is
+constant in Rn, which together with (4.8) implies v ≡ 0. This is yields a
+contradiction against (4.10), and the proof is finished.
+□
+Next we extend the above lemma to local (1 + ε)-minimizers.
+Lemma 4.3. For any σ ∈ (0, 1), there exists ε, δ > 0, depending only on n,
+p, and σ, such that for any q ∈ (1, ∞) with |p − q| < δ, and any local local
+(1 + ε)-minimizer u ∈ W 1,p∧q(Q1) satisfying (4.1), one has, with σ+ = σ
+and σ− = 1 − (1 − σ)p
+q, that
+1
+rn+σ+p
+ˆ
+Qr
+(u+)p dx +
+1
+rn+σ−q
+ˆ
+Qr
+(u−)q dx ≤ c,
+∀r ∈ (0, 1),
+where c > 1 depends only on n, p, and σ.
+Proof. As already observed in the proof of Proposition 3.3, the assumption
+u(0) = 0 implies (3.11) for every r ∈ (0, 1). Hence, the assumption (4.1)
+implies (3.4). The rest of the proof is the same with that of Lemma 3.4. More
+exactly, given σ ∈ (0, 1) and p > 1, we first choose δ > 0 sufficiently small
+such that Lemma 4.2 holds with 1+σ
+2
+in place of σ, for all local minimizers
+for functional Jp,q for any q ∈ (1, ∞) with |p − q| < δ. Then we can take
+ε > 0 small enough such that Lemma 3.4 holds with τ as in (3.11), σ+ = σ
+and σ− = 1 − (1 − σ)p
+q, ¯σ+ = 1+σ
+2
+> σ = σ−, and ¯σ− = 1 − (1−σ
+2 )p
+q >
+1 − (1 − σ)p
+q = σ−. We skip the details.
+□
+We are ready to prove the almost Lipschitz regularity for almost mini-
+mizers, when |p − q| ≪ 1.
+
+18
+SUNGHAN KIM AND HENRIK SHAHGHOLIAN
+Proof of Theorem 1.3. With the same (and simpler) compactness argument,
+we can also prove that local (1 + ε)-minimizers for Jp,p(w) ≡
+´
+|Dw|p dx is
+of class C0,σ, for any σ ∈ (0, 1) and every ε ∈ (0, εσ), since p-harmonic
+functions are of class C1,α ⊂ C0,1. Moreover, we can obtain a uniform C0,σ-
+estimates, with this compactness argument, and the smallness constant εσ
+depends only on n, p, and σ. Thus, the passage from Lemma 4.3 to Theorem
+1.3 is standard. We shall not present the obvious details here.
+□
+Declarations
+Data availability statement: All data needed are contained in the man-
+uscript.
+Funding and/or Conflicts of interests/Competing interests: The
+authors declare that there are no financial, competing or conflict of interests.
+References
+[AT15]
+M. D. Amaral and E. V. Teixeira, Free transmission problems, Comm. Math.
+Phys. 337 (2015), 1465–1489.
+[CKS21] M. Colombo, S. Kim and H. Shahgholian, A transmission problem for (p, q)-
+Laplacian, to appear in Comm. in Partial Differential Equations.
+[DET19] G. David, M. Engelstein, and T. Toro, Free boundary regularity for almost min-
+imizers, Adv. Math. 350 (2019), 1109–1192.
+[DS20]
+D. De Silva and O. Savin, Almost minimizers of the one-phase free boundary
+problem, Comm. Partial Differential Equations 45 (2020), 913–930.
+[DJS22] D. De Silva, S. Jeon and H. Shahgholian, Almost minimizers for a singular system
+with free boundary, J. Differential Equations 336 (2022), 167–203.
+[GG82]
+M. Giaquinta and E. Giusti, On the regularity of the minimizers of variational
+integrals, Acta Math. 148 (1982), 31–46.
+[Giu03]
+E. Giusti, Direct methods in the Calculus of Variations, World Scientific, 2003.
+[Lin19]
+P. Lindqvist, Notes on the Stationary p-Laplace Equation, Springer International
+Publishing, 2019.
+Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden
+Email address: [email protected]
+Department of Mathematics, Royal Institute of Technology, 100 44 Stock-
+holm, Sweden
+Email address: [email protected]
+

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+IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+1
+A Survey on Federated Recommendation Systems
+Zehua Sun∗, Yonghui Xu∗, Yong Liu, Wei He, Yali Jiang, Fangzhao Wu, Lizhen Cui†
+Abstract—Federated learning has recently been applied to
+recommendation systems to protect user privacy. In federated
+learning settings, recommendation systems can train recom-
+mendation models only collecting the intermediate parameters
+instead of the real user data, which greatly enhances the user
+privacy. Beside, federated recommendation systems enable to
+collaborate with other data platforms to improve recommended
+model performance while meeting the regulation and privacy
+constraints. However, federated recommendation systems faces
+many new challenges such as privacy, security, heterogeneity
+and communication costs. While significant research has been
+conducted in these areas, gaps in the surveying literature still
+exist. In this survey, we—(1) summarize some common privacy
+mechanisms used in federated recommendation systems and
+discuss the advantages and limitations of each mechanism; (2)
+review some robust aggregation strategies and several novel at-
+tacks against security; (3) summarize some approaches to address
+heterogeneity and communication costs problems; (4)introduce
+some open source platforms that can be used to build federated
+recommendation systems; (5) present some prospective research
+directions in the future. This survey can guide researchers and
+practitioners understand the research progress in these areas.
+Index Terms—Recommendation Systems, Federated Learning,
+Privacy, Security, Heterogeneity, Communication costs.
+I. INTRODUCTION
+I
+N recent years, recommendation systems have been widely
+used to model user interests so as to solve information over-
+load problems in many real-world fields, e.g., e-commerce [1]
+[2], news [3] [4] and healthcare [5] [6]. To further improve the
+recommendation performance, such systems usually collect as
+much data as possible, including a lot of private information of
+users, such as user attributes, user behaviors, social relations,
+and context information.
+Although these recommendation systems have achieved
+remarkable results in terms of accuracy, most of them require a
+central server to store collected user data, which exist potential
+privacy leakage risks because user data could be sold to
+a third party without user consent, or stolen by motivated
+attackers. In addition, due to privacy concerns and regulation
+restrictions, it becomes more difficult to integrate data from
+other platforms to improve recommendation performance. For
+example, regulations such as General Data Protection Reg-
+ulation (GDPR) [7] set strict rules on collecting user data
+and sharing data between different platforms, which may lead
+to insufficient data for recommendation systems and further
+affects the recommendation performance.
+Zehua Sun, Yonghui Xu, Wei He, Yali Jiang and Lizhen Cui are with Joint
+SDU-NTU Centre for Artificial Intelligence Research (C-FAIR) & Software
+School, Shandong University.
+Yong Liu are with Alibaba-NTU Singapore Joint Research Institute,
+Nanyang Technological University, Singapore.
+Fangzhao Wu are with Microsoft Research Asia, China.
+∗Zehua Sun and Yonghui Xu are Co-First authors.
+†Corresponding author: [email protected].
+Federated learning is a privacy-preserving distributed learn-
+ing scheme proposed by Google [8], which enables par-
+ticipants to collaboratively train a machine learning model
+by sharing intermediate parameters (e.g., model parameters,
+gradients) to the central server instead of their real data.
+Therefore, combining federated learning with recommendation
+systems becomes a promising solution for privacy-preserving
+recommendation systems. In this paper, we term it federated
+recommendation system (FedRS).
+A. Challenges
+While FedRS avoid direct exposure of real user data and
+provides a privacy-aware paradigm for model training, there
+are still some core challenges that need to be addressed.
+Challenge 1: Privacy concerns. Privacy protection is often
+the major goal of FedRS. In FedRS, each participant jointly
+trains a global recommendation model by sharing interme-
+diate parameters instead of their real data, which makes an
+important step towards privacy-preserving recommendation
+systems. However, a curious sever can still infer some sensitive
+information (e.g., user behavior, ratings) from the intermediate
+parameters [9] [10].
+Challenge 2: Security attacks. In FedRS, participants may
+be malicious. They can attack the security of FedRS by
+poisoning the local training samples or the intermediate pa-
+rameters uploaded. As a result, attackers can increase/decrease
+the exposure ratio of specific items [12] or degrade the
+overall performance of the recommendation model [11]. In
+addition, some attackers try to use well-designed constraints to
+approximate the patterns of benign participants, which further
+increases the difficulty of defense and detection [55].
+Challenge 3: Heterogeneity. FedRS also faces the problem
+of system, statistical and privacy heterogeneity. System hetero-
+geneity means that the storage, computing, and communication
+capabilities of clients usually vary greatly, clients with limited
+capability may become stragglers and affect training efficiency.
+Statistical heterogeneity means that data in different clients
+is often not independent and identically distributed (Non-
+IID), which significantly affects global model convergence
+and personalization of recommendation results. Privacy het-
+erogeneity means that users and information usually have
+different privacy constraints, thus using the same privacy
+budgets for them will bring unnecessary loss of accuracy and
+efficiency.
+Challenge 4: Communication costs. To achieve satisfac-
+tory recommendation performance, clients need to commu-
+nicate with the central server for multiple rounds. However,
+the real-world recommendation systems are usually built on
+complex deep learning models and millions of intermediate
+parameters needs to be communicated [13]. Therefore, clients
+arXiv:2301.00767v1  [cs.IR]  27 Dec 2022
+
+IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+2
+Fig. 1: Communication architecture of FedRS.
+may be hard to afford severe communication costs, which lim-
+its the application of FedRS in large-scale industrial scenarios.
+B. Related Surveys
+There are many surveys that have focused on recommenda-
+tion systems or federated learning. For example, Adomavicius
+et al. [14] provide a detailed categorization of recommenda-
+tion methods and introduce various limitations of each method.
+Yang et al. [15] give the definition of federated learning
+and discuss its architectures and applications. And Li et al.
+[16] summarize the unique characteristics and challenges of
+federated learning. However the existing surveys usually treat
+recommendation systems and federated learning separately,
+and few work surveyed specific problems in FedRS [17].
+Yang et al. [17] categorize FedRS from the aspect of the
+federated learning and discuss the algorithm-level and system-
+level challenges for FedRS. However, they do not provide
+comprehensive methods to address privacy, security, hetero-
+geneity, and communication costs challenges.
+C. Our Contribution
+Compared with the previous surveys, this paper makes
+the following contributions: (1) We provide a comprehensive
+overview of FedRS from the perspectives of definition, com-
+munication architectures and categorization. (2) We summarize
+the state-of-the-art studies of FedRS in terms of privacy,
+security, heterogeneity and communication costs areas. (3) We
+introduce some open source platforms for FedRS, which can
+help engineers and researchers develop algorithms and deploy
+applications of FedRS. (4) We discuss the promising future
+directions for FedRS.
+The rest of the paper is organized as follow: Section II
+discusses the overview of FedRS. Section III-Section VI sum-
+marize the state-of-the-art studies of FedRS from the aspects
+of privacy, security, heterogeneity and communication costs.
+Section VII introduces the existing open source platforms.
+Section VIII presents some prospective research directions.
+Finally, Section IX concludes this survey.
+II. OVERVIEW OF FEDERATED RECOMMENDATION
+SYSTEMS
+A. Definition
+FedRS is a technology that provides recommendation ser-
+vices in a privacy preserving way. To protect user privacy, the
+participants in FedRS collaboratively train the recommenda-
+tion model by exchanging intermediate parameters instead of
+sharing their own real data. In ideal case, the performance of
+recommendation model trained in FedRS should be closed to
+the performance of the recommendation model trained in the
+data centralized setting, which can be formalized as:
+|VF ED − VSUM| < δ.
+(1)
+where VF ED is the recommendation model performance in
+FedRS , VSUM is the recommendation model performance
+in traditional recommendation systems for centralized data
+storage, and δ is a small positive numbers.
+B. Communication Architecture
+In FedRS, the data of participants is stored locally, and
+the intermediate parameters are communicated between the
+server and participants. There are two major communication
+architectures used in the study of FedRS, including client-
+server architecture and peer-peer architecture.
+Client-Server Architecture. Client-server architecture is
+the most common communication architecture used in FedRS,
+as shown in Fig. 1(a), which relies on a trusted central server
+to perform initialization and model aggregation tasks. In each
+round, the server distributes the current global recommenda-
+tion model to some selected clients. Then the selected clients
+use the received model and their own data for local training,
+and send the updated intermediate parameters (e.g., model
+parameters, gradients) to the server for global aggregation. The
+
+Server
+① Initialization
+① Initialization
+②
+①
+② Download model
+④
+② Local update
+Participant 1
+③ Local update
+③ Send parameters
+②
+@ Send parameters
+@ Aggregation
+4
+4
+②
+2)
+? Aggregation
+③
+③
+3
+3
+3
+①
+①
+④
+④
+Participant 1
+Participant 2
+Participant N
+Participant 2
+Participant N
+(a) Client-Server ahictecture
+(b) Peer-Peer ahictectureIEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+3
+Fig. 2: Categorization of federated recommendation systems.
+client-server architecture requires a central server to aggregate
+the intermediate parameters uploaded by the clients. Thus,
+once the server has a single point of failure, the entire training
+process will be seriously affected [18]. In addition, the curious
+server may infer the clients’ privacy information through the
+intermediate parameters, leaving potential privacy concerns
+[10].
+Peer-Peer Architecture. Considering the single point of
+failure problem for client-server architecture in FedRS, Hegeds
+et al. [19] design a peer-peer communication architecture
+with no central server involved in the communication process,
+which is shown in Fig. 1(b). During each communication
+round, each participant broadcasts the updated intermediate
+parameters to some random online neighbors in the peer to
+peer network, and aggregates received parameters into its
+own global model. In this architecture, the single point of
+failure and privacy issues associated with a central server
+can be avoided. However, the aggregation process occurs on
+each client, which greatly increases the communication and
+computation overhead for clients [20].
+C. Categorization
+In FedRS, the participants are responsible for the local
+training process as the data owners. They can be different
+mobile devices or data platforms. Considering the unique
+properties for different types of participants, FedRS usually
+have different application scenarios and designs. Besides, there
+are also some differences between different recommendation
+models in the federation process. Thus, we summarize the
+current FedRS and categorize them from the perspectives of
+participant type and recommendation model. Fig. 2 shows the
+summary of the categorization of FedRS.
+1) Participant Type: Based on the type of participants,
+FedRS can be categorized into cross-device FedRS and cross-
+platform FedRS.
+Cross-device FedRS. In cross-device FedRS, different mo-
+bile devices are usually treated as participants [21] [22].
+The typical application of cross-device FedRS is to build a
+personal recommendation model for users without collecting
+their local data. In this way, users can enjoy recommend
+service while protecting their private information. The number
+of participants in cross-device FedRS is relative large and each
+participant keeps a small amount of data. Considering the
+limited computation and communication abilities for mobile
+devices, cross-device FedRS cannot handle very complex
+training tasks. Besides, due to the power and the network
+status, the mobile devices may drop out of the training process.
+Thus, the major challenges for cross-device FedRS are how
+to improve the efficiency and deal with straggler problem of
+devices during training process.
+Cross-platform FedRS. In cross-platform FedRS, different
+data platforms are usually treated as participants who want
+to collaborate to improve recommendation performance while
+meeting regulation and privacy constraints [23] [24] [25].
+For example, In order to improve the recommendation per-
+formance, recommendation systems often integrate data from
+multiple platforms (e.g., e-commercial platforms , social plat-
+forms). However, due to the privacy and regulation concerns,
+the different data platforms are often unable to directly share
+their data with each other. In this scenario, cross-platform
+FedRS can be used to collaboratively train recommendation
+models between different data platforms without directly ex-
+changing their users’ data. Compared to cross-device FedRS,
+the number of participants in cross-platform FedRS is rela-
+tively small, and each participant owns relative large amount of
+data. An important challenge for cross-platform FedRS is how
+to design a fair incentive mechanism to measure contributions
+and benefits of different data platforms. Besides, it is hard
+to find a trusted server to manage training process in cross-
+platform FedRS, so a peer to peer communication architecture
+can be a good choice in this case.
+2) Recommendation Model:
+According to the different
+recommendation models used in FedRS, FedRS can be cat-
+egorized into matrix factorization model based FedRS, deep
+learning model based FedRS and meta learning model FedRS.
+Matrix factorization model based FedRS. Matrix fac-
+torization [26] is the most common model used in FedRS,
+which formulates the user-item interaction or rating matrix
+R ∈ RN×M as a linear combination of user profile matrix
+U ∈ RN×K and item profile matrix V ∈ RM×K:
+R = UV T .
+(2)
+then uses the learned model to recommendation new items to
+the user according to the predicted value. In matrix factoriza-
+tion model based FedRS, the user factor vectors are stored and
+updated locally on the clients, and only the item factor vectors
+[27] or the gradients of item factor vectors [21] [22] [10]
+[28] [29] are uploaded to the server for aggregation. Matrix
+factorization model based FedRS can simply and effectively
+capture user tastes with the interaction and rating information
+between users and items. However it still has many limitations
+such as sparsity (the number of ratings to be predicted is much
+smaller than the known ratings) and cold-start (new users and
+new items lacks ratings information) problems [14].
+Deep learning model based FedRS. To learn more com-
+plex representations of users and items and improve the
+recommendation performance, deep learning technology has
+been widely used in recommendation systems. However, as
+privacy regulations get stricter, it becomes more difficult for
+recommendation systems to collect enough user data to build
+a high performance deep learning model. To make the full
+
+Cross-DeviceFedRS
+Participant Type
+Cross-PlatformFedRS
+Federated
+Recommendation
+System
+Matrix Factorization Mode
+Based FedRS
+Recommendation
+Deep Learning Model
+Model
+Based FedRS
+Meta Learning Model
+Based FedRSIEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+4
+use of user data while meeting privacy regulations, many ef-
+fective deep learning model based FedRS have been proposed
+[30] [31] [32]. Considering different model structures, deep
+learning model based FedRS usually adopt different model
+update and intermediate parameter transmit processes. For
+examples, Perifanis et al. [30] propose a federated neural
+collaborative filtering (FedNCF) framework based on NCF
+[33]. In FedNCF, the clients locally update the network
+weights as well as the user and item profiles, then upload
+the item profile and network weights after masking to the
+server for aggregation. Wu et al. [31] propose a federated
+graph neural network (FedGNN) framework based on GNN.
+In FedGNN, the clients locally train GNN models and update
+the user/item embeddings from their local sub-graph, then send
+the perturbed gradients of GNN model and item embedding to
+the central server for aggregation. Besides, Huang et al. [34]
+propose a federated multi-view recommendation framework
+based on Deep Structured Semantic Model (DSSM [35]). In
+FL-MV-DSSM, each view i locally trains the user and item
+sub-models based on their own user data and local shared item
+data, then send the perturbed gradients of both user and item
+sub-models to server for aggregation. Although deep learning
+model based FedRS achieve outstanding performance in terms
+of accuracy, the massive model parameters of deep learning
+models bring huge computation and communication overhead
+to the clients, which presents a serious challenge for real
+industrial recommendation scenarios.
+Meta learning model based FedRS. The most of existing
+federated recommendation studies are built on the assump-
+tion that data distributed on each client is independent and
+identically (IID). However, learning a unified federated rec-
+ommendation model often performs poorly when handling the
+Non-IID and highly personalized data on clients. Meta learning
+model can quickly adapt to new tasks while maintaining good
+generalization ability [36], which makes it particularly suitable
+for FedRS. In meta learning model based FedRS, the server
+aggregates the intermediate parameters uploaded by clients to
+learn a model parameter initialization, and the clients fine-tune
+the initialed model parameters in local training phase to fit to
+their local data [37] [38]. In this way, meta learning model
+based FedRS can adapt the clients’ local data to provide more
+personalized recommendations. Although the performance of
+meta learning model based FedRS are generally better than
+learning a unified global model, the private information leak-
+age can still occur during the learning process of model
+parameter initialization [37].
+III. PRIVACY OF FEDERATED RECOMMENDATION
+SYSTEMS
+In the model training process of FedRS, the user data is
+stored locally and only the intermediate parameters are up-
+loaded to a server, which can further protect user privacy while
+keeping recommendation performance. Nevertheless, several
+research works show that the central server can still infer
+some sensitive information based on intermediate parameters.
+For examples, a curious server can identify items the user
+has interacted with according to the non-zero gradients sent
+by the client [31]. Besides, the server can also infer the user
+ratings as long as obtaining the user uploaded gradients in two
+consecutive rounds [10]. To further protect privacy of FedRS,
+many studies have incorporated other privacy protection mech-
+anisms into the FedRS, including pseudo items, homomorphic
+encryption, secret sharing and differential privacy. This section
+introduces the application of each privacy mechanism used in
+FedRS, and compare their advantages and limitations.
+A. Pseudo Items
+To prevent the server from inferring the set of items that
+users have interacted with based on non-zero gradients, some
+studies utilize pseudo items to protect user interaction behav-
+iors in FedRS. The key idea of pseudo items is that the clients
+not only upload gradients of items that have been interacted
+with but also upload gradients of some sampled items that
+have not been with.
