Absence of replica symmetry breaking in the transverse and longitudinal random field Ising model

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1 arxiv: v6 [math-ph] 28 Dec 2017 Absence of replica symmetry breaking in the transverse and longitudinal random field Ising model C. Itoi Department of Physics, GS & CST, Nihon University October 13, 2018 Abstract It is proved that replica symmetry is not broken in the transverse and longitudinal random field Ising model. In this model, the variance of spin overlap of any component vanishes in any dimension almost everywhere in the coupling constant space in the infinite volume it. The weak Fortuin-Kasteleyn-Ginibre property in this model and the Ghirlanda-Guerra identities in artificial models in a path integral representation based on the Lie-Trotter-Suzuki formula enable us to extend Chatterjee s proof for the random field Ising model to the quantum model. 1 Introduction Replica symmetry breaking is known to be a non-trivial phenomenon in systems with quenched disorder. This phenomenon in mean field spin glass models has been studied deeply, since Talagrand proved the Parisi conjecture [23] for the Sherrington-Kirkpatrick (SK) model [25] in a mathematically rigorous manner [27]. When replica symmetry is broken, the observed value of an observable in a typical sample differs from its sample expectation with finite probability, even though all samples in the sample ensemble are synthesized using exactly the same method. Theoretical physicists and mathematicians have been seeking this phenomenon also in more realistic short range spin glass models, such as the Edwards-Anderson (EA) model [13], however, only a few rigorous results for the replica symmetry breaking have been obtained in low temperature region in short range systems. Nishimori and Sherrington showed that the replica symmetry breaking does not occur on the Nishimori line located out of the spin glass phase in the EA model [20, 21]. Recently, Chatterjee proved a remarkable theorem that replica symmetry is not broken in the Ising model with a longitudinal Gaussian random field in any dimension almost everywhere in the coupling constant space [3]. It was shown that the variance of overlap vanishes in the system with the Fortuin-Kasteleyn-Ginibre (FKG) property using the Ghirlanda-Guerra identities [1, 15]. In the present paper, we extend his argument to quantum systems with the weak FKG property. This is a first rigorous result for replica symmetry breaking in quantum disordered systems with short range interactions. 1

2 2 Definitions and main result We study disordered quantum spin systems on d-dimensional cubic lattice V L := [1,L] d Z d and their corresponding classical spin systems on (d + 1)-dimensional cubic lattice W L,M = V L T M, where T M := [1,M] Z with positive integers L and M. Let B L be a collection of interaction bonds which are translations of a pair of sites in V L. One of the most important example is given by nearest neighbor bonds B L = {{x,y} x,y V L, x y = 1}. A spin operator S i x (i = 1,2,3) at a site x V L on a Hilbert space H := x V L H x is defined by a tensor product of the Pauli matrix 1 2 σi acting on H x C 2 and unities. These operators are self-adjoint and satisfies the commutation relation [S 1 x,s2 y ] = iδ x,ys 3 x, [S2 x,s3 y ] = iδ x,ys 1 x, [S3 x,s1 y ] = iδ x,ys 2 x, and the spin at each site x has a fixed magnitude We study the following Hamiltonian 3 (Sx j )2 = j=1 H V (S,g) := A(S 1,g 1 )+B(S 3,g 3 ), (1) consisting of non-commuting two terms A and B defined by A(S 1,g 1 ) := x V L J 1 g 1 x S1 x, (2) B(S 3,g 3 ) := {x,y} B L S 3 x S3 y x V L (J 3 g 3 x +c)s3 x, (3) where (gx) i x VL,i=1,3 are standard Gaussian i.i.d. random variables and J 1,J 3,c R are coupling constants. Here, we define Gibbs state for the Hamiltonian. For a positive β, the partition function is defined by Z V (J,g) := Tre βh V (S,g) (4) where the trace is taken over the Hilbert space H. Let f be an arbitrary function of spin operators S i x,(x V L,i = 1,2,3). The expectation of f in the Gibbs state is given by f(s i ) = 1 Z V (J,g) Trf(Si )e βh V (S,g). (5) Here, we introduce a fictitious time t [0,1] and define a time evolution of operators with the Hamiltonian. Let O be an arbitrary self-adjoint operator, and we define an operator valued function O(t) of t [0,1] by O(t) := e th Oe th. (6) Furthermore,wedefinetheDuhamelexpectationoftimedependentoperatorsO 1 (t 1 ),,O k (t k ) by (O 1,O 2,,O k ) D := dt 1 dt k T[O 1 (t 1 )O 2 (t 2 ) O k (t k )], [0,1] k 2

3 where the symbol T is a multilinear mapping of the chronological ordering. If we define a partition function with arbitrary self adjoint operators O 0,O 1,,O k and real numbers x 1,,x k [ ] k Z(x 1,,x k ) := Trexpβ O 0 + x i O i, the Duhamel expectation of k operators represents the k-th order derivative of the partition function [12, 16, 24] β k (O 1,,O k ) D = 1 Z i=1 k Z x 1 x k. To study replica symmetry breaking, we consider n replicated spin model defined by the following Hamiltonian n H V (S α,g). (7) α=1 The overlap operator Rα,β i (i = 1,2,3) between different replicated spins is defined by R i α,β = 1 V L Sx i,α S i,β x V L for α,β = 1,2,n and α β. It is well-known that quantum spin systems on a d-dimensional lattice can be represented as (d + 1)-dimensional classical Ising systems [26]. The Lie-Trotter-Suzuki formula for the Hamiltonian (1) x, e βa βb = M (e βa/m e βb/m ) M and inserting M resolutions of unity in eigenstates of 2S 3 x on H 1 = σ S V σ 3 3 σ, (8) where we define Sx i σ i = σ x 2 σ i. S V := { 1,1} V L is a set of eigenvalue configurations, enable us to represent the d- dimensional quantum spin system in the following(d+1)-dimensional classical spin system Z V (J,g) = C W e βhw(σ,g), (9) M σ S W where the summation is taken over spin configurations S W := { 1,1} W L,M on the (d+1)- dimensional lattice W L,M and the factor C W is independent of spin configurations. In this representation, we impose the periodic boundary condition on spin configuration σ x,t+m = σ x,t with respect to t T M and free boundary condition with respect to x V L. For instance in the transverse field Ising model with longitudinal random field [9, 10, 12], the Hamiltonian is given by H W (σ,g) = t T M [ 1 4M σ x,t σ y,t + 1 2M {x,y} B L 3 ] (J 3 gx+c)σ 3 x,t + K x σ x,t σ x,t+1, (10) x V L x V L

4 where and the factor is given by tanhβk x = e βj 1g 1 x/m, C W = x V L 1 2 sinh βj 1gx 1 M M 2. (11) To obtain our main result, we consider an artificial (d + 1)-dimensional random field Ising model with quenched i.i.d standard Gaussian random variables (h i x,t) (x,t) WL,M,i=1,3, and arbitrary numbers b i,c R for i = 1,3. We define the following perturbed Hamiltonian where H(b 1,b 3,c,h 1,h 3 ) := t T M ( 1 4M 1 2M σ x,t σ y,t + K x,t σ x,t σ x,t+1 ) {x,y} B L x V L (J 3 gx 3 +b 3 Mh 3 x,t +c)σ x,t, (12) (x,t) W L,M tanhβk x,t = e β(j 1g 1 x +b 1h 1 x,t )/M, such that H(0,0,c,0,0) is identical to H W (σ,g) defined by (10). This model has operator representation Z V (b) = Tr M (e βat/m e Bt/M ), (13) M where t=1 A t (S 1,g 1,h 1 ) := x V L (J 1 g 1 x +b 1h 1 x,t )S1 x, (14) B t (S 3,g 3,h 3 ) := Sx 3 S3 y (J 3 gx 3 +b 3 Mh 3 x,t +c)sx 3, {x,y} B L x V L For lighter notation, we denote and define partition function Z i (b) by and define functions ψ i,l and p i,l by H 1 (b) := H(b,0,c,h,0), H 3 (b) := H(0,b,c,0,h) (15) Z i (b) := σ S W e βhi(b), ψ i,l (b) := 1 V L logz i(b), p i,l (b) := Eψ i,l (b), where a sample expectation E denotes expectation over all random fields (g i x ) x V L,i=1,3, (h i x,t ) (x,t) W L,M,i=1,3. Note that Z V = M C WZ i (0), 4