+For example, Lin et at. [22] propose a federated recom-
+mendation framework for explicit feedback scenario named
+FedRec, in which they design an effective hybrid filling strat-
+egy to generated virtual ratings of unrated items by following
+equation:
+r
+′
+ui =
+�
+�
+�
+�m
+k=1 yukruk
+�m
+k=1 yuk
+, t < Tpredict
+ˆrui, t ≥ Tpredict
+(3)
+where t denotes the number of current training iteration,
+and Tpredict denotes the iteration number when chooses the
+average value or predict value as virtual rating value to a
+sampled item i. However, the hybrid filling strategy in FedRec
+introduces extra noise to the recommendation model, which in-
+evitably affects the model performance. To tackle this problem,
+Feng et at. [39] design a lossless version of FedRec named
+FedRec++. FedRec++ divides clients into ordinary clients and
+denoising clients. The denosing clients collect noisy gradients
+from ordinary clients and send the summation of the noisy
+gradients to server to eliminate the gradient noise.
+Although pseudo items can effectively protect user interac-
+tion behaviors in FedRS, it does not modify the gradients of
+rated items. The curious server can still infer user ratings on
+the gradients uploaded by users [10].
+B. Homomorphic Encryption
+To further protect the user ratings in FedRS, many studies
+attempt to encrypt intermediate parameters before uploading
+them to the server. Homomorphic encryption mechanism al-
+lows mathematical operation on encrypted data [40], so it
+is well suited for the intermediate parameters upload and
+aggregation processes in FedRS.
+For example, Chai et at. [10] propose a secure feder-
+ated matrix factorization framework named FedMF, in which
+clients use Paillier homomorphic encryption mechanism [41]
+to encrypt the gradients of item embedding matrix before
+uploading them to the server, and the server aggregates
+gradients on the cipher-text. Due to the characteristics of
+homomorphic encryption, FedMF can achieve the same rec-
+ommendation accuracy as the traditional matrix factorization.
+
+IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+5
+TABLE I: Comparison between different privacy mechanism.
+Privacy Mechanisms
+Ref
+Main Protect Object
+Accuracy Loss
+Communication/Computation Costs
+Pseudo Items
+[22] [39] [31] [44] [45]
+Interaction Behaviors
+�
+Low Costs
+Homomorphic Encryption
+[10]
+Ratings
+�
+High Computation Costs
+[31]
+High-order Graph
+[42]
+Social Features
+Secret Sharing
+[29] [45]
+Ratings
+�
+High Communication Costs
+Local Differential Privacy
+[27] [31] [44]
+Ratings
+�
+Low Costs
+However, FedMF causes serious computation overheads since
+all computation operations are performed on the ciphertext
+and most of system’s time is spent on server updates. Besides,
+FedMF assumes that all participants are honest and will not
+leak the secret key to the server, which is hard to guarantee
+in reality.
+Besides, many studies also utilize homomorphic encryption
+mechanism to integrate private information from other par-
+ticipants to improve recommendation accuracy [31] [42]. For
+examples, Wu et al. [31] use homomorphic encryption mecha-
+nism to find the anonymous neighbors of users to expanse local
+user-item graph. And Perifanis et al. [42] use Cheon-Kim-
+Kim-Song (CKKS) fully homomorphic encryption mechanism
+[43] to incorporate learned parameters between user’s friends
+after the global model is generated.
+Homomorphic encryption mechanism based FedRS can
+effectively protect user ratings while maintaining recommen-
+dation accuracy. Besides, it can prevent privacy leaks when
+integrating information from other participants. However, ho-
+momorphic encryption brings huge computation costs during
+operation process. And it is also a serious challenge to keep
+the secret key not be obtained by the server or other malicious
+participants.
+C. Secret Sharing
+As another encryption mechanism used in FedRS, secret
+sharing mechanism breaks intermediate parameters up into
+multiple pieces, and distributes the pieces among participants,
+so that only when all pieces are collected can reconstruct the
+intermediate parameters.
+For example, Ying [29] proposes a secret sharing based
+federated matrix factorization framework named ShareMF.
+The participants divide the item matrix gradients gplain into
+several random numbers that meet:
+gplain = gsub1 + gsub2 + ... + gsubt.
+(4)
+Each participant keeps one of the random numbers and send
+the rest to t − 1 sampled participants, then uploads the sum
+of received and kept numbers as hybrid gradients to the
+server for aggregation. ShareMF protects the user ratings and
+interaction behaviors from being inferred by the server, but
+the rated items can still be leaked to other participants who
+received the split numbers. To tackle this problem, Lin et al.
+[45] combine secret sharing and pseudo items mechanisms to
+provide stronger privacy guarantee.
+Secret sharing mechanism based FedRS can protect user
+ratings while maintaining recommendation accuracy, and have
+lower computation costs compared to homomorphic encryp-
+tion based FedRS. But the exchange process of pieces between
+participants greatly increases the communication costs.
+D. Local Differential Privacy
+Considering the huge computation or communication costs
+caused by encryption based mechanisms, many studies try
+to use perturbation based mechanisms to adapt to large-scale
+FedRS for the industrial scenarios. Local differential privacy
+(LDP) mechanism allows to statistical computations while
+guaranteeing each individual participant’s privacynoise [46]
+[47], which can be used to perturb the intermediate parameters
+in FedRS.
+For example, Dolui et al. [27] propose a federated matrix
+factorization framework, which applies differential privacy
+on item embedding matrix before sending it to server for
+weighted average. However, the server can still infer which
+items the user has rated just by comparing the changes in
+item embedding matrix.
+In order to achieve a more comprehensive privacy protection
+during model training process, Wu et al. [31] combines
+pseudo items and LDP mechanisms to protect both user
+interaction behaviors and ratings in FedGNN. Firstly, to protect
+user interaction behaviors in FedGNN, the clients randomly
+sample N items that they have not interacted with, then
+generate the virtual gradients of item embeddings by using
+a same Gaussian distribution as the real embedding gradients.
+Secondly, to protect user ratings in FedGNN, the clients apply
+a LDP module to clip the gradients according to their L2-norm
+with a threshold δ and perturb the gradients by adding zero-
+mean Laplacian noise. The LDP module of FedGNN can be
+formulated as follow:
+gi = clip(gi, δ) + Laplace(0, λ).
+(5)
+where λ is the Laplacian noise strength. However, the gradi-
+ent magnitude of different parameters varies during training
+process, thus it is usually not appropriate to perturb gradients
+at different magnitudes with a constant noise strength. So Liu
+
+IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+6
+et al. [44] propose to add dynamic noise according to the
+gradients, which can be formulated as follow:
+gi = clip(gi, δ) + Laplace(0, λ · mean(gi)).
+(6)
+Local differential privacy mechanism doesn’t bring heavy
+computation and communication overhead to FedRS, but the
+additional noise inevitably affects the performance of the
+recommendation model. Thus, in the actual application sce-
+nario, we must consider the trade-off between the privacy and
+recommendation accuracy.
+E. Comparison
+To protect stronger privacy guarantee, many privacy mech-
+anisms (i.e., pseudo items, homomorphic encryption, differ-
+ential privacy privacy and secret sharing) have been widely
+used in FedRS, and the comparison between these mechanisms
+is shown in Table I. Firstly, the main protect objects of
+these mechanisms are different: pseudo items mechanism is
+to protect user interaction behaviors, and the rest mechanisms
+is to protect user ratings. Besides, homomorphic encryption
+can aslo integrate data from other paritcipants in a privacy-
+preserving way. Secondly, homomorphic encryption and secret
+sharing are both encryption-based mechanisms, and they can
+protect privacy while keeping accuracy. However, the high
+computation cost of homomorphic encryption limits the appli-
+cation in large-scale industrial scenarios. Although the secret
+sharing mechanism reduces the computation costs, the commu-
+nication costs increase greatly. Pseudo items and differential
+privacy mechanisms protect privacy by adding random noise,
+which has low computation costs and don’t bring additional
+communication costs. But the addition of random noise will
+inevitably affect model performance to a certain extent.
+IV. SECURITY OF FEDERATED RECOMMENDATION
+SYSTEMS
+Apart from privacy leakage problems, traditional recom-
+mendation systems for centralized data storage are also vul-
+nerable to poisoning attacks (shilling attacks) [48] [49] [50]
+[51] [52] [53]. Attackers can poison recommendation systems
+and make recommendations as their desires by injecting well-
+crafted data into the training dataset. But most of these
+poisoning attacks assume that the attackers have full prior
+knowledge of entire training datasets. Such assumption may
+be not valid for FedRS since the data in FedRS is distributed
+and stored locally for each participant. Thus, FedRS provides
+a stronger security guarantee than traditional recommendation
+systems. However, the latest studies indicate that attackers can
+still conduct poisoning attacks on FedRS with limited prior
+knowledge [12] [11] [54] [55]. In this section, we summarize
+some novel poisoning attacks against FedRS and provide some
+defense methods.
+A. Poisoning Attacks
+According to the goal of attacks, the poisoning attacks
+against FedRS can be categorized into targeted attacks and
+untargeted attacks as shown in Table II.
+1) Target Poisoning Attacks: The goal of target attacks
+on FedRS is to increase or decrease the exposure chance of
+specific items, which are usually driven by financial profit.
+For example, Zhang et al. [12] propose a poisoning attack
+for item promotion (PipAttack) against FedRS by utilizing
+popularity bias. To boost the rank score of target items,
+PipAttack use popularity bias to align target items with popular
+items in the embedding space. Besides, to avoid damaging
+recommendation accuracy and be detected, PipAttack designs
+a distance constraint to keep modified gradients uploaded by
+malicious clients closed to normal ones.
+In order to further reduce the degradation of recommenda-
+tion accuracy caused by targeted poisoning attacks, and the
+proportion of malicious clients needed to ensure the attack
+effectiveness, Rong [54] propose a model poisoning attack
+against FedRS (FedRecAttack), which makes use of a small
+proportion of public interactions to approximate the user
+feature matrix, then uses it to generate poisoned gradients.
+Both PipAttack and FedRecAttack rely on some prior
+knowledge. For example, PipAttack assumes the attacker is
+available for popularity information, and FedRecAttack as-
+sumes the attacker can get public interactions. So the attack
+effectiveness is greatly reduced in the absence of prior knowl-
+edge, which makes both attacks not generic in all FedRS. To
+make attackers conduct effective poisoning attacks to FedRS
+without the prior knowledge, Rong et al. [55] design two
+methods (i.e., random approximation and hard user mining) for
+malicious clients to generate poisoned gradients. In particular,
+random approximation (A-ra) uses Gaussian distribution to
+approximate normal users’ embedding vectors, and hard user
+mining (A-hum) uses gradient descent to optimize users’
+embedding vectors obtained by A-ra to mine hard users. In this
+way, A-hum can still effectively attack FedRS with extremely
+small proportion of malicious users.
+2) Untarget Poisoning Attacks: The goal of untarget attacks
+on FedRS is to degrade the overall performance of recom-
+mendation model, which are usually conducted by competing
+companies. For example, Wu et al. [11] propose an untargeted
+poisoning attack to FedRS named FedAttack, which uses glob-
+ally hard sampling technique [62] to subvert model training
+process. More specifically, after inferring user’s interest from
+local user profiles, the malicious clients select candidate items
+that best match the user’s interest as negative samples, and
+select candidate items that least match the user’s interest as
+positive samples. FedAttack only modifies training samples,
+and the malicious clients are also similar to normal clients
+with different interests, thus FedAttack can effectively damage
+the performance of FedRS even under defense.
+B. Defense Methods
+To reduce the influence of poisoning attacks on FedRS,
+many defense methods have been proposed in the literature,
+which can be classified into robust aggregation and anomaly
+detection.
+1) Robust Aggregation: The goal of robust aggregation
+is to guarantee global model convergence when up to 50%
+of participants are malicious [63], which selects statistically
+
+IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+7
+TABLE II: Representative works on the security of FedRS. RA refers to robust aggregation and AD refers to anomaly
+detection.
+Works
+Ref
+Attack Type
+Poison Object
+Defense Type
+Goal
+Target
+Untarget
+Model
+Data
+RA
+AD
+PipAttack
+[12]
+�
+�
+Increase/decrease popularity of target items.
+FedRecAttack
+[54]
+�
+�
+Increase/decrease popularity of target items.
+A-ra/A-hum
+[55]
+�
+�
+Increase/decrease popularity of target items.
+FedAttack
+[11]
+�
+�
+Degrade the overall performance of FedRS.
+Median
+[56]
+�
+Guarantee global model convergence.
+Trimmed-Mean
+[56]
+�
+Guarantee global model convergence.
+(Multi-)Krum
+[57]
+�
+Guarantee global model convergence.
+Bulyan
+[58]
+�
+Guarantee global model convergence.
+Norm-Bounding
+[59]
+�
+Guarantee global model convergence.
+A-FRS
+[60]
+�
+Guarantee global model convergence.
+FSAD
+[61]
+�
+Identify and filter poisoned parameters.
+more robust values rather than the mean values of uploaded
+intermediate parameters for aggregation.
+Median [56] selects the median value of each updated
+model parameter independently as aggregated global model
+parameter, which can represent the centre of the distribution
+better. Specifically, the server ranks each i − th parameter of
+n local model update, and uses the median value as i − th
+parameter of global model.
+Trimmed-Mean [56] removes the maximum and minimum
+values of each updated model parameter independently, and
+then takes the mean value as aggregated global model param-
+eter. Specifically, the server ranks each i − th parameter of n
+local model update, remove β smallest and β largest values,
+and uses the mean value of remained n−2β as i−th parameter
+of global model. In this way, Trimmed-Mean can effectively
+reduce the impact of outliers.
+Krum and Multi-Krum [57]. Krum selects a local model
+that is the closest to the others as global model. Multi-Krum
+selects multiple local models by using Krum, then aggregates
+them into a global model. In this way, even if the selected
+parameter vectors are uploaded by malicious clients, their
+impact is still limited because they are similar to other local
+parameters uploaded by normal clients.
+Bulyan [58] is a combination of Krum and Trimmed-Mean,
+which iteratively selects m local model parameter vectors
+through Krum, and then performs Trimmed-Mean on these
+m parameter vectors for aggregation. With high dimensional
+and highly non-convex loss function, Bulyan can still converge
+to effectual models.
+Norm-Bounding [59] clips the received local parameters to
+a fixed threshold, then aggregates them to update the global
+model. Norm-Bounding can limit the contribution of each
+local model updates so as to mitigate the affect of poisoned
+parameters on the aggregated model.
+A-FRS
+[60]
+utilizes
+gradient-based
+Krum
+instead
+of
+model parameter-based Krum to filter malicious clients in
+momentum-based FedRS. A-FRS theoretically guarantees that
+if the selected gradient is closed to the normal gradient, the
+momentum and model parameters will also be close to the
+normal momentum and model parameters.
+Although these robust aggregation strategies provide conver-
+gence guarantees to some extent, most of them (i.e., Bulyan,
+Krum, Median and Trimmed-mean) greatly degrade the per-
+formance of FedRS. Besides, some noval attacks(i.e., PipAt-
+tack, FedAttack) [12] [11] utilize well-designed constraints
+to approximate the patterns of normal users and circumvent
+defenses, which further increases the difficulty of defense.
+2) Anomaly Detection: The purpose of anomaly detection
+strategy is to identify the poisoned model parameters uploaded
+by malicious clients and filter them during the global model
+aggregation process. For example, Jiang et al. [61] propose
+an anomaly detection strategy named federated shilling attack
+detector (FSAD) to detect poisoned gradients in federated
+collaborative filtering scenarios. FSAD extracts 4 novel fea-
+tures according to the gradients uploaded by clients, then uses
+the gradient-based features to train a semi-supervised bayes
+classifier so as to identify and filter the poisoned gradients.
+However, in FedRS, the interests of different users vary widely,
+thus the parameters they uploaded are usually quite different,
+which increases the difficulty of anomaly detection [54].
+V. HETEROGENEITY OF FEDERATED RECOMMENDATION
+SYSTEMS
+Compared with traditional recommendation systems, Fe-
+dRS face more severe challenges in terms of heterogeneity,
+which are mainly reflected in system heterogeneity, statistical
+heterogeneity and model heterogeneity, as shown in Fig. 3.
+
+IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+8
+Fig. 3: Heterogeneity of federated recommendation systems.
+System heterogeneity refers to client devices have signif-
+icantly different storage, computation, and communication
+capabilities. Devices with limited capabilities greatly affects
+training efficiency, and further reduces the accuracy of the
+global recommendation model. [64]; Statistical heterogeneity
+refers to the data collected by different clients is usually
+not independent and identically distributed (non-IID). As a
+result, simply training a single global model is difficult to
+generalize to all clients, which affects the personalization of
+recommendations [65]; Privacy heterogeneity means that the
+privacy constraints of different users and information vary
+greatly, so simply treating them with the same privacy budgets
+will carry unnecessary costs [66]. This section introduces some
+effective approaches to address the heterogeneity of FedRS.
+A. System Heterogeneity
+In FedRS, the hardware configuration, network bandwidth
+and battery capacity of participating clients varies greatly,
+which results in diverse computing capability, communication
+speed, and storage capability [16]. During the training process,
+the clients with limited capacity could become stragglers, and
+even drop out of current training due to network failure, low
+battery and other problems [18]. The system heterogeneity
+significantly delays the training process of FedRS, further
+reducing the recommendation accuracy of the global model. To
+make the training process compatible with different hardware
+structures and tolerate the straggling and exit issues of clients,
+the most common methods are asynchronous communication
+[67] [18] and clients selection [68].
+Asynchronous
+communication.
+Considering
+the
+syn-
+chronous communication based federated learning must wait
+for straggler devices during aggregation process, many asyn-
+chronous communication strategies are presented to improve
+training efficiency. For examples, FedSA [67] proposes a
+semi-asynchronous communication method, where the server
+aggregates the local models based on their arrival order of
+each round. FedAsync [18] uses a weighted average strategy
+to aggregate the local models based on staleness, which assigns
+less weight to delayed feedback in update process.
+Clients selection. Client selection approach selects clients
+for updates based on resource constraints so that the server can
+aggregate as many local updates as possible at the same time.
+For example, in FedCS [68], the server sends a resource re-
+quest to each client so as to get their resource information, then
+estimates the required time of model distribution, updating
+and uploading processes based on the resource information.
+According to the estimated time, the server determine which
+clients can participant in training process.
+B. Statistical Heterogeneity
+Most of the existing federated recommendation studies
+are built on the assumption that data in each participant is
+independent and identically distributed (IID). However, the
+data distribution of each client usually varies greatly, hence
+training a consistent global model is difficult to generalized
+to all clients under non-IID data and inevitably neglects
+the personalization of clients [66]. To address the statistical
+heterogeneity problem of FedRS, many effective strategies
+have been proposed, which are mainly based on meta learning
+[38] [69] and clustering [70] [71].
+Meta learning. As known as “learning to learn”, meta
+learning technology aims to quickly adapt the global model
+learned by other tasks to a new task by using only a few
+number of samples [36]. The rapid adaptation and good
+generalization abilities makes it particularly well-suited for
+building personalized federated recommendation models. For
+examples, FedMeta [38] uses Model-Agnostic Meta-Learning
+(MAML) [73] algorithm to learn a well-initialized model that
+can be quickly adapted to clients, and effectively improve the
+personalization and convergence of FedRS. However, FedMeta
+
+Privacy
+Heterogenity
+8
+Private
+Public
+Private
+Public
+8
+user
+item
+attributes
+Statistical
+Heterogenity
+battery
+Data Distribution
+Data Distribution
+l
+4/5G
+Wi-Fi
+System
+Heterogenity
+Participant 2
+Participant 2IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+9
+needs to compute the second-order gradients, which greatly
+increases computation costs. Besides, the data split process
+also brings a huger challenge for clients with limited samples.
+Based on FedMeta, Wang et al. [69] propose a new meta
+learning algorithm called Reptile which applies the approx-
+imate first-order derivatives for the meta-learning updates,
+which greatly reduces the computation overloads of clients.
+Moreover, Reptile doesn’t need a data split process, which
+makes it also suitable for clients with limited samples.
+Clustering. The core idea of clustering is training person-
+alized models jointly with the same group of homogeneous
+clients. For examples, Jie et al. [70] uses historical parameter
+clustering technology to realize personalized federated recom-
+mendation, in which the server aggregates local parameters to
+generate global model parameters and clusters the local pa-
+rameters to generate clustering parameters for different client
+groups. Then the clients combine the clustering parameters
+with the global parameters to learn personalized models. Luo
+et al. [71] propose a personalized federated recommendation
+framework named PerFedRec, which constructs a collaborative
+graph and integrates attribute information so as to jointly learn
+the user representations by federated GNN. Based on the
+learned user representations, clients are clustered into different
+groups. And each cluster learns a cluster-level recommen-
+dation model. At last, each client can obtain a personalized
+model by merging the global recommendation model, the
+cluster-level recommendation model, and the fine-tuned local
+recommendation model. Although clustering based approaches
+can alleviate statistical heterogeneity, the clustering and com-
+bination process greatly increase the computation costs.