5 given by (12) and (15). Hereafter, f b,i denotes the Gibbs expectation of a function f : S W R with the Hamiltonian H i (b) Note that for i = 1,3 f(σ) b,i := 1 Z i (b) σ S W f(σ)e βhi(b), f(2s 3 ) = M f(σ) 0,i in the representations (12) and (15) for H i (b). In the present paper, we obtain the following main theorem for the transverse and longitudinal random field Ising model. Theorem 2.1 Consider the transverse and longitudinal random field Ising model defined by the Hamiltonian (1) and its replicated model (7). Almost everywhere in the coupling constant space, the infinite volume it E Ri 1,2, exists for any i = 1,2,3 and the variance of the overlap operator calculated in the replica symmetric Gibbs state vanishes E (Ri 1,2 E Ri 1,2 )2 = 0. (16) Theorem 2.1 shows that the overlap operator R1,2 i is self-averaging in this model. This impliesthattheobservedvalueoftheoverlapoperatorr1,2 i convergesinprobabilitytoward its Gibbs and sample expectation E R1,2 i. Since the replica symmetric Gibbs expectation of the overlap operator is spin glass order parameter, the phase diagram should be unique if the sample is synthesized in the same method. There are two key techniques to prove Theorem 2.1: the weak FKG property of the transverse and longitudinal random field Ising model in the (d + 1)-dimensional representation and continuity of an artificial perturbative (d + 1)-dimensional model with the Ghirlanda-Guerra identities [1, 15]. Since a straightforward extension of the Ghirlanda- Guerra identities to quantum systems is not sufficient to judge absence or appearance of replica symmetry breaking in quantum systems unlike the classical system [3], we utilize the classical Ghirlanda-Guerra identities in artificial models given by Hamiltonians (12) and (15). We prove that the expectation of the overlap operator is a continuous function of the perturbation parameter. These results enable us to prove Theorem 2.1 which shows absence of replica symmetry breaking in the transverse and longitudinal random field Ising model. 3 Proof Here, we consider the perturbed model defined by the Hamiltonian(15) in d+1 dimension. For this model, there are useful lemmas proved in the literature. Here we present them as Lemma without proofs. Lemma 3.1 is proved as in [2, 8, 11, 18], Lemma 3.2 and Lemma 3.3 are proved in [18, 19], and Lemma 3.4 is proved in [14]. 5

6 Lemma 3.1 The following infinite volume it independent of boundary conditions exists p i (b) = Eψ i,l(b). M for each (β,j 1,J 3,b,c) [0, ) R 4. Lemma 3.2 For any (β,j 1,J 3,b,c) [0, ) R 4, there exists a positive number C independent of L, such that the variance of ψ L is bounded from the above as follows E[ψ i,l(b) p i,l (b)] 2 C M V L. Here, we define two types of deviations of of an arbitrary operator O by δo := O O, O := O E O. And, also define two types of deviations an arbitrary function f : S W R by δf(σ) := f(σ) f(σ) b,i, f(σ) := f(σ) E f(σ) b,i. Define an order parameter m i L by m i L := 1 V L x V L S i x and define the corresponding order parameter µ i L by µ 1 L := 1 4 V L M µ 3 L := 1 2 V L M (1 σ x,t σ x,t+1 ), (17) x V L,t T M σ x,t. (18) x V L,t T M Lemma 3.3 For any (β,j 1,J 3,b,c) [0, ) R 4 with βj i 0, there exists a positive number C independent of L, such that M E δµi 2 L b,i C 1 βj i V L. (19) Lemma 3.3 gives an upper bound of Duhamel product for b i = 0 E(δm i L,δmi L ) D C 1 βj i V L. (20) We say that the system satisfies the weak Fortuin-Kasteleyn-Ginibre (FKG) condition, if the one point function S 3 x is monotonically increasing function of g3 y at any site y V L [2]. The weak FKG condition is equivalent to the positive semi-definiteness of truncated Duhamel function (S 3 x,s 3 y) D S 3 x S 3 y 0, 6