+C. Privacy Heterogeneity
+In reality, the privacy restrictions of different participants
+and information vary greatly, thereby using the same high level
+of privacy budget for all participants and information is unnec-
+essary, which even increases the computation/communication
+costs and degrades the model performance.
+Heterogeneous user privacy. In order to adapt the privacy
+needs for different users, Anelli et al. [72] present a user
+controlled federated recommendation framework named Fed-
+eRank. FedeRank introduces a probability factor π ∈ [0, 1] to
+control the proportion of interacted item updates and masks
+the remain interacted item update by setting them to zero.
+In this way, FedeRank allows users decide the proportion of
+data they want to share by themselves, which addresses the
+heterogeneity of user privacy.
+Heterogeneous information privacy. In order to adapt the
+privacy needs of different information components, HPFL [66]
+designs a differentiated component aggregation strategy. To
+obtain the global public information components, the server
+directly weighted aggregates the local public components with
+same properties. And to obtain the global privacy information
+components, the user and item representations are kept locally,
+and the server only aggregates the local drafts without the need
+to align the presentations. With the differentiated component
+aggregation strategy, HPFL can safely aggregate components
+with heterogeneous privacy constraints in user modeling sce-
+narios.
+VI. COMMUNICATION COSTS OF FEDERATED
+RECOMMENDATION SYSTEMS
+To achieve satisfactory recommendation performance, Fe-
+dRS requires multiple communications between server and
+clients. However, the real-world recommendation systems are
+usually conducted by complexity deep learning models with
+large model size [74], and millions of parameters needs to
+be updated and communicated [13], which brings severe
+communication overload to resource limited clients and further
+affects the application of FedRS in large-scale industrial sce-
+narios. This section summarizes some optimization methods to
+reduce communication costs of FedRS, which can be classified
+into importance-based updating [75] [20] [76] [77], model
+compression [78] [79], active sampling [80] and one shot
+learning [81].
+A. Importance-based Model Updating
+Importance-based model updating selects importance parts
+of the global model instead of the whole model to update and
+communicate, which can effectively reduce the communicated
+parameter size in each round.
+For examples, Qin et al. [75] propose a federated frame-
+work named PPRSF, which uses 4-layers hierarchical structure
+for reducing communication costs, including the recall layer,
+ranking layer, re-ranking layer and service layer. In the recall
+layer, the server roughly sorts the large inventory by using
+public user data, and recalls relatively small number of items
+for each client. In this way, the clients only need to update and
+communicate the candidate item embeddings, which greatly
+reduces the communication costs between server and clients,
+and the computation costs in the local model training and
+inference phases. However, the recall layer of PPRSF need
+to get some public information of users, which raises certain
+difficulty and privacy concerns.
+Yi et al. [20] propose an efficient federated news recom-
+mendation framework called Efficient-FedRec, which breaks
+the news recommendation model into a small user model and
+a big news model. Each client only requests the user model and
+a few news representations involved in their local click history
+for local training, which greatly reduces the communication
+and computation overhead. To further protect specific user
+click history against the server, they transmit the union news
+representations set involved in a group of user click history
+by using a secure aggregation protocol [82].
+Besides, Khan et al. [76] propose a multi-arm bandit
+method (FCF-BTS) to select part of the global model that
+contains a smaller payload to all clients. The rewards of
+selection process is guided by Bayesian Thompson Sampling
+(BTS) [83] approach with Gaussian priors. Experiments show
+that FCF-BTS can reduce 90% model payload for highly
+sparse datasets. Besides, the selection process occurs in the
+server side, thus avoiding additional computation costs on the
+clients. But FCF-BTS causes 4% - 8% loss in recommendation
+accuracy.
+To achieve a better balance between recommendation ac-
+curacy and efficiency, Ai et al. [77] propose an all-MLP
+
+IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+10
+network that uses a Fourier sub-layer to replace the self-
+attention sub-layer in a Transformer encoder so as to filter
+noise data components unrelated to the user’s real interests,
+and adapts an adaptive model pruning technique to discard
+the noise model components that doesn’t contribute to model
+performance. Experiments show that all-MLP network can
+significantly reduce communication and computation costs,
+and accelerates the model convergence.
+Importance-based model updating strategies can greatly
+reduce communication and computation costs at the same time,
+but only selecting the important parts for updating inevitably
+reduces the recommendation performance.
+B. Model Compression
+Model Compression is a well-known technology in dis-
+tributed learning [84], which compresses the communicated
+parameters per round to be more compact.
+For examples, Konen et al. [78] propose two methods
+(i.e., structured updates and sketched updates) to decrease the
+uplink communication costs under federated learning settings.
+Structured updates method directly learns updates from a
+pre-specified structure parameterized using fewer variables.
+Sketched updates method compresses the full local update
+using a lossy compression way before sending it to server.
+These two strategies can reduce the communication costs by
+2 orders of magnitude.
+To reduce the uplink communication costs in deep learning
+based FedRS, JointRec [79] combines low-rank matrix factor-
+ization [85] and 8-bit probabilistic quantization [86] methods
+to compress weight update. Supposing the weight update ma-
+trix of client n is Ha×b
+n
+, a ≤ b, low-rank matrix factorization
+decomposes Ha×b
+n
+into two matrices: Ha×b
+n
+= U a×k
+n
+V k×b
+n
+,
+where k = b/N and N is a positive number that influences the
+compression performance. And 8-bit probabilistic quantization
+method transforms the position of matrix value into 8-bit value
+before send it to server. Experiments demonstrate that JointRec
+can realize 12.83× larger compression ratio while maintaining
+recommendation performance.
+Model compression methods achieve significant results in
+reducing the uplink communication costs. However, the re-
+duction of communication cost sacrifices the computation
+resources of the clients, so it’s necessary to consider the trade-
+off between computation and communication costs when using
+model compression.
+C. Client Sampling
+In traditional federated learning frameworks [8], the server
+randomly selects clients to participate in the training process
+and simply aggregates the local models by average, which
+requires a large number of communications to realize satis-
+factory accuracy. Client sampling utilizes efficient sampling
+strategies so as to improve the training efficiency and reduce
+the communication rounds.
+For example, Muhammad et al. [80] propose an effective
+sampling strategy named FedFast to speed up the training
+efficiency of federated recommendation models while keeping
+more accuracy. FadFast consists of two efficient components:
+ActvSAMP and ActvAGG. ActvSAMP uses K-means algo-
+rithm to cluster users based on their profile, and samples
+clients in equal proportions from each cluster. And ActvAGG
+propagates local updates to the other clients in the same
+cluster. In this way, the learning process for these similar
+users is greatly accelerated and overall efficiency of the FedRS
+is consequently improved. Experiments show that FedFast
+reduces communication rounds by 94% compared to FedAvg
+[8]. However, FedFast is faced with the cold start problem
+because it requires a number of users and items for training.
+Besides, FedFast needs to retrain the model to support new
+users and items.
+D. One Shot Federated Learning
+The goal of one shot federated learning mechanism is to
+reduce communication rounds of FedRS [87] [88], which lim-
+its communication to a single round to aggregate knowledge
+of local models. For example, Eren et al. [81] implement
+an one-shot federated learning framework for cross-platform
+FedRS named FedSPLIT. FedSPLIT aggregates model through
+knowledge distillation [89], which can generate client specific
+recommendation results with just a single pair of communica-
+tion rounds between the server and clients after a small initial
+communication. Experiments show that FedSPLIT realizes
+similar root-mean-square error (RMSE) compared with multi-
+round communication scenarios, but it is not applicable to the
+scenario where the participant is a individual user.
+VII. OPEN SOURCE PLATFORMS
+This section introduces five open source platforms that can
+be used to build FedRS: Federated AI Technology Enabler
+(Fate)1, Tensorflow Federated (TFF)2, Pysyft3, PaddleFL4, and
+FederatedScope5. The comparison among some existing open
+source platforms is shown in Table III.
+A. Federated AI Technology Enabler
+Federated AI Technology Enabler (Fate) [90] is the first
+open source platform for federated learning around the world,
+which aims to enable companies and organizations to collab-
+orate on data while keeping data privacy and security. Fate
+supports various machine learning algorithms under federated
+learning settings, including logistic regression, XGBOOST,
+deep learning and transfer learning. Besides, Fate integrates
+homomorphic encryption, differential privacy and secret shar-
+ing mechanisms to protect privacy against the curious server.
+The structure of FATE consists of seven major modules:
+FederatedML, EggRoll, FATE-FLow, FATE-Board, FATE-
+Serving, KubeFATE and FATE-cloud. FederatedML imple-
+ments privacy-preserving federated machine learning algo-
+rithms; EggRoll manages the distributed computation frame-
+work; FATE-FLow coordinates the execution of the algorithm
+1https://github.com/FederatedAI/FATE
+2https://github.com/tensorflow/federated
+3https://github.com/OpenMined/PySyft
+4https://github.com/PaddlePaddle/PaddleFL
+5https://github.com/alibaba/FederatedScope
+
+IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+11
+components; FATE-Board provides visualization for building
+and evaluating models; FATE-Serving provides online infer-
+ence for the users; KubeFATE helps deploy Fate platform by
+using cloud native technologies; FATE-cloud provide cross-
+cloud deployment and management services.
+Fate provides a federated recommendation module (Federat-
+edRec) to solve the recommendation problems of rate predic-
+tion and item ranking tasks. FederatedRec implements many
+common recommendation algorithms under federated learning
+settings, including factorization machine, matrix factorization,
+SVD, SVD++ and generalized matrix factorization.
+B. Tensorflow Federated
+Tensorflow Federated (TFF) [91] is a lightweight system
+developed by Google, which provides the building blocks to
+enables developers to implement own federated models based
+on TensorFlow. Besides, developers can plug any existing
+Keras model into TFF with just a few lines of code. To
+enhancing privacy guarantees for federated learning, TFF
+integrates differential privacy mechanism.
+The interfaces of TFF are organized in two layers API (i.e.,
+Federated Learning API and Federated Core API). Federated
+Learning API implements high-level interfaces for developers
+to make training and evaluation process of federated learning.
+Federated Core API provides lower-level interfaces to express
+novel federated learning algorithms by using TensorFlow and
+distributed communication operators.
+Based on TFF, Singhal et al. [92] implements a model-
+agnostic framework for fast partial local federated learning,
+which is suitable for large-scale collaborative filtering recom-
+mendation scenarios.
+C. Pysyft
+PySyft [93] is developed by Open-Mined, which also pro-
+vides the building blocks for developers to implement own
+federated recommendation algorithms. Compared with TFF,
+PySyft can work with both Tensorflow and Pytorch. For
+privacy protection, PySyft can flexibly and simply integrate
+homomorphic encryption, differential privacy and secret shar-
+ing mechanisms so as to defend against the honest-but-curious
+server and participants. However, Pysyft doesn’t disclose the
+detailed interface design or system architecture.
+D. PaddleFL
+PaddleFL [94] is an open source federated learning platform
+developed by Baidu, which integrates both differential privacy
+and secret sharing mechanisms to provide privacy guarantees.
+PaddleFL contains two major components: Data Parallel and
+Federated Learning with MPC (PFM). Data Parallel is respon-
+sible for defining, distributing and training a federated learning
+task. PFM implements secure multi-party computation to
+ensure training and inference security.
+PaddleFL provides many federated recommendation algo-
+rithms that can be used directly. For example, PaddleFL
+implements a classical session-based recommendation model
+Gru4rec [95] under the federated learning settings and provide
+simulated experiments on real world dataset. But the simulated
+experiment suppose all datasets in different organizations are
+homogeneous, which is only satisfied under ideal case. In
+addition, PaddleFL also provides a strategy to train a Click-
+Through-Rate(CTR) model by using FedAvg [8] algorithm.
+E. FederatedScope
+FederatedScope [96], developed by Alibaba, is a flexible
+federated learning platform for heterogeneity. FederatedScope
+employs an event-driven architecture to support asynchronous
+training, and coordinate participants with personalized be-
+haviors and multiple goals into federated learning scenarios.
+FederatedScope can easily support different machine learning
+libraries such as Tensorflow and Pytorch. Besides, Federat-
+edScope enables various kinds of plug-in components and
+operations thar can be used for efficient further development.
+For privacy protection plug-ins, FederatedScope integrates ho-
+momorphic encryption, differential privacy and secret sharing
+mechanisms to enhance privacy guarantees. In the federated
+recommendation scenario, FederatedScope has built in matrix
+factorization models, datasets (Netflix and MovieLen) and
+trainer under different federated learning settings.
+VIII. FUTURE DIRECTIONS
+This section presents and discusses many prospective re-
+search directions in the future. Although some directions have
+been covered in above sections, we believe they are necessary
+for FedRS, and need to be further researched.
+Decentralized FedRS. Most of current FedRS are based on
+client-server communication architecture, which faces single-
+point-of-failure and privacy issues caused by the central server
+[97]. While much work has been devoted to decentralized
+federated learning [98] [99], few decentralized FedRS have
+been studied. A feasible solution is to replace client-server
+communication architecture with peer-peer communication
+architecture to achieve fully decentralized federated recom-
+mendation. Hegeds et al. [19] propose a fully decentralized
+matrix factorization framework based on gossip learning [100],
+where each participant sends their copy of the global recom-
+mendation model to random online neighbors in the peer to
+peer network.
+Incentive mechanisms in FedRS. FedRS collaborate with
+multiple participants to train a global recommendation model,
+and the recommendation performance of global model is
+highly dependent on the quantity and quality of data provided
+by the participants. Therefore, it is significant to design an
+appropriate incentive mechanism to inspire participants to
+contribute their own data and participate in collaborative
+training, especially in the cross-organization federated recom-
+mendation scenarios. The incentive mechanisms must be able
+to measure the clients’ contribution to the global model fairly
+and efficiently.
+Privacy of serving phase. Although many studies have
+combined different privacy mechanisms to protect user privacy
+in the training phase of FedRS, the privacy protection for
+the serving phase is still underexplored. To prevent user
+recommendation results from leaking, most of the current
+
+IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 14, NO. 8, DECEMBER 2022
+12
+TABLE III: The comparison among some existing open source platforms.
+Platforms
+Fate
+TFF
+Pysyft
+PaddleFL
+FederatedScope
+Publisher
+WeBank
+Google
+OpenMined
+Baidu
+Alibaba
+Audience
+Academia
+�
+�
+�
+�
+�
+Industry
+�
+�
+Models
+Neural Network
+�
+�
+�
+�
+�
+Tree Model
+�
+�
+Linear Model
+�
+�
+�
+�
+�
+Privacy
+Homomorphic encryption
+�
+�
+�
+Differential Privacy
+�
+�
+�
+�
+�
+Secret Sharing
+�
+�
+�
+Libraries
+Tensorflow
+�
+�
+�
+Pytorch
+�
+�
+studies assume local serving, where the server sends the entire
+set of candidate items to clients, and clients generate rec-
+ommendation results locally [10] [21]. However, such design
+brings enormous communication, computation and memory
+costs for clients since there are usually millions of items in
+real-world recommendation systems. Another feasible solution
+is online serving, where clients send encrypted or noised user
+embedding to the server to recall top-N candidate items, then
+clients generate personalized recommendation results based
+one these candidate items [101]. Nevertheless, there is a risk
+of privacy leakage associated with online serving, because
+recalled items are known to the server.
+Cold start problem in FedRS. The cold start problem
+means that recommendation systems cannot generate satisfac-
+tory recommendation results for new users with little history
+interactions. In federated settings, the user data is stored
+locally, so it is more difficult to integrate other auxiliary
+information (e.g., social relationships) to alleviate the cold
+start problem. Therefore, it is a challenging and prospective
+research direction to address the cold start problem while
+ensuring user privacy.
+Secure FedRS. In the real world, the participants in the
+FedRS are likely to be untrustworthy. Therefore, participants
+may upload poisoned intermediate parameters to affect rec-
+ommendation results or destroy recommendation performance.
+Although some robust aggregation strategies [57] and detec-
+tion methods [61] have been proposed to defense poisoning
+attacks in federated learning settings, most of them does not
+work well in FedRS. One one hand, some strategies such
+as Krum, Median and Trimmed-mean degrade the recom-
+mendation performance to a certain extend. One the other
+hand, some novel attacks [11] use well-designed constraints to
+mimic the patterns of normal users, extremely increasing the
+difficulty to be detected and defensed. Currently, there is still
+no effective defense methods against these poisoning attacks
+while maintaining recommendation accuracy.
+IX. CONCLUSION
+A lot of effort has been devoted to federated recommen-
+dation systems. A comprehensive survey is significant and
+meaningful. This survey summarizes the latest studies from
+aspects of the privacy, security, heterogeneity and commu-
+nication costs. Based on these aspects, we also make a
+detailed comparison among the existing designs and solutions.
+Moreover, we present many prospective research directions to
+promote development in this field. FedRS will be a promising
+field with huge potential opportunities, which requires more
+efforts to develop.
+ACKNOWLEDGMENTS
+This research is partially supported by the National Key
+R&D Program of China No.2021YFF0900800, the NSFC
+No.91846205, the Shandong Provincial Key Research and De-
+velopment Program (Major Scientific and Technological Inno-
+vation Project) (No.2021CXGC010108), the Shandong Provin-
+cial Natural Science Foundation (No.ZR202111180007), the
+Fundamental Research Funds of Shandong University, and
+the Special Fund for Science and Technology of Guangdong
+Province under Grant (2021S0053).
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+Zehua Sun is currently pursuing his master’s degree
+in the School of Software of Shandong University.
+He received his bachelor’s degree in software engi-
+neering from the School of Software of Shandong
+University in 2017. His research interests include
+federated learning, recommendation systems and
+data mining.
+Yonghui Xu is a full professor at Joint SDU-NTU
+Centre for Artificial Intelligence Research (C-FAIR),
+Shandong University. Before that, he was a research
+fellow in the Joint NTU-UBC Research Centre of
+Excellence in Active Living for the Elderly (LILY),
+Nanyang Technological University, Singapore. He
+received his Ph.D. from the School of Computer Sci-
+ence and Engineering at South China University of
+Technology in 2017 and BS from the Department of
+Mathematics and Information Science Engineering
+at Henan University of China in 2011. His research
+areas include various topics in Trustworthy AI, knowledge graphs, expert
+systems and their applications in e-commerce and healthcare. He has been
+invited as reviewer of top journals and leading international conferences, such
+as, TKDE, TNNLS, IEEE Transactions on Cybernetics, Knowledge-Based
+System, TKDD, IJCAI and AAAI.
+Yong Liu is a Senior Research Scientist at Alibaba-
+NTU Singapore Joint Research Institute, Nanyang
+Technological University (NTU). He was a Data
+Scientist at NTUC Enterprise, and a Research Sci-
+entist at Institute for Infocomm Research (I2R),
+A*STAR, Singapore. He received his Ph.D. degree
+in Computer Engineering from NTU in 2016 and
+B.S. degree in Electronic Science and Technology
+from University of Science and Technology of China
+(USTC) in 2008. His research interests include rec-
+ommendation systems, natural language processing,
+and knowledge graph. He has been invited as a PC member of major
+conferences such as KDD, SIGIR, ACL, IJCAI, AAAI, and reviewer for
+IEEE/ACM transactions.
+Wei He is a associate professor at Shandong univer-
+sity. He received bachelor and master degrees from
+computer science department of shandong university
+in 1994 and 1999 respectively, and received Ph.d.
+from engineering of shandong university in 2009.
+He won the progress first prize in science and
+technology of shandong province and the progress
+second prize in science and technology of shandong
+province, and excellent achievement in computer
+application. He has published more than 20 papers
+in the computer journal, journal of software of
+domestic and international journals conference. More papers were recorded
+by SCI, EI.
+Yali Jiang is currently a Lecturer in the School
+of Software, Shandong University. She received her
+B.Sc., M.Sc. and Ph.D. degrees from Shandong
+University in 1999, 2002 and 2011, respectively.
+She is engaged in information security and cryp-
+tography research, her main research areas are pub-
+lic key security authentication system and lattice
+based cryptographic algorithm design and analysis,
+including cloud computing security, big data privacy
+protection, IoT security, etc. She has participated
+in the National 863 Program, Shandong Provincial
+Excellent Young and Middle-aged Research Award Fund, Shandong Provincial
+Natural Science Foundation and joint research projects of enterprises.
+Fangzhao Wu is a Principal Researcher at Microsoft
+Research Asia, President of AAAI2022 and senior
+member of China Computer Society. He received
+the Ph.D. and B.S. degrees both from Electronic
+Engineering Department of Tsinghua University in
+2017 and 2012 respectively. He published more than
+100 academic papers and was cited nearly 3000
+times He has won NLPCC2019 Excellent Paper
+Award, WSDM 2019 Outstanding PC and AAAI
+2021 Best SPC. His research mainly focuses on
+responsible AI, privacy protection, natural language
+processing, and recommender systems. The research results have been applied
+in Microsoft News, Bing Ads and other Microsoft products.
+LiZhen Cui (IET Fellow, IEEE Senior Member)
+is the Dean at School of Software, Shandong Uni-
+versity. He is the Co-Director of Joint SDU-NTU
+Centre for Artificial Intelligence Research (C-FAIR)
+and Research Center of Software & Data Engi-
+neering, Shandong University. He is the Associate
+Director of National Engineering Laboratory for E-
+Commerce Technologies. He is a Professor with
+the School of Software and the Joint SDU-NTU
+Centre for Artificial Intelligence Research (C-FAIR),
+Shandong University, and also a Visiting Professor
+with Nanyang Technological University, Singapore. He was a Visiting Scholar
+with Georgia Tech, Atlanta, GA, USA. He received his bachelor’s, M.Sc.,
+and Ph.D. degrees from Shandong University, Jinan, China, in 1999, 2002
+and 2005, respectively. He has authored or coauthored over 200 articles in
+journals and refereed conference proceedings. His research interests include
+big data management and analysis and AI theory and application.
+

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+arXiv:2301.03113v1  [math.OC]  8 Jan 2023
+Accelerated Randomized Block-Coordinate Algorithms for
+Co-coercive Equations and Applications
+Quoc Tran-Dinh
+Department of Statistics and Operations Research
+The University of North Carolina at Chapel Hill
+318 Hanes Hall, UNC-Chapel Hill, NC 27599-3260.
+Email: [email protected].
+July 2022
+Abstract
+In this paper, we develop an accelerated randomized block-coordinate algorithm to
+approximate a solution of a co-coercive equation. Such an equation plays a central role
+in optimization and related fields and covers many mathematical models as special cases,
+including convex optimization, convex-concave minimax, and variational inequality prob-
+lems. Our algorithm relies on a recent Nesterov’s accelerated interpretation of the Halpern
+fixed-point iteration in [48].
+We establish that the new algorithm achieves O
+�
+1/k2�
+-
+convergence rate on E
+�
+∥Gxk∥2�
+through the last-iterate, where G is the underlying co-
+coercive operator, E [·] is the expectation, and k is the iteration counter. This rate is signif-
+icantly faster than O (1/k) rates in standard forward or gradient-based methods from the
+literature. We also prove o
+�
+1/k2�
+rates on both E
+�
+∥Gxk∥2�
+and E
+�
+∥xk+1 − xk∥2�
+. Next,
+we apply our method to derive two accelerated randomized block coordinate variants
+of the forward-backward splitting and Douglas-Rachford splitting schemes, respectively
+for solving a monotone inclusion involving the sum of two operators. As a byproduct,
+these variants also have faster convergence rates than their non-accelerated counterparts.
+Finally, we apply our scheme to a finite-sum monotone inclusion that has various appli-
+cations in machine learning and statistical learning, including federated learning. As a
+result, we obtain a novel federated learning-type algorithm with fast and provable con-
+vergence rates.
+1
+Introduction
+Monotone inclusion provides a powerful tool to model several problems in optimization,
+nonlinear analysis, mechanics, and machine learning, among many other areas, see, e.g.,
+[5, 9, 17, 41, 44, 45, 46]. Though it is a classical mathematical tool [5, 28, 45, 46], there
+has been a notable research surge of this topic in the last few years due to new applications
+in modern machine learning and data science. Methods for solving monotone inclusions often
+generalize existing optimization algorithms, and exploit structures of the underlying operators
+1
+
+such as splitting property. Classical methods include gradient or forward, extragradient, past-
+extragradient, proximal-point, forward-backward splitting, forward-backward-forward split-
+ting, Douglas-Rachford splitting, projective splitting methods, and their variants, see, e.g.,
+[5, 12, 14, 28, 17, 31, 42, 52]. However, developing accelerated [block-]coordinate methods
+with fast convergence rates for lare-scale monotone inclusions is still a challenging task.
+In this paper, we focus on a very basic model of monotone inclusions, which is called a
+co-coercive equation of the form:
+Find x⋆ ∈ Rp such that:
+Gx⋆ = 0,
+(CE)
+where G : Rp → Rp is a co-coercive operator (see Section 2 for definition). For our convenience,
+we assume that the solution set zer(G) := G−1(0) = {x⋆ ∈ Rp : Gx⋆ = 0} of (CE) is nonempty.
+The co-coercive equation (CE) though looks simple, it is equivalent to the problem of
+finding a fixed-point x⋆ of a nonexpansive operator T := I − G, i.e. x⋆ = Tx⋆, where I is the
+identity operator (see [5]). Therefore, it covers many fundamental problems in different fields
+by appropriately reformulating them into special cases of (CE), or equivalently, fixed-point
+problems (see, e.g., [13, 40] and also Sections 4 and 5 below).
+Motivation and related work. We are interested in the case that G in (CE) lives in a
+high-dimensional space Rp such that operating on full-dimensional vectors x of Rp is expensive
+or even prohibited. Such models are ubiquitous in large-scale modern machine learning and
+data science applications [8, 23, 47]. One common approach to tackle these models is block-
+coordinate methods, which iteratively update one or a small number of blocks of the model
+parameters instead of the full parameter vector. Such an approach is though very classical
+[4, 38], it has attracted a huge attention in recent years in optimization, monotone inclusions,
+and fixed-point problems, see, e.g., [6, 11, 24, 36, 37, 40, 43, 53]. However, developing efficient
+variants of the block-coordinate method to solve co-coercive equation (CE) remains largely
+elusive. Most existing works focus on special cases of (CE) such as optimization, convex-
+concave minimax, and supervised learning models, see e.g., [6, 24, 37, 36, 43, 53].
+Our goal in this paper is to advance a recent development of accelerated methods and
+apply it to randomized [block-]coordinate schemes. Unlike non-accelerated algorithms, it has
+been recognized that [2, 54] generalizing accelerated methods from convex minimization to
+monotone inclusions is not straightforward. Recent attempt on designing accelerated methods
+for monotone inclusions and variational inequality (VIPs) has been made, see, e.g., in [2, 10,
+20, 30, 50].
+These algorithms often achieve a faster convergence rate than their classical
+counterparts on the gradient norm or some appropriate operator residual norms. Typical
+rates on the square of a residual norm are usually O
+�
+1/k2�
+(or faster, o
+�
+1/k2�
+) compared
+to O (1/k) (or o (1/k)) in non-accelerated methods, where k is the iteration counter. The
+O
+�
+1/k2�
+rate matches the convergence rate lower bound in different settings, see [21, 35, 39]
+for some concrete examples.
+As mentioned earlier, since the problem of approximating a
+solution of (CE) can be reformulated equivalently to a fixed-point problem of a non-expansive
+operator [5], theory and solution methods from one field can be applied to another and vice
+versa. Due to its generality, (CE) can cover many common applications in scientific computing
+as discussed, e.g., in [40, 46]. For instance, it can be customized to handle linear systems,
+[composite] smooth and nonsmooth convex optimization, feasibility problems, decentralized
+2
+
+optimization, federated learning, among others. To avoid repetition, we do not present these
+applications in this paper, but refer to [40, 46] for more details on how to reformulate them
+into a fixed-point problem, or equivalently, a co-coercive equation of the form (CE).
+Motivated by applications in high-dimensional spaces, we aim at developing an accelerated
+randomized block-coordinate method to solve (CE).
+Our basic mathematical tool is the
+Halpern fixed-point iteration from [19] for solving (CE) and its recent development in, e.g., [16,
+25, 27, 54]. Our central idea is to represent the accelerated Halpern fixed-point method into
+a two-step iterative scheme (in Nesterov’s accelerated sense) using two consecutive iterates as
+discussed in [48]. Then, we combine this resulting scheme and a randomized block-coordinate
+strategy to derive a novel randomized block-coordinate algorithm for solving (CE).
+Contribution. Our concrete contribution can be summarized as follows. Firstly, we pro-
+pose a new accelerated randomized block-coordinate algorithm to solve (CE) which achieves a
+O
+�
+1/k2�
+last-iterate convergence rate, or even a o
+�
+1/k2�
+-rate on E
+�
+∥Gxk∥2�
+. Our algorithm
+is very simple to implement and significantly different from existing methods. To the best
+of our knowledge, this is the first randomized block-coordinate algorithm for (CE) achieving
+o
+�
+1/k2�
+-fast convergence rates. Next, we utilize a change of variable to develop a practical
+variant of our method, which can avoid full-dimensional operations on the iterates. Alter-
+natively, we apply our algorithm to the forward-backward splitting and Douglas-Rachford
+splitting methods to obtain new accelerated randomized block-coordinate variants for solving
+monotone inclusions involving the sum of two maximally monotone operators. As a byprod-
+uct of our convergence analysis, these variants also achieve faster convergence rates than their
+classical counterparts. Finally, we apply our method to tackle a class of finite-sum mono-
+tone inclusions which forms the basis of many supervised machine learning tasks, including
+federated learning [22, 26, 32, 33]. It leads to a new federated learning-type algorithm with
+O
+�
+1/k2�
+and o
+�
+1/k2�
+- convergence rates for a general class of finite-sum monotone inclusions.
+Let us highlight the following points of our contribution and discuss its limitation. Firstly,
+one of the most related works to our method is [40], which extends the asynchronous ran-
+domized block-coordinate method to (CE). Though their method is asynchronous, it is non-
+accelerated, and therefore, in our context, achieves O (1/k) and at most o (1/k) convergence
+rates on the squared norm of the residual mapping. Note that the form of our algorithm is also
+different from [40], while achieving O
+�
+1/k2�
+and o
+�
+1/k2�
+faster rates. Unfortunately, asyn-
+chronous variants of our method remain open. Secondly, unlike methods for convex problems,
+convergence analysis of algorithms for monotone inclusions, including (CE) is fundamentally
+different, including the construction of a potential or Lyapunov function. Moreover, it remains
+unclear if some recent techniques, e.g., in [16, 25, 27, 54] can be extended to [randomized
+block-] coordinate variants. In this paper, we follow a different approach compared to those,
+including convergence analysis technique. Thirdly, our randomized block-coordinate variants
+for splitting schemes in Section 4 are also very different from the ones in [11] since their
+methods rely on standard splitting methods. However, as a limitation of our new forward-
+backward splitting method, it still requires a co-coercive assumption of one operator. Finally,
+our application to a finite-sum monotone inclusion in Section 5 is new compared to [13] since
+our problem setting is more general than that of [13], and our scheme relies on an accelerated
+Douglas-Rachford splitting scheme instead of a forward-type method as in [13].
+3
+
+Paper organization. The rest of this paper is organized as follows. In Section 2 we
+briefly review some background related to (CE) and recall some preliminary results used in
+this paper. Our main result is in Section 3, where we develop a new algorithm and establish its
+convergence rate guarantees. We also show how to apply our method to fixed-point problems
+and derive its practical variant. Section 4 presents two applications of our method to the
+forward-backward and Douglas-Rachford splitting schemes for solving monotone inclusions.
+Section 5 is an application of our method to a general finite-sum monotone inclusion which
+potentially has many applications in machine learning and networks. We close this paper
+with some concluding remarks.
+2
+Background and Preliminary Results
+We first review some background on monotone operators and related concepts. Then, we
+recall the Halpern fixed-point iteration from [19] and its relation to Nesterov’s accelerated
+methods.
+2.1
+Monotone operators and related concepts
+We work with a finite dimensional space Rp equipped with the standard inner product ⟨·, ·⟩ and
+Euclidean norm ∥ · ∥. For a set-valued mapping G : Rp ⇒ 2Rp, dom(G) = {x ∈ Rp : Gx ̸= ∅}
+denotes its domain, graph(G) = {(x, y) ∈ Rp × Rp : y ∈ Gx} denotes its graph, where 2Rp is
+the set of all subsets of Rp. The inverse of G is defined by G−1y := {x ∈ Rp : y ∈ Gx}. For
+x = [x1, · · · , xn] ∈ Rp, we define a weighted norm ∥x∥w :=
+��n
+i=1 wi∥xi∥2�1/2, where xi is
+the i-the block of x and wi > 0 is a given weight (i = 1, · · · , n).
+Monotonicity. For a set-valued mapping G : Rp ⇒ 2Rp, we say that G is monotone if
+⟨u − v, x − y⟩ ≥ 0 for all x, y ∈ dom(G), u ∈ Gx, and v ∈ Gy. G is said to be µG-strongly
+monotone (or sometimes called coercive) if ⟨u − v, x − y⟩ ≥ µG∥x − y∥2 for all x, y ∈ dom(G),
+u ∈ Gx, and v ∈ Gy, where µG > 0 is called a strong monotonicity parameter. If G is single-
+valued, then these conditions reduce to ⟨Gx−Gy, x−y⟩ ≥ 0 and ⟨Gx−Gy, x−y⟩ ≥ µG∥x−y∥2
+for all x, y ∈ dom(G), respectively. We say that G is maximally monotone if graph(G) is not
+properly contained in the graph of any other monotone operator. Note that G is maximally
+monotone, then αG is also maximally monotone for any α > 0, and if G and H are maximally
+monotone, and dom(F) ∩ int (dom(H)) ̸= ∅, then G + H is maximally monotone.
+Lipschitz continuity and co-coerciveness. A single-valued operator G is said to be
+L-Lipschitz continuous if ∥Gx − Gy∥ ≤ L∥x − y∥ for all x, y ∈ dom(G), where L ≥ 0 is a
+Lipschitz constant. If L = 1, then we say that G is nonexpansive, while if L ∈ [0, 1), then we
+say that G is L-contractive, and L is its contraction factor. We say that G is 1
+L-co-coercive if
+⟨Gx−Gy, x−y⟩ ≥ 1
+L∥Gx−Gy∥2 for all x, y ∈ dom(G). If L = 1, then we say that G is firmly
+nonexpansive. If G is 1
+L-cocoercive, then it is also monotone and L-Lipschitz continuous (by
+using the Cauchy-Schwarz inequality), but the reverse statement is not true in general.
+Resolvent operator. The operator JGx := {y ∈ Rp : x ∈ y + Gy} is called the resolvent
+of G, often denoted by JGx = (I+G)−1x, where I is the identity mapping. Clearly, evaluating
+JG requires solving a strongly monotone inclusion 0 ∈ y−x+Gy. If G is monotone, then JG is
+singled-valued, and if G is maximally monotone then JG is singled-valued and dom(JG) = Rp.
+If G is monotone, then JG is firmly nonexpansive [5, Proposition 23.10].
+4
+
+2.2
+The Halpern fixed-point iteration and its variants
+Let us recall the following Halpern fixed-point iteration from [19] for approximating a fixed-
+point x⋆ of a non-expansive operator T : Rp → Rp (i.e. x⋆ = Tx⋆):
+xk+1 := βkx0 + (1 − βk)Txk,
+where
+βk :=
+1
+k+2.
+(1)
+As proven in [27], this scheme achieves ∥xk−Txk∥2 = O
+� 1
+k2
+�
+rate guarantee, which is optimal.
+Now, for a given operator G : Rp → Rp, G is 1
+L-co-coercive if and only if T := I − 2
+LG is
+nonexpansive [5, Proposition 4.11]. Therefore, the Halpern fixed-point method (1) applying
+to approximate a solution x⋆ of the co-coercive equation Gx⋆ = 0 can be written as
+xk+1 := βkx0 + (1 − βk)
+�
+xk − 2
+LGxk�
+= βkx0 + (1 − βk)xk − ηkGxk,
+(2)
+where ηk := 2(1−βk)
+L
+. As shown in [16], this scheme also achieves an optimal convergence rate,
+i.e. ∥Gxk∥2 = O
+�
+1/k2�
+. Clearly, (1) and (2) are equivalent.
+Next, it has been shown in [48] that if we eliminate x0 in (2) using two consecutive updates
+xk and xk+1, then we obtain the following scheme:
+xk+1 := xk + θk(xk − xk−1) −
+�
+ηkGxk − γkGxk−1�
+,
+(3)
+where θk := βk(1−βk)
+βk−1
+and γk := βkηk−1
+βk−1 .
+Finally, if we additionally introduce yk+1 := xk − αkGxk, then we can equivalently trans-
+form (3) into the following form (see [48] for details):
+�
+yk+1 := xk − αkGxk,
+xk+1 := yk+1 + θk(yk+1 − yk) + νk(xk − yk+1),
+(4)
+where αk :=
+ηk
+1−βk and νk :=
+βk
+βk−1.
+The scheme (4) shows a connection between the Halpern-type method [19] and Nesterov’s
+accelerated algorithms [2, 29, 34, 35]. Compared to Nesterov’s accelerated methods for solving
+smooth convex optimization problems, (4) has an additional correction term νk(xk −yk+1). It
+is also related to Ravine’s method as shown in [3]. Note that, both (3) and (4) can be applied
+to proximal-point, forward-backward splitting, Douglas-Rachford splitting, and three-operator
+splitting schemes for solving monotone inclusions, variational inequality, and convex-concave
+saddle-point problems, see, e.g., [2, 7, 20, 30, 48] for more details.
+3
+Accelerated Randomized Block-Coordinate Algorithms
+In this section, we develop a new randomized block-coordinate variant of (3) to solve (CE).
+We assume that the variable x of (CE) is decomposed into n-blocks as x = [x1, x2, · · · , xn]
+(1 ≤ n ≤ p), where xi ∈ Rpi for i ∈ [n] := {1, 2, · · · , n}. For the operator G, we denote
+[Gx]i as the i-the block coordinate of Gx such that Gx = [[Gx]1, · · · , [Gx]n]. We also denote
+G[i]x = [0, · · · , 0, [Gx]i, 0, · · · , 0] so that only the i-th block is computed, while others are
+zero.
+Throughout this paper, we assume that G in (CE) satisfies the following assumption.
+5
+
+Assumption 3.1. The operator G in (CE) is L−1-block-coordinate-wise co-coercive, i.e. for
+any x, y ∈ dom(G), there exist Li ∈ [0, +∞) (∀i ∈ [n]) such that
+⟨Gx − Gy, x − y⟩ ≥
+n
+�
+i=1
+1
+Li ∥[Gx]i − [Gy]i∥2 ≡ ∥Gx − Gy∥2
+L−1,
+(CP)
+where L−1 := ( 1
+L1 , · · · , 1
+Ln ). Moreover, dom(G) = Rp and zer(G) := {x⋆ ∈ Rp : Gx⋆ = 0} ̸= ∅.
+Clearly, Assumption 3.1 extends the standard co-coerciveness [5] of a monotone operator
+to block-coordinate-wise settings, and therefore, it is still very common. Moreover, due to
+the equivalence between the co-coercive equation (CE) and the fixed-point problem as we
+mentioned earlier, our setting appears to be sufficiently general to cover many applications.
+Since we will develop randomized methods operating on blocks xi of x for some i ∈ [n],
+we introduce the following probability model for selecting block coordinates of x. Let ik be a
+random variable on [n] := {1, 2, · · · , n} that satisfies the following probability distribution:
+Prob (ik = i) = pi,
+for all i ∈ [n],
+(5)
+where pi > 0 for all i ∈ [n] and �n
+i=1 pi = 1. We also denote pmin := mini∈[n] pi > 0. If
+pi =
+1
+n, then ik is a uniformly random variable. Otherwise, we also cover non-uniformly
+randomized block-coordinate methods.
+To define convergence guarantees of our methods, we denote Fk to be the smallest σ-
+algebra generated by the random set {x0, x1, · · · , xk} collecting all iterate vectors up to the
+k-the iteration of our algorithm. We also use Ek [X] := Eik [X | Fk] to denote the conditional
+expectation of X taken overall the randomness generated by the random variable ik ∈ [n]
+(and therefore xk) conditioned on Fk, and E [·] for the total expectation.
+3.1
+Accelerated RBC Method and Convergence Analysis: Main Result
+Inspired by our expression (3), we propose the following Accelerated Randomized Block-
+Coordinate (ARBC) scheme for solving (CE).
+Starting from x0 ∈ Rp, we set x−1 := x0,
+and at each iteration k ≥ 0, randomly generate ik ∈ [n] following the probability law (5) and
+update:
+xk+1 := xk + θk(xk − xk−1) −
+ψ
+pik
+�
+ηkG[ik]xk − γkG[ik]xk−1�
+,
+(ARBC)
+where θk > 0, ψ > 0, ηk > 0, and γk ≥ 0 are given parameters, which will be determined later,
+and G[i]xk = [0, · · · , 0, [Gxk]i, 0, · · · , 0] such that [Gxk]i is the i-the block of Gxk (i ∈ [n]).
+The scheme (ARBC) requires two block-coordinate evaluations [Gxk]ik and [Gxk−1]ik of
+G at the two consecutive iterates xk and xk−1, respectively. Clearly, it is different from ex-
+isting randomized [block]-coordinate methods in the literature, including methods for convex
+optimization [6, 18, 24, 36, 37, 43, 53]. However, due to the extrapolation term θk(xk −xk−1),
+(ARBC) still requires full vector update at each iteration. This is unavoidable in accelerated
+methods as in [18, 36, 37]. We will further discuss this point in Subsection 3.2.