7 for any two sites x,y V L. The d-dimensional transverse and longitudinal random field Ising model satisfies weak FKG condition, because of the following lemma for the corresponding (d + 1)-dimensional classical model with positive semi-definite exchange interactions. To explain the FKG inequality, we define a partial order over the set S W of spin configurations and generalized monotonicity for a function of spin configurations. For two spin configurations σ,τ S W, we denote σ τ, if σ x τ x for all x W L,M. We say that a function f : S W R is monotonically increasing in a general sense, if σ τ implies f(σ) f(τ). The following FKG inequality can be proved [14]. Note that the artificial Hamiltonians H i (a,b) given by (12) and (15) satisfy the FKG condition as well. Therefore, the one point function σ x,s b,i is monotonically increasing function of h 3 y,t. Lemma 3.4 Let f and g be monotonically increasing functions of spin configurations on W L,M in a general sense. In the random field Ising model with positive semi definite exchange interactions, f and g satisfy the Fortuin-Kasteleyn-Ginibre inequality where a truncated correlation function is defined by f(σ);g(σ) b,i 0, (21) f(σ);g(σ) b,i := f(σ)g(σ) b,i f(σ) b,i g(σ) b,i. (22) Lemma 3.5 For arbitrary sites w,x,y,z W L,M, Proof. Define functions f ± : S W W 2 L,M R by σ x σ y ;σ w σ z b,i (σ x +σ y );(σ w +σ z ) b,i (23) f ± (σ,w,x) := (σ w ±1)(1±σ x ). For arbitrary fixed w,x,y,z W L,M, functions f + (,w,x) and f (,y,z) of spin configurations are monotonically increasing in general sense. From the FKG inequality, and also σ x σ y ;σ w σ z b,i + (σ x +σ y );(σ w +σ z ) b,i = ( f + (σ,w,x);f + (σ,y,z) b,i + f (σ,w,x);f (σ,y,z) b,i )/2 0, (24) σ x σ y ;σ w σ z b,i + (σ x +σ y );(σ w +σ z ) b,i, = ( f + (σ,w,x);f (σ,y,z) b,i + f (σ,w,x);f + (σ,y,z) b,i )/2 0. (25) These inequalities give the inequality (23). Next we evaluate Gibbs expectation of functions of the overlap operators in the path integral representation with the Hamiltonian H i (bc). In these representations, we denote ρ 1 α,β := 1 (1 σx,s α 16 V L M σα x,s+1 )(1 σβ x,s σβ x,s+1 ), (x,s) W L,M ρ 3 α,β := 1 σx,s α 4 V L M σβ x,s, (26) (x,s) W L,M for α β. Note that for α β,γ δ and for i,j = 1,3. ρ i α,β ρj γ,δ 0,i = (R i α,β,rj γ,δ ) D, 7