+To establish the convergence of (ARBC), we introduce the following potential function:
+Vk := 2ψtkηk−1
+�
+⟨Gxk−1, xk−1 − x⋆⟩ − �n
+i=1
+1
+Li ∥[Gxk−1]i∥2�
++ ∥xk−1 − x⋆ + tk(xk − xk−1)∥2 + µk∥xk−1 − x⋆∥2,
+(6)
+6
+
+where x⋆ ∈ zer(G), and tk > 0 and µk ≥ 0 are given parameters, which will be determined
+later. It is obvious that under Assumption 3.1, {xk} is well-defined, and we have Vk ≥ 0 for
+all k ≥ 0 regardless the choice of xk−1 and xk. We first prove the following key result.
+Lemma 3.1. Suppose that Assumption 3.1 holds for (CE). Let {xk} be generated by (ARBC)
+and Vk be defined by (6). Suppose further that the parameters in (ARBC) and (6) satisfy
+
+
+
+
+
+
+
+
+
+tkηk−1 ≥
+�
+1 −
+1
+tk−µk
+�
+tk+1ηk,
+θk
+:= tk−µk−1
+tk+1
+,
+γk
+:=
+tk+1θk
+tk+1θk+1 · ηk.
+(7)
+Then, the following inequality holds:
+Vk − Ek
+�
+Vk+1�
+≥ µk(2tk − µk − 1)∥xk − xk−1∥2 − (µk+1 − µk)∥xk − x⋆∥2
++ �n
+i=1
+ψtk+1[2pi(tk+1θk+1)−ψLitk+1ηk]
+ηkLipi
+��ηk[Gxk]i − γk[Gxk−1]i
+��2.
+(8)
+Proof. First of all, let us introduce dk := ηkGxk − γkGxk−1. Then, from (ARBC), we have
+xk+1 − xk = θk(xk − xk−1) −
+ψ
+pik dk
+[ik]. Hence we can expand the second term of (6) as
+T[1]
+:=
+∥xk + tk+1(xk+1 − xk) − x⋆∥2
+(ARBC)
+=
+∥xk − x⋆ + tk+1θk(xk − xk−1) − ψtk+1p−1
+ik dk
+[ik]∥2
+=
+∥xk − x⋆∥2 + t2
+k+1θ2
+k∥xk − xk−1∥2 + ψ2t2
+k+1∥p−1
+ik dk
+[ik]∥2 − 2ψtk+1⟨p−1
+ik dk
+[ik], xk − x⋆⟩
++ 2tk+1θk⟨xk − xk−1, xk − x⋆⟩ − 2ψt2
+k+1θk⟨p−1
+ik dk
+[ik], xk − xk−1⟩.
+Alternatively, we can also expand
+∥xk−1 + tk(xk − xk−1) − x⋆∥2 = ∥xk − x⋆ + (tk − 1)(xk − xk−1)∥2
+= ∥xk − x⋆∥2 + 2(tk − 1)⟨xk − xk−1, xk − x⋆⟩
++ (tk − 1)2∥xk − xk−1∥2.
+Moreover, we also have the following elementary expression
+µk∥xk−1 − x⋆∥2 − µk+1∥xk − x⋆∥2 = µk∥xk − xk−1∥2 − 2µk⟨xk − xk−1, xk − x⋆⟩
+− (µk+1 − µk)∥xk − x⋆∥2.
+Now, let us consider the following function:
+Qk := ∥xk−1 + tk(xk − xk−1) − x⋆∥2 + µk∥xk−1 − x⋆∥2.
+(9)
+Then, combining the last three expressions, and using the definition (9) of Qk, we have
+Qk − Qk+1 =
+�
+(tk − 1)2 − θ2
+kt2
+k+1 + µk
+�
+∥xk − xk−1∥2 − ψ2t2
+k+1∥p−1
+ik dk
+[ik]∥2
++ 2 (tk − 1 − θktk+1 − µk) ⟨xk − x⋆, xk − xk−1⟩ − (µk+1 − µk)∥xk − x⋆∥2
++ 2ψtk+1⟨p−1
+ik dk
+[ik], xk − x⋆⟩ + 2ψt2
+k+1θk⟨p−1
+ik dk
+[ik], xk − xk−1⟩.
+(10)
+7
+
+Next, using the fact that dk
+[ik] = [0, · · · , 0, dk
+ik, 0, · · · , 0] and (5), we can easily show that
+Ek
+�
+∥p−1
+ik dk
+[ik]∥2�
+= �n
+i=1 p−1
+i ∥dk
+i ∥2,
+Ek
+�
+⟨p−1
+ik dk
+[ik], xk − x⋆⟩
+�
+= �n
+i=1⟨dk
+i , xk
+i − x⋆
+i ⟩,
+Ek
+�
+⟨p−1
+ik dk
+[ik], xk − xk−1⟩
+�
+= �n
+i=1⟨dk
+i , xk
+i − xk−1
+i
+⟩.
+(11)
+Taking conditional expectation Ek [·] both sides of (10) and then using (11) and dk = ηkGxk −
+γkGxk−1 into the resulting expression, and rearranging it, we can derive
+Qk − Ek
+�
+Qk+1�
+=
+�
+(tk − 1)2 − θ2
+kt2
+k+1 + µk
+�
+∥xk − xk−1∥2 − (µk+1 − µk)∥xk − x⋆∥2
++ 2 (tk − 1 − θktk+1 − µk) ⟨xk − x⋆, xk − xk−1⟩
++ 2ψtk+1ηk
+�n
+i=1⟨[Gxk]i, xk
+i − x⋆
+i ⟩ − 2ψtk+1γk
+�n
+i=1⟨[Gxk−1]i, xk−1
+i
+− x⋆
+i ⟩
+− ψ2t2
+k+1
+�n
+i=1
+1
+pi ∥ηk[Gxk]i − γk[Gxk−1]i∥2
++ 2ψt2
+k+1θkηk
+�n
+i=1⟨[Gxk]i − [Gxk−1]i, xk
+i − xk−1
+i
+⟩
++ 2ψtk+1 [tk+1θk(ηk − γk) − γk] �n
+i=1⟨[Gxk−1]i, xk
+i − xk−1
+i
+⟩.
+Utilizing the condition (CP) of G as
+�n
+i=1⟨[Gxk]i − [Gxk−1]i, xk
+i − xk−1
+i
+⟩ ≥ �n
+i=1
+1
+Li ∥[Gxk]i − [Gxk−1]i∥2,
+into the last expression, we arrive at
+Qk − Ek
+�
+Qk+1�
+≥
+�
+(tk − 1)2 − θ2
+kt2
+k+1 + µk
+�
+∥xk − xk−1∥2 − (µk+1 − µk)∥xk − x⋆∥2
++ 2
+�
+tk − 1 − θktk+1 − µk
+�
+⟨xk − x⋆, xk − xk−1⟩ + 2ψtk+1ηk⟨Gxk, xk − x⋆⟩
+− 2ψtk+1γk⟨Gxk−1, xk−1 − x⋆⟩ − ψ2t2
+k+1
+�n
+i=1
+1
+pi ∥ηk[Gxk]i − γk[Gxk−1]i∥2
++ 2ψt2
+k+1θkηk
+�n
+i=1
+1
+Li ∥[Gxk]i − [Gxk−1]i∥2
++ 2ψtk+1
+�
+tk+1θk(ηk − γk) − γk
+�
+⟨Gxk−1, xk − xk−1⟩.
+Rearranging this inequality, we get
+Qk − Ek
+�
+Qk+1�
+≥
+�
+(tk − 1)2 − θ2
+kt2
+k+1 + µk
+�
+∥xk − xk−1∥2 − (µk+1 − µk)∥xk − x⋆∥2
++ 2
+�
+tk − 1 − θktk+1 − µk
+�
+⟨xk − x⋆, xk − xk−1⟩
++ ψtk+1
+�n
+i=1
+�
+2(tk+1θkηk−γk)
+Li
+− ψtk+1γ2
+k
+pi
+�
+∥[Gxk−1]i∥2
+− 2ψt2
+k+1ηk
+�n
+i=1
+�
+2θk
+Li − ψγk
+pi
+�
+⟨[Gxk]i, [Gxk−1]i⟩
++ ψtk+1ηk
+�n
+i=1
+�
+2(tk+1θk+1)
+Li
+− ψtk+1ηk
+pi
+�
+∥[Gxk]i∥2
++ 2ψtk+1ηk
+�
+⟨Gxk, xk − x⋆⟩ − �n
+i=1
+1
+Li ∥[Gxk]i∥2�
+− 2ψtk+1γk
+�
+⟨Gxk−1, xk−1 − x⋆⟩ − �n
+i=1
+1
+Li ∥[Gxk−1]i∥2�
++ 2ψtk+1
+�
+tk+1θk(ηk − γk) − γk
+�
+⟨Gxk−1, xk − xk−1⟩.
+(12)
+8
+
+Let us first impose the following condition as in the third line of (7):
+tk+1θk(ηk − γk) − γk = 0
+⇔
+γk :=
+tk+1θk
+tk+1θk+1 · ηk.
+(13)
+This condition leads to
+
+
+
+
+
+
+
+
+
+
+��
+Ak
+i := ψtk+1
+�
+2(tk+1θkηk−γk)
+Li
+− ψtk+1γ2
+k
+pi
+�
+= ψtk+1ηk[2pi(tk+1θk+1)−ψLitk+1ηk]
+Lipi
+·
+t2
+k+1θ2
+k
+(tk+1θk+1)2 ,
+Bk
+i := ψt2
+k+1ηk
+�
+2θk
+Li − ψγk
+pi
+�
+= ψtk+1ηk[2pi(tk+1θk+1)−ψLitk+1ηk]
+Lipi(tk+1θk+1)
+·
+tk+1θk
+(tk+1θk+1),
+Ck
+i := ψtk+1ηk
+�
+2(tk+1θk+1)
+Li
+− ψtk+1ηk
+pi
+�
+= ψtk+1ηk[2pi(tk+1θk+1)−ψLitk+1ηk]
+Lipi
+.
+Therefore, using these three coefficients Ak
+i , Bk
+i , and Ck
+i , we can show that
+T [i]
+[2] := ψtk+1
+�
+2(tk+1θkηk−γk)
+Li
+− ψtk+1γ2
+k
+pi
+�
+∥[Gxk−1]i∥2
+− 2ψt2
+k+1ηk
+�
+2θk
+Li − ψγk
+pi
+�
+⟨[Gxk]i, [Gxk−1]i⟩
++ ψtk+1ηk
+�
+2(tk+1θk+1)
+Li
+− ψtk+1ηk
+pi
+�
+∥[Gxk]i∥2
+=
+ψtk+1ηk
+�
+2pi(tk+1θk+1)−ψLitk+1ηk
+�
+Lipi
+��[Gxk]i −
+tk+1θk
+tk+1θk+1[Gxk−1]i
+��2.
+In this case, we can simplify (12) as
+Qk − Ek
+�
+Qk+1�
+≥
+�
+(tk − 1)2 − θ2
+kt2
+k+1 + µk
+�
+∥xk − xk−1∥2 − (µk+1 − µk)∥xk − x⋆∥2
++ 2
+�
+tk − 1 − θktk+1 − µk
+�
+⟨xk − x⋆, xk − xk−1⟩
++ 2ψtk+1ηk
+�
+⟨Gxk, xk − x⋆⟩ − �n
+i=1
+1
+Li ∥[Gxk]i∥2�
+− 2ψtk+1γk
+�
+⟨Gxk−1, xk−1 − x⋆⟩ − �n
+i=1
+1
+Li ∥[Gxk−1]i∥2�
++ �n
+i=1
+ψtk+1ηk[2pi(tk+1θk+1)−ψLitk+1ηk]
+Lipi
+��[Gxk]i −
+tk+1θk
+tk+1θk+1[Gxk−1]i
+��2.
+(14)
+Let us impose the following two conditions:
+tk − µk − 1 = θktk+1
+and
+tkηk−1 ≥ tk+1γk.
+(15)
+The first equality leads to the choice of θk as in (7), i.e. θk := tk−µk−1
+tk+1
+.
+Next, let us check the second condition of (15). Using (13) and tk+1θk = tk − µk − 1 from
+(15), the second condition of (15) is equivalent to
+tkηk−1 ≥ tk+1ηk
+�
+1 −
+1
+tk−µk
+�
+,
+which is the first line of (7). Hence, the second condition of (15) holds.
+Finally, by (15), we can easily show that
+T[3] := µk + (tk − 1)2 − θ2
+kt2
+k+1 = (tk − 1)tk − (tk − µk)tk+1θk = µk(2tk − µk − 1).
+Using this expression, (CP), the conditions in (15),
+tk+1θk
+tk+1θk+1 = γk
+ηk from (13), and ηk − γk =
+ηk
+tk+1θk+1 into (14), and then using the definition of Vk from (6) for the resulting inequality,
+we obtain (8).
+9
+
+Now, we are ready to prove the convergence of (ARBC) in the following theorem.
+Theorem 3.1. Suppose that Assumption 3.1 holds for (CE).
+Let {xk} be generated by
+(ARBC) and Vk be defined by (6). For given ω > 3 and 0 < ψ < min
+�
+2pi
+Li : i ∈ [n]
+�
+, we
+update
+µk := 1,
+tk := k+2ω+1
+ω
+,
+θk := tk−2
+tk+1 ,
+ηk := tk−1
+tk+1 ,
+and
+γk := tk−2
+tk+1 = θk.
+(16)
+Then, we obtain the following bounds:
+�+∞
+k=0(k + 2ω + 2)2E
+�
+∥ηkGxk − γkGxk−1∥2�
+≤
+ω2
+ψC V0,
+�+∞
+k=0(k + ω + 1)E
+�
+∥xk − xk−1∥2�
+≤ 2
+ωV0,
+�+∞
+k=0(k + ω)E
+�
+∥Gxk−1∥2�
+≤
+16
+ψ2ωV0 +
+8C0
+ψ2(ω−3),
+�+∞
+k=1(k + 1)2E
+�
+∥Gxk − Gxk−1∥2�
+≤ C0,
+(17)
+where C := min
+i∈[n]
+�
+2
+Li − ψ
+pi
+�
+> 0 and C0 :=
+ψ2ω2(ω−1)2
+(ω+1)2
+∥Gx0∥2 +
+�
+8(ω−1)
+ω
++ ψω2(1−pmin)
+pminC
+�
+V0.
+Moreover, the following statements also hold:
+�
+E
+�
+∥xk+1 − xk∥2�
+= O
+� 1
+k2
+�
+and
+E
+�
+∥xk+1 − xk∥2�
+= o
+� 1
+k2
+�
+,
+E
+�
+∥Gxk∥2�
+= O
+� 1
+k2
+�
+and
+E
+�
+∥Gxk∥2�
+= o
+� 1
+k2
+�
+.
+(18)
+Theorem 3.1 establishes convergence rates of (ARBC) on two main criteria E
+�
+∥Gxk∥2�
+and E
+�
+∥xk+1 − xk∥2�
+, among other side results as stated in (17). However, the statement (18)
+does not show the independence of the rates on the number of blocks n. As we can observe
+from our proof below that these convergence rates depend on
+1
+ψ2 , where ψ is a given stepsize in
+Theorem 3.1. If we choose pi = 1
+n, i.e. uniformly random, and assume that Li = L for i ∈ [n],
+then ψ is proportional to 1
+n, leading to E
+�
+∥xk+1 − xk∥2�
+= O
+�
+n2
+k2
+�
+and E
+�
+∥Gxk∥2�
+= O
+�
+n2
+k2
+�
+.
+The dependence of the convergence rates on n2 has been observed in Nesterov’s accelerated
+methods for convex optimization, see, e.g. [1, 18, 36]. In (18), we state both Big-O and
+small-o convergence rates where Big-O rates are often proved when k ≤ O (n), while small-o
+rates are achieved when k is sufficiently large.
+The proof of Theorem 3.1. Firstly, we fix µk := 1 for all k ≥ 0 and since θk is updated by
+θk := tk−µk−1
+tk+1
+= tk−2
+tk+1 as in (7), if we choose ηk := tk−µk
+tk+1
+= tk−1
+tk+1 , then the first condition
+of (7) reduces to 1 ≤ tk−1−1
+tk−2 , which holds if tk−1 − tk + 1 ≥ 0. Let us choose tk := k+2ω+1
+ω
+for some ω > 3.
+Clearly, we have tk−1 − tk + 1 =
+ω−1
+ω
+> 0.
+Moreover, we also have
+γk = tk+1θkηk
+tk+1θk+1 = tk−2
+tk+1 = θk as shown in (16).
+Next, under the choice of parameters as above, (8) reduces to
+Vk − Ek
+�
+Vk+1�
+≥ �n
+i=1
+ψt2
+k+1
+Lipi (2pi − ψLi) ∥ηk[Gxk]i − γk[Gxk−1]i∥2
++ 2(tk − 1)∥xk − xk−1∥2.
+10
+
+Taking full expectation this inequality and using tk = k+2ω+1
+ω
+and tk−2
+tk−1 ≥
+1
+ω+1, we obtain
+E
+�
+Vk�
+− E
+�
+Vk+1�
+≥ �n
+i=1
+ψ(k+2ω+2)2
+Lipiω2
+(2pi − ψLi) E
+�
+∥ηk[Gxk]i − γk[Gxk−1]i∥2�
++ 2(k+ω+1)
+ω
+E
+�
+∥xk − xk−1∥2�
+.
+(19)
+Since E
+�
+Vk�
+≥ 0, summing up (20) from k := 0 to k := K ≥ 0, we obtain
+�K−1
+k=0
+ψ(k+2ω+2)2
+ω2
+�n
+i=1
+�
+2
+Li − ψ
+pi
+�
+E
+�
+∥ηk[Gxk]i − γk[Gxk−1]i∥2�
+≤ E
+�
+V0�
+,
+�K−1
+k=0 (k + ω + 1)E
+�
+∥xk − xk−1∥2�
+≤ 2
+ωE
+�
+V0�
+.
+(20)
+This implies the first two lines of (17) after taking the limit as K → +∞ and noting that
+E [V0] = V0 due to the certainty of x0 and x−1.
+Next, let us define the following full vector ¯xk+1 as
+¯xk+1 := xk + θk(xk − xk−1) − ψ
+�
+ηkGxk − γkGxk−1�
+= zk − ψdk,
+(21)
+where zk := xk +θk(xk −xk−1) and dk := ηkGxk −γkGxk−1 ≡ ηkGxk −θkGxk−1. Then, from
+(ARBC), we have xk+1 = zk −
+ψ
+pik dk
+[ik]. Therefore, for any uk independent of ik, we have
+Ek
+�
+∥xk+1 − uk∥2�
+= ∥zk − uk∥2 − 2ψEk
+�
+⟨p−1
+ik dk
+[ik], zk − uk⟩
+�
++ ψ2Ek
+�
+∥p−1
+ik dk
+[ik]∥2�
+= ∥zk − uk∥2 − 2ψ⟨dk, zk − uk⟩ + ψ2 �n
+i=1
+1
+pi ∥dk
+i ∥2
+= ∥zk − ψdk − uk∥2 + ψ2 �n
+i=1
+�
+1
+pi − 1
+�
+∥dk
+i ∥2
+= ∥¯xk+1 − uk∥2 + ψ2 �n
+i=1
+�
+1
+pi − 1
+�
+∥dk
+i ∥2.
+(22)
+Now, from (21), we have
+¯xk+1 − xk + ψηkGxk =
+θk
+ηk−1 (xk − xk−1 + ψηk−1Gxk−1) +
+�
+1 −
+θk
+ηk−1
+�
+· θk(1−ηk−1)
+ηk−1−θk (xk−1 − xk).
+Note also that since ω > 1, we have 0 <
+θk
+ηk−1 =
+tk−2
+tk−1−1 =
+k+1
+k+ω ≤ 1. Moreover, using the
+update rule (16), we can easily show that θk(1−ηk−1)
+ηk−1−θk
+= (ω+1)(k+1)
+ωk+2ω2−1 ≤ ω+1
+ω
+≤ 2 for all k ≥ 0.
+Hence, by convexity of ∥ · ∥2, we have
+∥¯xk+1 − xk + ψηkGxk∥2 ≤
+θk
+ηk−1∥xk − xk−1 + ψηk−1Gxk−1∥2 + 4(ω−1)
+k+ω ∥xk − xk−1∥2.
+Substituting uk := xk − ψηkGxk into (22) and combining the result with the last inequality
+and using max{ 1
+pi − 1 : i ∈ [n]} = 1−pmin
+pmin , we can show that
+Ek
+�
+∥xk+1 − xk + ψηkGxk∥2�
+≤
+θk
+ηk−1 ∥xk − xk−1 + ψηk−1Gxk−1∥2 + 4(ω−1)
+k+ω ∥xk − xk−1∥2
++ ψ2(1−pmin)
+pmin
+∥ηkGxk − θkGxk−1∥2.
+Taking full expectation of both sides of this inequality, we arrive at
+E
+�
+∥xk+1 − xk + ψηkGxk∥2�
+≤
+θk
+ηk−1 E
+�
+∥xk − xk−1 + ψηk−1Gxk−1∥2�
++ 4(ω−1)
+k+ω E
+�
+∥xk − xk−1∥2�
++ ψ2(1−pmin)
+pmin
+E
+�
+∥ηkGxk − θkGxk−1∥2�
+.
+11
+
+Multiplying this inequality by (k + ω)2 and rearranging the result, we obtain
+(k + ω)2E
+�
+∥xk+1 − xk + ψηkGxk∥2�
+≤ (k + ω − 1)2E
+�
+∥xk − xk−1 + ψηk−1Gxk−1∥2�
+− [(ω − 3)(k + ω) + 1] E
+�
+∥xk − xk−1 + ψηk−1Gxk−1∥2�
++ 4(ω − 1)(k + ω)E
+�
+∥xk − xk−1∥2�
++ ψ2(1−pmin)
+pmin
+(k + ω)2E
+�
+∥ηkGxk − θkGxk−1∥2�
+.
+This inequality also implies that limk→∞(k+ω)2E
+�
+∥xk+1 − xk + ψηkGxk∥2�
+exists. Summing
+up this inequality from k := 0 to k := K − 1, then using the first and second lines of (17) and
+x−1 := x0, we get
+(K + ω − 1)2E
+�
+∥xK − xK−1 + ψηK−1GxK−1∥2�
+≤ ψ2η2
+−1(ω − 1)2E
+�
+∥Gx0∥2�
++
+�
+8(ω−1)
+ω
++ ψω2(1−pmin)
+pminC
+�
+E
+�
+V0�
+.
+(23)
+Alternatively, we also have
+�K−1
+k=0 (k + ω)E
+�
+∥xk − xk−1 + ψηk−1Gxk−1∥2�
+≤ E[C0]
+ω−3 ,
+where C0 := ψ2η2
+−1(ω − 1)2E
+�
+∥Gx0∥2�
++
+�
+8(ω−1)
+ω
++ ψω2(1−pmin)
+pminC
+�
+E
+�
+V0�
+. Since E
+�
+V0�
+= V0,
+E
+�
+∥Gx0∥2�
+= ∥Gx0∥2, and η−1 = η0 =
+ω
+ω+1, we obtain C0 as in Theorem 3.1. The last
+inequality and the existence of limk→∞(k + ω + 1)2E
+�
+∥xk+1 − xk + ψηkGxk∥2�
+imply
+limk→∞(k + ω + 1)2E
+�
+∥xk+1 − xk + ψηkGxk∥2�
+= 0.
+(24)
+Next, noting that ηk ≥
+k+ω+1
+k+2ω+2 ≥ 1
+2, we can show that
+ψ2
+4 ∥Gxk−1∥2 ≤ ψ2η2
+k−1∥Gxk−1∥2 ≤ 2∥xk − xk−1∥2 + 2∥xk − xk−1 + ψηk−1Gxk−1∥2.
+(25)
+Therefore, we can easily obtain
+�K−1
+k=0 (k + ω)E
+�
+∥Gxk−1∥2�
+≤
+16
+ψ2ωE
+�
+V0�
++
+8E[C0]
+ψ2(ω−3).
+(26)
+This proves the third line of (17).
+Now, from xk+1 := xk + θk(xk − xk−1) − ψp−1
+ik dk
+[ik] of (ARBC), we have
+∥xk+1 − xk∥2 = θ2
+k∥xk − xk−1∥2 − 2ψθk⟨p−1
+ik dk
+[ik], xk − xk−1⟩ + ψ2∥p−1
+ik dk
+[ik]∥2.
+Taking conditional expectation Ek [·] of this expression and using (11) and θk = γk, we have
+Ek
+�
+∥xk+1 − xk∥2�
+= θ2
+k∥xk − xk−1∥2 − 2ψθk⟨dk, xk − xk−1⟩ + ψ2 �n
+i=1
+1
+pi ∥dk
+i ∥2
+= θ2
+k∥xk − xk−1∥2 + ψ2 �n
+i=1
+1
+pi ∥ηk[Gxk]i − θk[Gxk−1]i∥2
+− 2ψθ2
+k⟨Gxk − Gxk−1, xk − xk−1⟩ − 2ψθk(ηk − θk)⟨Gxk, xk − xk−1⟩.
+12
+
+Utilizing (CP) and the Young inequality into the last expression, we can show that
+Ek
+�
+∥xk+1 − xk∥2�
+≤
+�
+θ2
+k + 2θk(ηk − θk)
+�
+∥xk − xk−1∥2 + ψ2θk(ηk−θk)
+2
+∥Gxk∥2
+− 2ψθ2
+k
+�n
+i=1
+1
+Li ∥[Gxk]i − [Gxk−1]i∥2
++ ψ2 �n
+i=1
+1
+pi ∥ηk[Gxk]i − θk[Gxk−1]i∥2.
+Multiplying this inequality by ω2t2
+k+1 = (k + 2ω + 2)2 and noting that ω2t2
+k+1[θ2
+k + 2θk(ηk −
+θk)] = ω2t2
+k−2ω(k+2ω+1) and ω2t2
+k+1θk(ηk−θk) = ω(k+1), and then taking full expectation
+of the resulting inequality, we obtain
+ω2t2
+k+1E
+�
+∥xk+1 − xk∥2�
+≤ ω2t2
+kE
+�
+∥xk − xk−1∥2�
+− 2ω(k + 2ω + 1)E
+�
+∥xk − xk−1∥2�
++
+ψ2ω2t2
+k+1
+pmin
+E
+�
+∥ηkGxk − θkGxk−1∥2�
++ ψ2ω(k+1)
+2
+E
+�
+∥Gxk∥2�
+− 2ψ(k+1)2
+Lmax
+E
+�
+∥Gxk − Gxk−1∥2�
+,
+where Lmax := max{Li : i ∈ [n]}. This inequality leads to
+T[3] := (k + 2ω + 2)2E
+�
+∥xk+1 − xk∥2�
++ 2ω(k + 2ω + 1)E
+�
+∥xk − xk−1∥2�
++ 2ψ(k+1)2
+Lmax
+E
+�
+∥Gxk − Gxk−1∥2�
+≤ (k + 2ω + 1)2E
+�
+∥xk − xk−1∥2�
++
+ψ2
+pmin(k + 2ω + 2)2E
+�
+∥ηkGxk − θkGxk−1∥2�
++ ψ2ω(k+1)
+2
+∥Gxk∥2.
+Utilizing this inequality, (26), and the second line of (17), we can conclude that limk→∞(k +
+2ω + 2)2E
+�
+∥xk+1 − xk∥2�
+exists. Summing up the inequality T[3] from k := 0 to k := K − 1
+and noting that 4ω2 ≥ (4ω − 1)ω and x0 = x−1, we obtain
+(K + 2ω + 1)2 E
+�
+∥xK − xK−1∥2�
++ 2ω �K−1
+k=0 (k + 2ω + 1)E
+�
+∥xk − xk−1∥2�
++
+2ψ
+Lmax
+�K−1
+k=0 (k + 1)2E
+�
+∥Gxk − Gxk−1∥2�
+≤
+ψ2
+pmin
+�K−1
+k=0 (k + 2ω + 2)2∥ηkGxk − θkGxk−1∥2
++ ψ2ω
+2
+�K−1
+k=0 (k + 1)E
+�
+∥Gxk∥2�
+(17),(26)
+≤
+�
+ψω2
+pminC + 8
+�
+E
+�
+V0�
++ 4ωE[C0]
+(ω−3) .
+(27)
+This inequality shows that E
+�
+∥xk+1 − xk∥2�
+= O
+�
+1/k2�
+as in (18). Combining the existence
+of limk→∞(k + 2ω + 2)2E
+�
+∥xk+1 − xk∥2�
+and �k
+k=0(k + ω + 1)E
+�
+∥xk − xk−1∥2�
+< +∞, we
+obtain limk→∞(k + 2ω + 2)2E
+�
+∥xk+1 − xk∥
+�2 = 0, which proves the o-rate in (18).
+Finally, from (23), (25), and (27), we have
+(k + ω)2E
+�
+∥Gxk∥2�
+(25)
+≤
+8(k+ω)2
+ψ2
+E
+�
+∥xk+1 − xk∥2�
++ 8(k+ω)2
+ψ2
+E
+�
+∥xk+1 − xk + ψηkGxk∥2�
+(23),(27)
+≤
+8
+ψ2
+�
+ψω2
+pminC + 8(ω−1)
+ω
++ ψω2(1−pmin)
+pminC
++ 8
+�
+E
+�
+V0�
++ 32ωE[C0]
+ψ2(ω−3) .
+This inequality shows that E
+�
+∥Gxk∥2�
+= O
+�
+1/k2�
+. The o-rate E
+�
+∥Gxk∥2�
+= o
+� 1
+k2
+�
+immedi-
+ately follows from the first line of this inequality, the first line of (18), and (24).
+13
+
+Application to fixed-point problems.
+Let us apply (ARBC) to approximate a fixed-
+point x⋆ of a nonexpansive operator T : Rp → Rp, i.e. x⋆ = Tx⋆. As mentioned earlier, if we
+define G := I − T, then G is firmly nonexpansive, or equivalently, 1-co-coercive. Moreover,
+x⋆ is a fixed-point of T iff Gx⋆ = 0.
+Therefore, we can apply (ARBC) to solve Gx⋆ =
+0. For simplicity of presentation, we assume that ik is generated uniformly randomly, i.e.
+Prob (ik = i) = 1
+n for all i ∈ [n]. In this case, (ARBC) reduces to
+xk+1
+i
+:=
+�
+xk
+i + ˆηkxk
+i − ˆγkxk−1
+i
++ ˆψ
+�
+ηk[Txk]ik) − γk[Txk−1]ik
+�
+, if i = ik,
+xk
+i + θk(xk
+i − xk−1
+i
+),
+otherwise,
+(28)
+where ˆψ := nψ, ˆηk := θk − ˆψηk, and ˆγk := θk − ˆψγk. Clearly, our new scheme (28) is different
+from existing methods for approximating a fixed-point x⋆ of a non-expansive operator T. The
+convergence rates of the residual E
+�
+∥xk − Txk∥2�
+and E
+�
+∥xk+1 − xk∥2�
+of (28) are guaranteed
+by Theorem 3.1. However, we omit them here to avoid repetition.
+3.2
+Practical variant of ARBC
+Let us derive an alternative form of (ARBC) so that it is easier to implement in practice. First,
+from (ARBC), we have xk+1 − xk = θk(xk − xk−1) −
+ψ
+pik dk
+[ik], where dk := ηkGxk − γkGxk−1.
+Let us assume that θk =
+τk+1
+τk
+for a given positive sequence {τk}.
+This relation leads to
+τk+1 = τkθk. Moreover, we can write (ARBC) as
+1
+τk+1(xk+1 − xk) = 1
+τk (xk − xk−1) −
+ψ
+pik τk+1dk
+[ik].
+Now, if we introduce wk := 1
+τk (xk − xk−1), then (ARBC) can be rewritten as
+xk := xk−1 + τkwk
+and
+wk+1 := wk −
+ψ
+pik τk+1dk
+[ik].
+By induction, we can show that xk = x0 + �k
+i=1 τiwi. Let us express this representation as
+xk = x0 − τ1(w2 − w1) − (τ1 + τ2)(w3 − w2) − (τ1 + τ2 + τ3)(w4 − w3)
+− · · · − (τ1 + · · · + τk−1)(wk − wk−1) + (τ1 + · · · + τk−1 + τk)wk.
+Therefore, if we define ck := ��k
+i=1 τi with a convention that c0 := 0, and ∆wk := wk+1 − wk,
+then we can write xk as
+xk = x0 − �k−1
+i=1 ci∆wi + ckwk.
+If we introduce zk := x0 − �k−1
+i=1 ci∆wi, then we get zk = zk−1 − ck−1∆wk−1 with z0 := x0,
+and hence xk = zk + ckwk. Therefore, we can summarize our derivation above as
+
+
+
+
+
+
+
+
+
+wk+1 := wk −
+ψ
+pikτk+1 dk
+ik,
+zk+1
+:= zk − ck∆wk = zk +
+ckψ
+pikτk+1 dk
+ik
+xk+1
+= zk+1 + ck+1wk+1.
+14
+
+Eliminating xk and xk−1 from the last scheme, we can write (ARBC) equivalently to
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+dk
+i
+:= ηk[G(zk + ckwk)]i − γk[G(zk−1 + ck−1wk−1)]i,
+if i = ik,
+wk+1
+i
+:=
+� wk
+i −
+ψ
+piτk+1dk
+i , if i = ik,
+wk
+i ,
+otherwise,
+zk+1
+i
+:=
+� zk
+i +
+ψck
+piτk+1 dk
+i , if i = ik,
+zk
+i ,
+otherwise.
+(29)
+Here, x0 ∈ Rp is given, z0 = z−1 := x0, and w0 = w−1 := 0. Moreover, the parameters τk
+and ck are respectively updated as
+τk+1 := τkθk
+and
+ck := ck−1 + τk,
+(30)
+where c0 = c−1 := 0 and τ0 := 1.
+The scheme (29) is though different from accelerated randomized block-coordinate meth-
+ods for convex optimization such as [1, 18, 36, 49], it has some common features as those
+methods such as the block-coordinate evaluations of G at zk + ckwk and zk−1 + ck−1wk−1,
+respectively. One notable property of (29) is that it does not require full dimensional updates
+of wk and zk. Note that one can also extend our method (ARBC) (or equivalently, (29)) to up-
+date multiple blocks by randomly choosing a subset Sk ⊂ [n] such that Prob (i ∈ Sk) = pi > 0
+for i ∈ [n].
+4
+Applications to Monotone Inclusions
+In this section, we derive two variants of (ARBC) to approximate a solution of the following
+monotone inclusion involving the sum of two monotone operators:
+Find x⋆ ∈ Rp such that:
+0 ∈ Ax⋆ + Bx⋆,
+(MI)
+where A, B : Rp ⇒ Rp are maximally monotone operators. Moreover, we assume that Ax =
+[A1x1, A2x2, · · · , Anxn] is a separable operator compounded by n independent blocks.
+We apply (ARBC) to two common methods for solving (MI): the forward-backward split-
+ting (FBS) and the Douglas-Rachford (splitting (DRS) schemes.
+4.1
+ARBC Forward-Backward Splitting Method
+Let us first reformulate (MI) equivalently to (CE) by using the following forward-backward
+(FB) residual mapping:
+Gβx := β−1(x − JβA(x − βBx)),
+(31)
+where β > 0 is given and JβA := (I + βA)−1 is the resolvent of βA. As shown in [48], if B
+is 1
+L-co-coercive and 0 < β < 4
+L, then Gβ(·) defined by (31) is β(4−βL)
+4
+-co-coercive. Moreover,
+x⋆ ∈ zer(A + B) is a solution of (MI) iff Gβx⋆ = 0. The latter is exactly a special case of
+(CE). If dom(B) = Rp, then Gβ satisfies Assumption 3.1 with Li = β(4−βL)
+4
+for i ∈ [n]. Note
+that we can extend our results to the case B is L−1-block coordinate-wise co-coercive as in
+(CP). However, we omit this extension here.
+15
+
+Our goal is to specify (ARBC) to solve Gβx⋆ = 0. In this case, we obtain the following
+variant of (ARBC):
+xk+1 := xk + θk(xk − xk−1) −
+ψ
+pik
+�
+ηkGβ
+[ik]xk − γkGβ
+[ik]xk−1�
+,
+where θk, ψ, ηk, and γk are updated as in (ARBC). Clearly, by taking into account the
+separable structure of A and using (31), we can explicitly write the block-coordinate of Gβx
+as
+[Gβx]i = 1
+β(xi − JβAi(xi − β[Bx]i)).
+Combining the last two expressions, we can write the new variant of (ARBC) as follows:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ˆdk−1
+i
+:= xk−1
+i
+− JβAi(xk−1
+i
+− β[Bxk−1]i),
+if i = ik,
+dk
+i
+:= xk
+i − JβAi(xk
+i − β[Bxk]i),
+if i = ik,
+xk+1
+i
+:=
+
+
+
+xk
+i + θk(xk
+i − xk−1
+i
+) −
+ψ
+βpi
+�
+ηkdk
+i − γk ˆdk−1
+i
+�
+, if i = ik,
+xk
+i + θk(xk
+i − xk−1
+i
+),
+otherwise,
+(32)
+where x0 ∈ Rp is a given initial point, x−1 := x0, and ik ∈ [n] is randomly generated based
+on the probability law (5), i.e. Prob (i = ik) = pi for i ∈ [n].
+The scheme (32) requires two block-coordinate evaluations [Bxk−1]i and [Bxk]i of B and
+two evaluations of JβAi at each iteration k. Therefore, it essentially costs as twice as existing
+standard block-coordinate FBS methods. However, its convergence rate is significantly faster
+than those standard block-coordinate FBS methods, typically O
+�
+1/k2�
+compared to O (1/k).
+Finally, we specify Theorem 3.1 (without proof) to obtain convergence results of (32).
+Corollary 4.1. Let B be 1
+L-co-coercive on Rp, A be maximally monotone, and zer(A+B) ̸= ∅
+in (MI). Let {xk} be generated by (32) using pi :=
+1
+n (∀i ∈ [n]). For given ω > 3 and
+0 < β < 4
+L, we choose 0 < ψ <
+8
+nβ(4−βL) and update θk, ηk, and γk as in (16). Then, both
+quantities E
+�
+∥Gβxk∥2�
+and E
+�
+∥xk+1 − xk∥2�
+simultaneously achieve O
+�
+1/k2�
+and o
+�
+1/k2�
+convergence rates.
+In fact, if {xk} is generated by (32), then the bounds in (17) of Theorem 3.1 still hold for
+{xk} and Gβxk defined by (31). However, we only state the convergence rates in Corollary
+4.1.
+4.2
+ARBC Douglas-Rachford Splitting Method
+We consider the case B in (MI) is just maximally monotone. In this case, we consider the DR
+residual mapping of (MI) defined as follows (see also [48]):
+Eβu := 1
+β (JβBu − JβA(2JβBu − u)) ,
+(33)
+where β > 0 is given, and JβA and JβB are the resolvents of βA and βB, respectively. As
+shown in [48], Eβ(·) is β-co-coercive and dom(Eβ) = Rp. Moreover, x⋆ ∈ zer(A + B) is a
+solution of (MI) if and only if there exists u⋆ ∈ Rp such that Eβu⋆ = 0 and x⋆ = JβBu⋆.
+16
+
+Now, if we directly apply (ARBC) to solve Eβu⋆ = 0, then by exploiting the separable
+structure of A, we obtain the following scheme for solving (MI):
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ˆvk−1
+i
+:= [JβBuk−1]i,
+if i = ik,
+vk
+i
+:= [JβBuk]i,
+if i = ik,
+ˆdk−1
+i
+:= ˆvk−1
+i
+− JβAi(2ˆvk−1
+i
+− uk−1
+i
+),
+if i = ik,
+dk
+i
+:= vk
+i − JβAi(2vk
+i − uk
+i ),
+if i = ik,
+uk+1
+i
+:=
+
+
+
+uk
+i + θk(uk
+i − uk−1
+i
+) − ψ
+pi
+�
+ηkdk
+i − γk ˆdk−1
+i
+�
+, if i = ik,
+uk
+i + θk(uk
+i − uk−1
+i
+),
+otherwise,
+(34)
+where u0 ∈ Rp is given, u−1 := u0, and ik ∈ [n] is randomly generated based on (5).
+Unlike (32), which operates directly on the sequence
+�
+xk�
+, (34) generates an intermediate
+sequence {uk}. To recover an approximate solution xk of (MI), we can compute xk := JβBuk
+at the end of the algorithm. Again, our new scheme (34) is very different from existing ones
+in the literature, including [11]. Note that (34) requires two block-coordinate evaluations
+[JβBxk]i and [JβBxk−1]i of JβB, and two evaluations of JβAi at each iteration k. Hence, its
+per-iteration complexity costs as twice as the method in [11]. However, we believe that the
+convergence rate of (34) is significantly faster than the one in [11].
+Finally, similar to Corollary 4.1, we specify Theorem 3.1 to obtain convergence of (34).
+Corollary 4.2. Assume that zer(A + B) ̸= ∅ and both A and B in (MI) are maximally
+monotone. Let {uk} be generated by (34) using pi := 1
+n for all i ∈ [n]. For given ω > 3,
+β > 0, and 0 < ψ < 2β
+n , we update θk, ηk, and γk as in (16). Then, both E
+�
+∥Eβuk∥2�
+and E
+�
+∥uk+1 − uk∥2�
+simultaneously achieve O
+�
+1/k2�
+and o
+�
+1/k2�
+convergence rates. If, in
+addition, B is single-valued and xk := JβBuk, then both E
+�
+∥Gβxk∥2�
+and E
+�
+∥xk+1 − xk∥2�
+simultaneously achieve O
+�
+1/k2�
+and o
+�
+1/k2�
+convergence rates, where Gβxk is given by (31).
+This corollary is a direct consequence of Theorem 3.1, and we omit its proof. The last
+conclusion of Corollary 4.2 can easily be obtained by using the relation between Eβu and Gβu
+as stated in [48, Lemma 2].