8 Lemma 3.6 In the model defined by the Hamiltonian (15) with J 1 0 and J 3 0, the following expectations calculated in the replica symmetric Gibbs state vanish for any i = 1,3. Proof. First, consider i = 3 E[ ρ 3 1,22 b,3 ρ 3 1,2 2 b,3] = = 1 16 V L 2 M V L 2 M V L 2 M 2 M E[ ρi 2 1,2 b,i ρ i 1,2 2 b,i ] = 0, (27) 1 16 V L 2 M 2 x,y W L,M E[ σ x σ y 2 b,3 σ x 2 b,3 σ y 2 b,3] x,y V L E σ x σ y b,3 σ x b,3 σ y b,3 σ x σ y b,3 + σ x b,3 σ y b,3 x,y V L E σ x ;σ y b,3 x,y V L E σ x ;σ y b,3. (28) The final line is nonnegative because of the FKG inequality. Therefore, M E[ ρ3 2 1,2 b,3 ρ 3 1,2 2 b,3] M E δµ3 2 L b,3 where we have used Lemma 3.3. For i = 1, E[ ρ 1 1,2ρ 1 1,2 b,1 ρ 1 1,2 2 1 b,1] E M 128 V L 2 M 2 1 M 128 V L 2 M 2 x,y V L 1 M 32 V L 2 M 2 x,y V L 1 = M 32 V L 2 M 2 x,y V L 1 = M 8 E δµ3 2 L b,1 x,y V L E (σ x,s +σ x,s+1 );(σ y,t +σ y,t+1 ) b,1 s,t T M σ x,s ;σ y,t b,1 s,t T M C βj 3 [VL, (29) s,t T M E σ x,s σ x,s+1 ;σ y,t σ y,t+1 b,1 E σ x,s ;σ y,t b,1 s,t T M C βj 3 [VL. (30) We have used the inequality (23) in Lemma 3.5, Lemma 3.3. These bounds give the it. In the original model with b 1 = b 3 = 0, Lemma 3.6 implies for any i = 1,3. E[(Ri 1,2,R1,2) i D R1,2 i 2 ] = 0, (31) 8

9 Here we regard ψ i,l (h) := 1 V L logz i(b) (32) as a function of disorder h = (h w ) w WL,M. Let h and (h ) be i.i.d. standard Gaussian random variables, and define square root interpolating random variables with v [0, 1] by vhw + 1 vh w, (33) for w W L,M. Then, we define a generating function γ i (v) of the parameter v i [0,1] by γ i (v) := E[E ψ i,l ( vh+ 1 vh )] 2, (34) where E and E denote expectation in h and h, respectively. This generating function γ i is a generalization of a function introduced by Chatterjee [5]. Lemma 3.7 For any (β,j 1,J 3,b,c) [0, ) R 4, any positive integer L, any positive integer k and any v 0 [0,1), an upper bound on the k-th order derivative of the function γ i is given by d k γ i dv i k (v 0) (k 1)! (1 v 0 ) k 1 β 2 b 2 4 V L. (35) For an arbitrary v [0,1], the k-th order derivative of γ i is represented in the following d k γ i dv k (v) = E [ E ψ i,l,w1,,w k ( vh+ 1 vh ) ] 2. (36) w 1 W L,M w k W L,M Here we denote ψ i,l,w1,,w k (h) := k ψ i,l (h) h wk h w1. Proof. We obtain the formula (36) with k times use of integration by parts. This implies non negativity of all coefficients of the Taylor series of the function γ i (v) around any v = v 0 [0,1). Then, k-th derivatives are monotonically increasing in v. From Taylor s theorem, there exists v 1 (v 0,1) such that n 1 γ i (v) = k=0 (v v 0 ) k k! γ (k+1) i Each term in this series is bounded from the above by This completes the proof. γ i(1) = β2 b 2 4 V L 2 M (v 0 )+ (v v 0) n γ (n+1) i (v 1 ). n! w W L,M E σ w 2 b,i β2 b2 4 V L. 9

10 We define a term of the energy density with random field h 1 L := 1 4 V L M h 3 L := 1 2 V L M h x,t (1 σ x,t σ x,t+1 ), (37) (x,t) W L,M h x,t σ x,t. (38) w W L,M Lemma 3.8 For any βb 0, we have E δh i L2 C b,i β 2 b 2 V L + C V L, (39) where C and C are positive constant independent of L Proof. For i = 3 integration by parts gives E δh 3 L2 b,3 = = 1 4 V L 2 M 1 4 V L 2 M Eh x h y σ x ;σ y b,3 x,y W L,M [ E 2 σ x ;σ y b,3 + h x,y W x h y L,M E σ x ;σ x b.c,3 x W L,M 1 E 4 ψ 3,L V L β 2 b 2 h 2 2 x,y W x h + 1 y 4 V L L,M 1 [ V V L β 2 b 2 L 2 M 2 E 4 ψ ] 2 3,L 1 h x h y 4 V L x,y W L,M γ (4) 3 (0) + 1 β 2 b 2 V L 3 β 2 b 2 2 V L V L, (40) The bound for i = 1 is obtained in the same way. ] Lemma 3.9 For almost all b R, we have p i b = M βe hi L β 2 b b,i = M 4 (1 E ρi 1,2 b,i), (41) for p i (b) := M p i,l(b), and for b 0, M E hi L b,i = 0. (42) Proof. This can be shown in the standard convexity argument to obtain the Ghirlanda- Guerra identities in classical and quantum systems [4, 6, 7, 18, 19, 22, 28]. Note that ψ i,l, p i,l and p i are convex functions of b and c. To show the first equality (41), regard p i,l p i and ψ i,l as functions of b for lighter notation. By Lemma 3.7, we have Eψ i,l (b) 2 p i,l (b) 2 C V L, 10

11 where C is a positive number independent of L. Define the following functions w L (ǫ) := 1 ǫ [ ψ i,l(b+ǫ) p i,l (b+ǫ) + ψ i,l (b ǫ) p i,l (b ǫ) + ψ i,l (b) p i,l (b) ] e L (ǫ) := 1 ǫ [ p i,l(b+ǫ) p i (b+ǫ) + p i,l (b ǫ) p i (b ǫ) + p i,l (b) p i (b) ], for ǫ > 0. Note that the assumption on ψ i,l gives Ew L (ǫ) 3 C ǫ V L, (43) for any ǫ > 0. Since ψ i,l, p i,l and p i are convex functions of b, we have ψ i,l b (b) p i b (b) 1 ǫ [ψ i,l(b+ǫ) ψ i,l (b)] p i b 1 ǫ [ψ i,l(b+ǫ) p i,l (b+ǫ)+p i,l (b+ǫ) p i,l (b)+p i,l (b) ψ i,l (b) p i (b+ǫ)+p i (b+ǫ)+p i (b) p i (b)] p i b (b) 1 ǫ [ ψ i,l(b+ǫ) p i,l (b+ǫ) + p i,l (b) ψ i,l (b) + p i,l (b+ǫ) p i (b+ǫ) + p i,l (b) p i (b) ]+ 1 ǫ [p i(b+ǫ) p i (b)] p i b (b) w L (ǫ)+e L (ǫ)+ p i b (b+ǫ) p i b (b). As in the same calculation, we have Then, ψ i,l b (b) p i b (b) 1 ǫ [ψ i,l(b) ψ i,l (b ǫ)] p i b (b) w L (ǫ) e L (ǫ)+ p i b (b ǫ) p i b (b). E ψ i,l b (b) p i b (b) 3 C ǫ V L +e L(ǫ)+ p i b (b+ǫ) p i b (b ǫ). Convergence of p i,l in the infinite volume it implies β h E i L b,i p i M b (b) p i b (b+ǫ) p i b (b ǫ), The right hand side vanishes, since the convex functionp i (b) is continuously differentiable almost everywhere and ǫ > 0 is arbitrary. Jensen s inequality gives Eβ h i L b,i p i M b (b) = 0, (44) for almost all b. This leads the first equality (41). The equality (44) implies also M E hi L b,i = 0. 11