+5
+Application to Finite-Sum Monotone Inclusions
+Many machine learning applications and optimization models over networks, including fed-
+erated learning, can be formulated into the following finite-sum monotone inclusion [13, 40]:
+Find x⋆ ∈ dom(A) ∩ dom(B) such that
+0 ∈ 1
+n
+n
+�
+i=1
+Aix⋆ + Bx⋆ ≡ Ax⋆ + Bx⋆,
+(35)
+where Ai : Rp ⇒ Rp (∀i ∈ [n]) are maximally monotone and B : Rp ⇒ Rp is also maximally
+monotone. Here, we also assume that zer(A + B) ̸= ∅. Note that A := 1
+n
+�n
+i=1 Ai in (35) is
+the average of a finite-sum operator, and it is different from the block separable operator in
+(MI). Therefore, we cannot directly apply the methods in Section 4 to solve (35).
+17
+
+One important special case of (35) is the optimality condition of the following finite-sum
+convex minimization problem which is ubiquitous in machine learning and statistical learning:
+min
+x∈Rp
+�
+F(x) := 1
+n
+n
+�
+i=1
+fi(x) + g(x)
+�
+,
+(36)
+where fi : Rp → R ∪ {+∞} and g : Rp → R ∪ {+∞} are proper, closed, and convex. The
+optimality condition of (36) can be written as 0 ∈ 1
+n
+�n
+i=1 ∂fi(x⋆) + ∂g(x⋆), which is covered
+by (35) by setting Ai := ∂fi and B := ∂g.
+To develop a new variant of (ARBC) for solving (35), we first reformulate (35) into (MI)
+by duplicating the variable x as x := [x1, x2, · · · , xn], where xi ∈ Rp for all i ∈ [n]. Then, we
+can reformulate (35) into the following monotone inclusion:
+0 ∈ Ax⋆ + Bx⋆ + ∂δL(x⋆), where Ax := [A1x1, · · · , Anxn],
+Bx := [nBx1, 0, · · · , 0], (37)
+and ∂δL is the subdifferential of the indicator of the linear subspace L := {x = [x1, · · · , xn] ∈
+Rnp : xi = x1, ∀i ∈ [n]}. It is obvious to show that x⋆ is a solution of (35) if and only if
+x⋆ = [x⋆, · · · , x⋆] solves (37), see, e.g., [51]. Moreover, A and B in (37) are separable.
+Let us apply the ARBC DR splitting scheme (34) to solve (37). Then we use the interpre-
+tation in Subsection 3.2 to obtain a practical variant. Here, we view A as A and B + ∂δL as
+B in (MI). Let us first compute the resolvent of β(B + ∂δL) at uk. This requires solving
+�
+0 ∈ nβBu1 + βs1 + u1 − uk
+1,
+0 = βsi + ui − uk
+i ,
+i = 2, · · · , n,
+(38)
+where s := [s1, · · · , sn] ∈ ∂δL(u) ≡ L⊥ := {s := [s1, · · · , sn] : �n
+i=1 si = 0}. The last line of
+(38) leads to ui = uk
+i − βzi if ui = u1 for i = 2, · · · , n. Therefore, we obtain
+(n − 1)u1 =
+n
+�
+i=2
+uk
+i − β
+n
+�
+i=2
+si =
+n
+�
+i=2
+uk
+i − β
+n
+�
+i=1
+si + βs1 =
+n
+�
+i=2
+uk
+i + βs1,
+due to the fact that �n
+i=1 si = 0. This equation implies that u1 + βs1 = nu1 − �n
+i=2 uk
+i . Sub-
+stituting this expression into the first line of (38), we get 0 ∈ nβBu1+nu1−�n
+i=1 uk
+i , or equiv-
+alently, 0 ∈ βBu1 + u1 − 1
+n
+�n
+i=1 uk
+i . Solving this inclusion, we obtain u1 = JβB
+� 1
+n
+�n
+i=1 uk
+i
+�
+.
+Let us defined ˆuk := JβB
+� 1
+n
+�n
+i=1 uk
+i
+�
+. Then, we have Jβ(B+∂δL)uk = [ˆuk, · · · , ˆuk]. Conse-
+quently, for any i ∈ [n], we obtain [Jβ(B+∂δL)uk]i = ˆuk.
+Next, we use the trick in Subsection 3.2 to eliminate uk
+i in (34). Since uk
+i = zk
+i + ckwk
+i ,
+we have ¯uk := 1
+n
+�n
+i=1 uk
+i = 1
+n
+�n
+i=1(zk
+i + ckwk
+i ) = ¯zk + ck ¯wk, where ¯zk := 1
+n
+�n
+i=1 zk
+i and
+¯wk := 1
+n
+�n
+i=1 wk
+i . However, at each iteration k, only wk
+ik and zk
+ik are updated, we have
+�
+¯zk+1 := 1
+n
+�n
+i=1 zk+1
+i
+= 1
+n
+�n
+i=1 zk
+i + 1
+n(zk+1
+ik
+− zk
+ik) = ¯zk + 1
+n∆zk+1
+ik
+,
+¯wk+1 := 1
+n
+�n
+i=1 wk+1
+i
+= 1
+n
+�n
+i=1 wk
+i + 1
+n(wk+1
+ik
+− wk
+ik) = ¯wk + 1
+n∆wk+1
+ik
+,
+where ∆zk+1
+ik
+:= zk+1
+ik
+− zk
+ik and ∆wk+1
+ik
+:= wk+1
+ik
+− wk
+ik.
+Now, we are ready to specify (34) to solve (37) as in Algorithm 1.
+18
+
+Algorithm 1 (Accelerated Federated Douglas-Rachford Algorithm (AccFedDR))
+1: Initialization: Input an initial point u0 ∈ Rp and set c0 := c−1 := 0 and τ0 := 1.
+2:
+Initialize each user i with z0
+i = z−1
+i
+:= u0 and w0
+i = w−1
+i
+= 0 for i ∈ [n].
+3:
+Initialize sever with ˆu0 = ˆu−1 := u0, ¯z0 := 0, and ¯w0 := 0.
+4: For k := 0, · · · , kmax do
+5:
+Sample an active user ik ∈ [n] following the probability law (5).
+6:
+[Communication] Server sends ˆuk and ˆuk−1 to user ik.
+7:
+[Local update] User ik updates its iterates wk+1
+ik
+and zk+1
+ik
+as
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ˆdk−1
+ik
+:= ˆuk−1 − JβAik (2ˆuk−1 − zk−1
+ik
+− ck−1wk−1
+ik
+),
+dk
+ik
+:= ˆuk − JβAik (2ˆuk − zk
+ik − ckwk
+ik),
+wk+1
+ik
+:= wk
+ik −
+ψ
+pik τk+1 (ηkdk
+ik − γk ˆdk−1
+ik
+),
+zk+1
+ik
+:= zk
+ik +
+ψck
+pikτk+1 (ηkdk
+ik − γk ˆdk−1
+ik
+).
+(39)
+8:
+[Communication] User ik sends ∆wk+1
+ik
+:= wk+1
+ik
+−wk
+ik and ∆zk+1
+ik
+:= zk+1
+ik
+−zk
+ik to server.
+9:
+[Server update] Server updates
+¯wk+1 := ¯wk + 1
+n∆wk+1
+ik
+,
+¯zk+1 := ¯zk + 1
+n∆zk+1
+ik
+, and ˆuk+1 := JβB(¯zk+1 + ck+1 ¯wk+1).
+10: End For
+Let us abbreviate Algorithm 1 by AccFedDR. Note that the parameters are updated as in
+(16) and (30). Clearly, AccFedDR is still synchronous, but it only requires the participation
+of one user ik at each communication round k. This scheme is also similar to SAGA [15] and
+a SAGA variant for co-coercive equations in [13]. However, our AccFedDR can solve a more
+general class of problems described by (35), where A is not necessarily co-coercive as in [13].
+This algorithm can also be applied to federated learning, see, e.g., [22, 26, 32, 33].
+To prove convergence of Algorithm 1, let us define the following residual operator:
+�
+ˆu
+:= JβB
+� 1
+n
+�n
+i=1 ui
+�
+,
+Eβu := 1
+β[ˆu − JβA1(2ˆu − u1), · · · , ˆu − JβAn(2ˆu − un)].
+(40)
+One can easily show that if Eβu⋆ = 0, then ˆu⋆ = JβB
+� 1
+n
+�n
+i=1 u⋆
+i
+�
+solves (35). Now, we can
+specify the convergence of AccFedDR as a consequence of Theorem 3.1.
+Corollary 5.1. Let Ai (i ∈ [n]) and B in (35) be maximally monotone and zer(A + B) ̸= ∅.
+Let {wk
+i } and {zk
+i } be generated by AccFedDR. Let uk := [uk
+1, · · · , uk
+n] with uk
+i := zk
+i + ckwk
+i
+for all i ∈ [n] and Eβu be defined by (40). For given ω > 3, β > 0, and 0 < ψ < 2βpmin, we
+update θk, ηk, and γk as in (16). Then, we obtain the following bounds:
+�+∞
+k=0(k + ω + 1)E
+�
+∥Eβuk−1∥2�
+< +∞,
+�+∞
+k=0(k + ω)E
+�
+∥uk − uk−1∥2�
+< +∞,
+�+∞
+k=0(k + 1)2E
+�
+∥Eβuk − Eβuk−1∥2�
+< +∞.
+(41)
+19
+
+Moreover, the following statements also hold:
+�
+E
+�
+∥uk+1 − uk∥2�
+= O
+� 1
+k2
+�
+and
+E
+�
+∥uk+1 − uk∥2�
+= o
+� 1
+k2
+�
+,
+E
+�
+∥Eβuk∥2�
+= O
+� 1
+k2
+�
+and
+E
+�
+∥Eβuk∥2�
+= o
+� 1
+k2
+�
+.
+(42)
+Note that Corollary 5.1 only shows convergence on {uk}. To form an approximate solution
+¯xk of (35), we simply compute ¯xk := JβB
+� 1
+n
+�n
+i=1 uk
+i
+�
+. Clearly, we can also easily show that
+E
+�
+∥¯xk − ¯xk−1∥2�
+= O
+� 1
+k2
+�
+and E
+�
+∥¯xk − ¯xk−1∥2�
+= o
+� 1
+k2
+�
+by the nonexpansiveness of JβB.
+Remark 5.1. Since our model (35) is more general than that of [13], our AccFedDR al-
+gorithm developed in this section can be applied to solve special cases as discussed in [13].
+However, we omit the details of these applications here to avoid repetition.
+6
+Concluding Remarks
+We have developed a novel accelerated randomized block-coordinate method for solving a co-
+coercive equation of the form (CE). The new algorithm achieves O
+�
+1/k2�
+and even o
+�
+1/k2�
+convergence rates on the squared norm of the underlying operator G and some other quantities
+in expectation. We have also derived a practical variant and investigated three applications
+of our method for more general problems to cope with broader classes of applications. Several
+research questions remain open to us. Firstly, how to develop an accelerated randomized block-
+coordinate method for (CE) under weaker assumptions: monotone and Lipschitz continuous?
+For example, how to extend our method to extra-anchored gradient schemes [54] or their
+variants [25, 50]? Secondly, how to extend such a type of methods to (MI) without the co-
+coerciveness of B? Thirdly, how to develop asynchronous variants of our method, including
+the variant of Algorithm 1. In addition, several practical and implementation aspects of our
+methods as well as numerical verification are still left out in this paper.
+We leave these
+research questions for our future research.
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+24
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+arXiv:2301.00339v1  [cond-mat.mtrl-sci]  1 Jan 2023
+The Origin of Two-dimensional Electron Gas in Zn1−xMgxO/ZnO Heterostructures
+Xiang-Hong Chen,1 Dong-Yu Hou,1 Zhi-Xin Hu,2 Kuang-Hong Gao,1, ∗ and Zhi-Qing Li1, †
+1Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology,
+Department of Physics, Tianjin University, Tianjin 300354, China
+2Center for Joint Quantum Studies and Department of Physics, Tianjin University, Tianjin 300354, China
+(Dated: January 3, 2023)
+Although the two-dimensional electron gas (2DEG) in (001) Zn1−xMgxO/ZnO heterostructures
+has been discovered for about twenty years, the origin of the 2DEG is still inconclusive. In the
+present letter, the formation mechanisms of 2DEG near the interfaces of (001) Zn1−xMgxO/ZnO
+heterostructures were investigated via the first-principles calculations method. It is found that the
+polarity discontinuity near the interface can neither lead to the formation of 2DEG in devices with
+thick Zn1−xMgxO layers nor in devices with thin Zn1−xMgxO layers. For the heterostructure with
+thick Zn1−xMgxO layers, the oxygen vacancies near the interface introduce a defect band in the
+band gap, and the top of the defect band overlaps with the bottom of the conduction band, leading
+to the formation of the 2DEG near the interface of the device. For the heterostructure with thin
+Zn1−xMgxO layers, the absorption of hydrogen atoms, oxygen atoms, or OH groups on the surface
+of Zn1−xMgxO film plays a key role for the formation of 2DEG in the device. Our results manifest
+the sources of 2DEGs in Zn1−xMgxO/ZnO heterostructures on the electronic structure level.
+Since
+the
+discovery
+of
+two-dimensional
+electron
+gas (2DEG) at the interface of LaAlO3/SrTiO3 het-
+erojunction [1],
+2DEG has been found at various
+oxide heterostructures, such as Zn1−xMgxO/ZnO [2–
+4],
+Al2O3/SrTiO3
+[5,
+6],
+EuO/KTaO3
+[7],
+(AlxGa1−x)2O3/Ga2O3
+[8]
+and
+LaAlO3/KTaO3
+[9].
+The 2DEG at oxide heterostructures not only provides
+a platform for fundamental research, but also promotes
+the development of novel all-oxide electronic devices.
+Among these oxide heterostructures, Zn1−xMgxO/ZnO
+heterostructures
+are
+particularly
+attractive
+due
+to
+their ultra-high Hall mobility (up to 106 cm2V−1s−1
+at low temperature [10]).
+However, the origin of the
+2DEG at the Zn1−xMgxO/ZnO interface is still unclear.
+Researchers only empirically attribute it to the polar
+discontinuity [11–14]: since Zn1−xMgxO (0 < x < 0.6)
+and ZnO have different spontaneous polarization, the
+polarization at the
+interface is
+discontinuous after
+they form heterojunctions.
+This discontinuity causes
+a large number of bound charges to be generated at
+the heterointerface, creating a built-in electric field
+throughout the
+heterostructure.
+This field drives
+electrons toward the interface to form 2DEG. In con-
+trast, some researchers believe that the 2DEG at the
+Zn1−xMgxO/ZnO interface originates from the donor on
+the Zn1−xMgxO surface [15, 16]. Experimentally, 2DEG
+can also be formed when the thickness of Zn1−xMgxO
+layer
+is
+greater
+than
+300 nm
+in
+Zn1−xMgxO/ZnO
+heterostructures [17–20].
+There would be no internal
+potential gradient in the aforementioned heterostructure
+with thick Zn1−xMgxO layer, and the contribution of
+surface donors to 2DEG could also be negligible [21, 22].
+Thus the formation of 2DEG in this case cannot be
+explained by the mechanisms mentioned above.
+On
+∗ Corresponding author, e-mail: [email protected]
+† Corresponding author, e-mail: [email protected]
+the whole, the origin of 2DEG at Zn1−xMgxO/ZnO
+heterointerface needs to be further studied. In this letter,
+the origin of 2DEG at Zn1−xMgxO/ZnO heterointerface
+is studied from the perspective of microscopic electronic
+structures by first-principles calculations. Interestingly,
+it is found that the polar discontinuity mechanism is
+not responsible for the formation of the 2DEG. For the
+heterostructures with thick Zn1−xMgxO layers, 2DEG
+mainly arises from oxygen vacancies, while the 2DEG
+originates from surface adsorption for heterostructures
+with thin Zn1−xMgxO layers.
+Considering
+that
+the
+2DEG
+can
+be
+formed
+in
+Zn1−xMgxO/ZnO
+heterostructures
+with
+both
+thick
+(∼100 to 500 nm) [23–27] and thin Zn1−xMgxO layers
+(∼10 to 30 nm) [14–16] experimentally, we construct the
+configurations as follows.
+For Zn1−xMgxO/ZnO het-
+erostructures with thick Zn1−xMgxO layers, we passi-
+vated the oxygen terminal of ZnO slab and the Zn-Mg
+terminal of Zn1−xMgxO slab by pseudo-H atoms with
+fractional charges. ZnO slab with passivated oxygen ter-
+minal can be used to simulate ZnO substrate, and the
+charge of H is taken as 0.48e with e being the elementary
+charge [21]. The charge of the pseudo-H atoms in the
+passivated Zn-Mg terminal is taken as 1.52e [21]. After
+passivation, the pseudo-H atoms not only saturate the
+surface dangling bonds but also make the passivated sur-
+face and the adjacent atomic layers exhibit bulk prop-
+erties [21, 23]. In this case, the Zn1−xMgxO and ZnO
+slabs can be treated as semi-infinite thick films.
+Con-
+sidering the Mg content x can be as high as 0.60 in
+Zn1−xMgxO/ZnO heterostructures experimentally [24],
+we set the Mg content x as 0.25 and 0.50, respectively.
+For each doping level, the Mg ions are uniformly doped
+into the ZnO film, which together with the ZnO sub-
+strate forms a heterostructure with a clear interface.
+For the Zn1−xMgxO/ZnO heterostructures with thin
+Zn1−xMgxO layers, the difference in the configuration is
+that there is no pseudo-H atom at the Zn-Mg terminal.
+
+2
+Generally, the unpassivated Zn1−xMgxO (001) surface
+is unstable and the surface adsorption or reconstruction
+is inevitable [25–31]. Thus, the surface adsorption and
+defects are considered to simulate the Zn1−xMgxO/ZnO
+heterostructures with thin Zn1−xMgxO layers [26]. As
+an example, in Fig. 1(a) we give the structure diagram
+of a Zn1−xMgxO/ZnO heterostructure with two surfaces
+passivated by pseudo-H atoms. The heterostructure con-
+tains a 2×2 in-plane (001) Zn0.75Mg0.25O/ZnO supercell
+and 18 Zn-Mg-O layers and 18 Zn-O layers. A 15-˚A-thick
+vacuum layer is added along the [001] direction to prevent
+any unintentional interactions between the slabs. From
+the interface to surface, the atomic layers on the ZnO side
+are labeled as L¯1, L¯2, · · · , L ¯17, and L ¯18, while the atomic
+layers on the Zn0.75Mg0.25O side are labeled as L1, L2,
+· · · , L17, and L18, respectively. The top view of Fig. 1(a)
+along the [001] direction is shown in Fig. 1(b).
+Three
+adsorption sites named On-top, Fcc-hollow, and Hcp-
+hollow, are indicated by the arrows. The positions of zinc
+atoms in each layer are numbered as 1, 2, 3, and 4, respec-
+tively. For the Mg doping level x = 0.25 case, the zinc
+atoms at position 1 are substituted by magnesium atoms
+in the odd layers, while the zinc atoms at position 3 are
+replaced in the even layers. For the x = 0.50 situation,
+the zinc atoms at positions 2 and 4 are replaced by mag-
+nesium atoms in each layer. All calculations are carried
+out in framework of density functional theory using the
+Viennaab initio Simulation Package (VASP) [32]. The
+in-plane lattice constants of Zn1−xMgxO/ZnO (x = 0.25
+and 0.50) heterostructures are fixed to those of ZnO dur-
+ing the calculations.
+FIG.
+1.
+(a)
+Schematic
+geometrical
+structure
+of
+Zn1−xMgxO/ZnO
+(x=0.25
+and
+0.50)
+heterostructure
+with two pseudo-H-passivated surfaces.
+(b) The top view
+of the heterostructure along the [001] direction.
+Here the
+“On-top, Fcc-hollow, and Hcp-hollow” are the adsorption
+sites for exotic atoms or groups.
+Figure
+2(a)
+shows
+the
+band
+structure
+of
+Zn0.75Mg0.25O/ZnO heterostructure shown in Fig. 1(a)
+(i.e., the heterostructure has 18 Zn-O and 18 Zn-Mg-O
+-1
+0
+1
+2
+3
+4
+0
+20
+40
+60
+80
+100
+-6
+-4
+-2
+0
+2
+4
+6
+ 
+ 
+Energy (eV)
+M
+K
+(a)
+(b)
+L18
+ Planar average
+ Macroscopic average
+L2
+ 
+ 
+Electrostatic potential (eV)
+Distance along the [001] (�
+)
+L18
+L4
+Interface
+ZnO
+Zn
+0. 75
+Mg
+0. 25
+O
+FIG.
+2.
+(a)
+The
+energy
+band
+structure
+of
+the
+Zn0.75Mg0.25O/ZnO heterostructure without oxygen vacan-
+cies and with two pseudo-H-atoms-passivated surfaces.
+(b)
+The plane average (solid curve) and macroscopic average
+(dash-dot curve) electrostatic potential (seen by electron)
+across the Zn0.75Mg0.25O/ZnO heterostructure along the [001]
+direction.
+layers and two pseudo-H-atoms-passivated surfaces).
+Clearly, the valence band maximum (VBM) and the con-
+duction band minimum (CBM) are both located at the Γ
+point, and the Fermi level lies in the band gap. Thus the
+energy band of the Zn0.75Mg0.25O/ZnO heterostructure
+exhibits direct-gap semiconductor characteristics (the
+calculated band gap is 1.45 eV) and no 2DEG is formed
+at the interface. For the x = 0.50 case, the band struc-
+ture is similar to that of the x = 0.25 and the calculated
+bad gap is 1.56 eV. Therefore, 2DEG cannot appear near
+the interfaces of the perfect Zn1−xMgxO/ZnO (x = 0.25
+and 0.50) heterostructures (without defects) with thick
+Zn1−xMgxO layers. We also calculated the electrostatic
+potential distribution for the above heterostructures,
+and Fig. 2(b) presents the results for the x = 0.25 case
+as an example.
+There is a conspicuous bulge in the
+macroscopic average potential curve near the interface
+(from the L¯4 Zn-O layer to the L2 Zn-Mg-O layer). In
+the atomic layers away from the interface, e.g., the Zn-O
+layers from L¯4 to L ¯18 or Zn-Mg-O layers from L2 to
+L18, the average potential almost retains a constant.
+Thus, a potential barrier rather than a quantum well
+is formed near the interface of the Zn0.75Mg0.25O/ZnO
+heterostructure.
+Similar phenomena are also observed
+in the macroscopic average potential curve of the
+Zn0.5Mg0.5O/ZnO
+heterostructure.
+This
+potential
+barrier should be caused by the polar discontinuity at
+the interface, which could induce a localized polarization
+field near the interface.
+The polarization field cannot
+cause the bottom of the conduction band to overlap
+with the top of the valence band as in the case of
+LaAlO3/SrTiO3 heterostructures [33]. Thus, the polar
+discontinuity alone cannot explain the observed 2DEG
+near the interface of Zn1−xMgxO/ZnO heterostructure
+with thick Zn1−xMgxO layers.
+Then,
+why
+the
+2DEGs
+can
+be
+formed
+in
+Zn1−xMgxO/ZnO
+heterostructures
+with
+thick
+
+wollod-qoH
+ECC-JOJJOM
+OU-tob
+(p) Lob AIGM sJoua e [ool]q!lGcrIo
+U
+M
+·H
+T18
+r18
+SUO
+(B)3
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+-0.4
+-0.2
+0.0
+0.2
+0.4
+ 
+ 
+Formation energy (eV)
+Zn O
+Zn
+0.75
+Mg
+0.25
+O
+In t erface
+(a)
+L1
+L6
+L12
+L12
+L6
+L1
+L1
+L6
+L12
+L12
+L6
+L1
+Formation energy (eV)
+ 
+In t erface
+Zn O
+Zn
+0.5
+Mg
+0.5
+O
+(b)
+FIG. 3. Formation energies of oxygen vacancies at different
+atomic layers in Zn1−xMgxO/ZnO heterostructures with two
+pseudo-H-passivated surfaces. (a) For the x = 0.25, and (b)
+for the x = 0.50 heterostructures.
+Zn1−xMgxO layers?
+It should be noticed that as
+intrinsic defects in ZnO and Zn1−xMgxO films, oxygen
+vacancies are inevitable during device fabrication and
+could play crucial roles for the formation of 2DEG in
+Zn1−xMgxO/ZnO heterostructures [34–37]. Next, we in-
+vestigate the effect of oxygen vacancies on the electronic
+structures of Zn1−xMgxO/ZnO (x = 0.25 and 0.50)
+heterostructures with thick Zn1−xMgxO layers.
+First,
+we calculate the formation energy of oxygen vacancies
+(Ef) in each atomic layer of the above heterostructures.
+In the oxygen-rich limit, Ef can be written as [38]
+Ef = E(VO) − (E0 − 0.5EO2),
+(1)
+where E(VO) and E0 are the calculated total energies of
+the Zn1−xMgxO/ZnO (x = 0.25 and 0.50) heterostruc-
+tures with and without oxygen vacancies, and EO2 is the
+calculated total energy of the single O2 molecule. For
+the configuration in Fig. 1, each in-plane supercell con-
+tains four oxygen atoms, whose positions are labeled as
+a, b, c, and d, respectively. The oxygen atoms at d po-
+sition are removed in a certain fixed layer to create oxy-
+gen vacancies in the calculations.
+Figure 3 shows the
+formation energies of the oxygen vacancies in each layer
+of the Zn0.75Mg0.25O/ZnO and Zn0.5Mg0.5O/ZnO het-
+erostructures with 18 Zn-O and 18 Zn-Mg-O layers, and
+two pseudo-H-passivated surfaces. Inspection of Fig. 3
+indicates that the overall variation trends of the Ef vs
+layer number curves for the two heterostructures are sim-
+ilar.
+Thus we only discuss the variation of Ef in the
+Zn0.75Mg0.25O/ZnO heterostructure. On the ZnO side,
+the value Ef keeps as a constant in the first two layers,
+and then sharply increases with increasing layer number,
+reaches its maximum at L¯4, then decreases with further
+increasing layer number, and tends to be saturated as the
+layer number is greater than 9. On the Zn0.75Mg0.25O
+side, the values of Ef near the interface (L1 to L6 Zn-
+Mg-O layers) vary between −0.3 eV and −0.1 eV, while
+those for the layers with layer number being greater than
+6 are almost fixed at −0.1 eV. Obviously, the oxygen va-
+cancies can be easily formed on the ZnO side, especially
+in the first two Zn-O layers near the interface.
+Considering the variation trends in electronic struc-
+tures with VO position for the x = 0.25 and 0.50 het-
+erostructures with two pseudo-H passivated surfaces are
+also similar, we only present and discuss the results ob-
+tained from the x = 0.25 ones.
+We first discuss the
+case that oxygen vacancies are located at the most eas-
+ily formed position (L¯1 layer). Figure 4(a) presents the
+band structure of this configuration. From this figure,
+one can see that the oxygen vacancies in the L¯1 Zn-O
+layer introduce a defect band in the band gap and the
+top of the defect band is higher than the Fermi level.
+At the same time, the Fermi level enters into the bot-
+tom of the conduction band, i.e., the conduction band
+overlaps with the defect band. Thus, part of the elec-
+trons in the defect band would be transferred into the
+conduction band and become conduction electrons. Fig-
+ure 4(b) shows the partial density of states (DOS) pro-
+jected onto atomic planes for the x = 0.25 heterostruc-
+ture with oxygen vacancies in the L¯1 Zn-O layer and two
+pseudo-H passivated surfaces. Clearly, only in L¯2 , L¯1,
+and L1 layers the DOS near the Fermi level is nonzero,
+i.e., the conduction electrons are concentrated in the two
+Zn-O layers and one Zn-Mg-O layer near the interface.
+These three layers occupy a space with thickness ∼8.4 ˚A,
+which indicates that the 2DEG is formed near the in-
+terface of the heterostructure. From the orbital DOS of
+L¯2 to L1 layers, it is found that these conduction elec-
+trons are mainly composed of Zn-4s and O-2p orbitals
+(not shown). In addition, it is found that when the oxy-
+gen vacancies are located in the L¯2 and L¯3 Zn-O lay-
+ers and the L1 to L6 Zn-Mg-O layers, their band struc-
+FIG. 4. (a) The band structure of Zn0.75Mg0.25O/ZnO het-
+erostructure with oxygen vacancies in the L¯1 Zn-O layer.
+(b) The partial DOS projected onto atomic planes for
+the Zn0.75Mg0.25O/ZnO heterostructure with oxygen vacan-
+cies in the L¯1 Zn-O layer.
+(c) The band structure of
+Zn0.75Mg0.25O/ZnO heterostructure with oxygen vacancies in
+the L ¯15 Zn-O layer.
+(d) The partial DOS projected onto
+atomic planes for the Zn0.75Mg0.25O/ZnO heterostructure
+with oxygen vacancies in the L ¯15 Zn-O layer.
+
+　K
+I
+M
+D02 (e)G)
+0
+0
+(C)
+S
+(b)
+L
+M
+L
+-3
+3
+5
+EUGL& (GA)
+EUGLS (GA)
+0
+0
+(g)
+(d)
+r18
+II
+I184
+tures are similar to that in Fig. 4(a). However, the band
+structures of the heterostructures would reveal semicon-
+ductor characteristics when the oxygen vacancies are far
+from the interface (i.e., behind the L¯3 Zn-O layer and
+L6 Zn-Mg-O layer).
+We take the Zn0.75Mg0.25O/ZnO
+heterostructure with oxygen vacancies in the L ¯15 Zn-O
+layer as an example. Figure 4(c) shows the band struc-
+ture of this configuration. The oxygen vacancies in the
+L ¯15 Zn-O layer also introduce a defect band in the gap,
+while the top of the defect band is located at 0.11 eV be-
+low the bottom of the conduction band. The Fermi level
+lies between the conduction band and the defect band.
+Therefore, the introduction of oxygen vacancies in the
+L ¯15 Zn-O layer cannot induce 2DEG at the interface of
+the heterostructure. Figure 4(d) shows the partial DOS
+projected onto atomic planes for the Zn0.75Mg0.25O/ZnO
+heterostructure with oxygen vacancies in the L ¯15 Zn-O
+layer. From this figure, one can see that the defect band
+of the oxygen vacancies is in fact composed of a large
+number of deep energy levels as far as the energy band
+of the inner atomic layer is concerned. These deep lev-
+els cannot overlap with the conduction band even if the
+bottom conduction band of the Zn-O layer near the in-
+terface is lower than that of the inner Zn-O layer. On
+the contrary, the defect levels of the oxygen vacancies
+near the interface layers are so shallow that the bottom
+of the conduction band overlaps with the top of the de-
+fect band [see Fig. 4(b)]. This is why 2DEG exists only
+when the oxygen vacancies are located near the interface
+of the heterostructure. On the other hand, the defect
+band formed by oxygen vacancies of inner Zn-O layers
+could enhance the conductivity of the heterostructure:
+the device will exhibit a thermal-activated form conduc-
+tance with activation energy Ea, where Ea is about half
+of the energy difference between the bottom of conduc-
+tion band and the top of the defect band. Summarizing
+the results mentioned above, one can readily conclude
+that the oxygen vacancies near the interface are the ori-
+gin of the 2DEGs in Zn1−xMgxO/ZnO (x = 0.25 and
+0.50) heterostructures with thick Zn1−xMgxO layers.
+Now,
+we
+study
+the
+origin
+of
+2DEGs
+in
+Zn1−xMgxO/ZnO (x
+=
+0.25 and 0.50) heterostruc-
+tures when the Zn1−xMgxO films are very thin. In this
+situation, we calculate the electronic structures of the
+heterostructures with 42 Zn-O and 18 Zn-Mg-O layers,
+in which the surface of the Zn1−xMgxO film is no longer
+passivated.
+The reason for choosing 42 Zn-O layers
+(instead of 18 layers) is to obtain the distribution range
+of 2DEG on the ZnO side.
+Since the results obtained
+from the x = 0.25 and 0.50 heterostructures are also
+similar, we only present and discuss the results for the
+x = 0.25 heterostructure. Figure 5(a) shows the electro-
+static potential of Zn0.75Mg0.25O/ZnO heterostructure
+(with 42 Zn-O and 18 Zn-Mg-O layers) in which only
+the surface of ZnO film is passivated by pseudo-H
+atoms.
+Obviously, the macroscopic average potential
+on the ZnO side is insensitive to the position, while it
+decreases with increasing distance to the interface on
+0
+40
+80
+120
+160
+-10
+-5
+0
+5
+10
+0
+40
+80
+120
+160
+-10
+-5
+0
+5
+10
+Interface
+ 
+ 
+Electrostatic potential (eV)
+Distance along the [001] direction (�)
+ZnO
+Zn
+0 .7 5
+Mg
+0 .2 5
+O
+(a)
+ Macroscopic-Average
+ Planar-Average
+Electrostatic potential (eV)
+ 
+Interface
+ZnO
+Zn
+0 .7 5
+Mg
+0 .2 5
+O
+L10
+L18
+(b)
+FIG. 5. (a) The plane average (solid curve) and macroscopic
+average (dash-dot curve) electrostatic potential (seen by elec-
+tron) along [001] direction for the Zn0.75Mg0.25O/ZnO het-
+erostructure, in which only the surface of ZnO film is pas-
+sivated by pseudo-H atoms.
+(b) The plane average (solid
+curve) and macroscopic average (dash-dot curve) electro-
+static potential (seen by electron) along [001] direction of the
+Zn0.75Mg0.25O/ZnO heterostructure with a Zn0.75Mg0.25O
+surface of H-atom adsorption.
+the Zn0.75Mg0.25O side.
+A macroscopic field perpen-
+dicular to the surface with the magnitude of 0.06 V/˚A
+is obtained by linear fitting the macroscopic average
+electrostatic potential.
+This kind of field or potential
+would lead to an instability of the (001) polar (so-called
+Tasker type III) surface [29, 39]. In the light of recent
+experimental and theoretical results, the polar oxide
+surfaces can be stabilized via charge transfer between the
+upper and lower surfaces [25, 26], adsorption of external
+atoms [25–29], and stoichiometry variations [25–28].
+For Zn1−xMgxO/ZnO heterostructures, we consider the
+effects of adsorption (hydrogen atoms, OH groups, and
+oxygen atoms) and stoichiometry variations (defects) on
+the electronic structures of Zn1−xMgxO/ZnO (x = 0.25
+and 0.50) heterostructures.
+Through structural relax-
+ations, it is found that the hydrogen atoms prefer to
+be adsorbed atop the zinc atom (On-top site), while
+the preferred adsorption sites for the OH groups and
+oxygen atoms are the Fcc-hollow sites [see Fig. 1(b)].
+Our results are consistent with those in Refs. [25–27].
+For the 2 × 2 in-plane (001) Zn1−xMgxO supercell, the
+numbers of the On-top and Fcc-hollow sites are both 4.
+In our calculations, the coverages of hydrogen atoms,
+OH groups, and oxygen atoms adsorbed on the surface
+of Zn1−xMgxO are 50%, 50%, and, 25%, respectively,
+while the concentration of vacancies on the Zn or Mg
+sites is 25% [25–27].
+Specifically, for the x = 0.25
+heterostructure, the absorption sites of the hydrogen
+atoms are set on the top of the zinc atoms at positions
+1 and 4 [see Fig. 1(b)]; the adsorption sites for the OH
+groups are set at Fcc-hollow positions located at the
+top of the arrow and the position of the black dot; the
+Fcc-hollow position at the top of the arrow is also set
+as the adsorption site of oxygen atoms; the Zn vacancies
+are obtained via removing the zinc atoms located at
+
+5
+H
+ads
+L42
+ZnO
+Zn
+0. 75
+Mg
+0. 25
+O
+L10
+L12
+ 
+ 
+Energy (eV)
+(a)
+(b)
+ZnO
+Zn
+0. 75
+Mg
+0. 25
+O
+L1
+L42
+H
+pass
+L10
+L15
+L1
+ 
+DOS (states/eV)
+ 
+Energy (eV)
+FIG. 6. The partial DOS projected onto the atomic layers for
+the Zn0.75Mg0.25O/ZnO heterostructures with surfaces of (a)
+H-atom absorption, and (b) O-atom absorption.
+position 1.
+Figure
+5(b)
+shows
+the
+electrostatic
+potential
+of
+Zn0.75Mg0.25O/ZnO
+heterostructures
+with
+hydrogen
+atoms adsorbed on the Zn0.75Mg0.25O surface (and with
+42 Zn-O and 18 Zn-Mg-O layers). The results for the
+Zn0.75Mg0.25O surface with oxygen atoms adsorption,
+OH groups adsorption, and Zn or Mg vacancies are simi-
+lar to that shown in Fig. 5(b). The macroscopic average
+potential on the ZnO side remains nearly a constant af-
+ter adsorption of hydrogen atoms. On the Zn0.75Mg0.25O
+side, the macroscopic average potential is almost insensi-
+tive to the position from L1 to L9 layers, and then slightly
+increases with increasing distance to the interface. An
+electrostatic field (with magnitude of ∼0.038 V/˚A) being
+opposite to that shown in Fig 5(b) exists between L10
+and L18 layers.
+Thus surface adsorption or metal ion
+vacancies could really stabilize the polar surfaces of the
+Zn1−xMgxO/ZnO (x = 0.25 and 0.50) heterostructures.
+The electronic structures of the Zn1−xMgxO/ZnO
+(x = 0.25 and 0.50) heterostructures with exotic-atoms-
+adsorbed surfaces or with surfaces having metal ion va-
+cancies, have been also calculated.
+It is found that
+the electronic structures of the heterostructures reveal
+semiconductor characteristics when the surface contains
+metal ion vacancies, while the electronic structure ex-
+hibits metallic characteristics as the hydrogen, oxygen
+atoms, and OH groups are adsorbed on the surface, re-
+spectively.
+Figure 6(a) show the partial DOS decom-
+posed to the atomic layers for the Zn0.75Mg0.25O/ZnO
+heterostructures (42 Zn-O and 18 Zn-Mg-O layers) with
+hydrogen atoms adsorption.
+For the oxygen-atom- or
+OH-groups-adsorption case, the partial DOS plot is sim-
+ilar to that in Fig. 6(a). Clearly, the adsorption of hy-
+drogen atoms on the surface introduces defect states in
+the gap. Although polarization field distributed in the
+L10 to L18 Zn-Mg-O layers has significantly lifted up
+the top of the valence band, the valence band is still far
+from overlapping with the conduction band. However,
+the defect states introduced by hydrogen atoms are lo-
+cated near the Fermi level, and partially higher than the
+Fermi level. As a result, the defect band overlaps with
+the conduction band, which renders the heterostructure
+to exhibit metallic characteristics in electronic structures.
+Inspection of Fig. 6(a) also indicates that the conduction
+electrons are distributed from L ¯12 to L10 layers, i.e., in
+the range of ∼5.53 nm near the interface. Thus, the het-
+erostructure would reveal 2D or quasi-2D behaviors in
+transport properties. We also calculated the electronic
+structures of the Zn1−xMgxO/ZnO (x = 0.25 and 0.50)
+heterostructures with oxygen vacancies and surfaces ad-
+sorbed with exotic atoms. It is found that the introduc-
+tion of an oxygen vacancy in a certain atomic layer of the
+2×2 in-plane supercell would also produce a defect band
+in the gap. However, the defect band does not overlap
+with the conduction band, i.e., the introduction of oxygen
+vacancies does not change the ultimate properties of the
+heterostructures. As an example, in Fig. 6(b) we give the
+partial DOS projected onto the atomic layers for the H-
+adsorbed Zn0.75Mg0.25O/ZnO heterostructure (with 42
+Zn-O and 18 Zn-Mg-O layers) with oxygen vacancies in
+L¯1. Comparing the partial DOS of the heterostructure
+without oxygen vacancies [Fig. 6(a)], one can see that
+oxygen vacancies in L¯1 introduce an extra defect band,
+whose maximum is located at 0.15 eV below the bottom
+of the conduction band.
+In this situation, the 2DEG
+distributes from L ¯15 to L10 layers (∼6.53 nm) and still
+originates from the adsorption of hydrogen atoms. The
+electronic structure of the heterostructures with oxygen-
+atoms-adsorbed or OH-groups-adsorbed surface is simi-
+lar to that for the H-adsorbed heterostructure. In ad-
+dition, the introduction of oxygen vacancies in Zn-O
+layer near the interface does not change the semicon-
+ductor characteristic of the electronic structure for the
+heterostructure with metal ion vacancies in the surface
+of Zn1−xMgxO film. Thus, for Zn0.75Mg0.25O/ZnO het-
+erostructure with thin Zn1−xMgxO film, the adsorption
+of hydrogen atoms, oxygen atoms, or OH groups on the
+surface of Zn1−xMgxO layer is responsible for the forma-
+tion of 2DEG near the interface.
+In summary, to explore the origin of 2DEGs in
+Zn1−xMgxO/ZnO heterostructures, we constructed the
+Zn1−xMgxO/ZnO (x
+=
+0.25 and 0.50) heterostruc-
+tures with different surfaces and investigated their elec-
+tronic structures by first-principles calculations.
+It is
+found that the polarity discontinuity near the interface
+can neither lead to the formation of 2DEGs in devices
+with thick Zn1−xMgxO layers nor in devices with thin
+Zn1−xMgxO layers. For the heterostructures with thick
+Zn1−xMgxO layers, the oxygen vacancies near the inter-
+face are the source of the 2DEGs. For the heterostruc-
+tures with thin Zn1−xMgxO layers, adsorption of hydro-
+gen atoms, oxygen atoms, or OH groups on the surface
+of Zn1−xMgxO films can not only stabilize the polar sur-
+face of Zn1−xMgxO layer, but also cause the formation
+
+-J
+-1
+03
+-56
+of 2DEGs near the interfaces of the devices.
+The calculation was conducted on the CJQS-HPC plat-
+form at Tianjin University. This work is supported by the
+National Natural Science Foundation of China through
+Grants No. 12174282.
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