12 This and Lemma 3.8 enable us to obtain since M E hi L b,i = 0, E h i L b,i = E δh i L + hi L b,i b,i E δh i L b,i+e h i L b,i E δh i L2 b,i +E h i L b,i. Therefore the identities are obtained from the above as in the random field Ising model [3]. Note that Lemma 3.9 implies the existence of M E ρi 1,2 b,i for b 0. Lemma 3.10 Let f : SW n R be a bounded function of n replicated spin configurations. The Gibbs and sample expectations of f and spin overlap in the model defined by the Hamiltonian (12), satisfy the following identity for almost all b R M [E fρi 1,n+1 b,i 1 n E f b,ie ρ i 1,2 b,c,i 1 n which provides the Ghirlanda-Guerra identities [1, 15]. Proof. From the identity (44) in Lemma 3.9, M E hi L f b,i = 0. Calculating the right hand side gives the identity. n E fρ i 1,α b,i] = 0, (45) α=2 Lemma 3.11 For almost all constant field c R, the expectation of the overlap in the infinite volume it is continuous at b = 0 b 0 M E ρi 1,2 b,i = M E ρi 1,2 0,i, (46) b 0 M 1,2 2 b,i = M 1,2 2 0,i. (47) Proof. Evaluate the following partial derivative b E ρ3 1,2 b,3 β 2 b = E σ 8 V L M 2 x ;σ y b,3 ( σ x σ y b,3 3 σ x b,3 σ y b,3 ) x,y W L,M β2 b 8 V L M 2 x,y W L,M E σ x ;σ y b,3 σ x σ y b,3 3 σ x b,3 σ y b,3 β2 b E σ 2 V L M 2 x ;σ y b,3 = β2 b 2 V x,y W L M 2 L,M x,y W L,M E σ x ;σ y b,3 2βb c E µ3 L b,3 (48) 12

13 The FKG inequality has been used. This bound enables us to evaluate the following integral c2 c2 b 2β dc c 1 b 0 c 1 dc = 2β 0 M [E ρ3 1,2 b,3 E ρ 3 1,2 0,3 ] = db b c M E µ3 1,2 b,3 c2 dc c 1 b 0 db b M E ρ3 1,2 b,3 db b M [E µ3 L b,3,c=c 2 E µ 3 L b,3,c=c 1 ]. (49) The boundedness of E µ 3 L b,3 gives the it c2 c 1 dc b 0 M [E ρ3 1,2 b,3 E ρ 3 1,2 0,3 ] = 0 (50) for arbitrary c 1,c 2 R. Therefore, the integrand in the left hand side vanishes for almost all c, and this implies the first equality (46) for i = 3. For i = 1, evaluate the partial derivative b E ρ1 1,2 b,1 β 2 b = E σ 64 V L M 2 x,s σ x,s+1 ;σ y,t σ y,t+1 b,1 ( σ x,s σ x,s+1 ;σ y,t σ y,t+1 b,1 x,y V L,s,t T M 3 σ x,s σ x,s+1 b,1 σ y,t σ y,t+1 b,1 +2) 3β2 b 32 V L M 2 3β2 b 32 V L M 2 3β2 b 8 V L M 2 3βb 2 x,y V L,s,t T M E σ x,s σ x,s+1 ;σ y,t σ y,t+1 b,1 x,y V L,s,t T M E (σ x,s +σ x,s+1 );(σ y,t +σ y,t+1 ) b,1 E σ x ;σ y b,1 x,y W L,M c E µ3 L b,1 (51) The inequality (23) in Lemma 3.5 has been used. This bound and the same argument as for i = 3 give the first equality (46) for i = 1. To show the second equality (47), the following representation obtained by the FKG 13

14 inequality is useful b E ρ3 1,2 2 b,3 = ( b E 1 σ w 2 b,3 4 V L M w W L,M = β 2 b 16 V L 2 M 2 ) 2 x,y,z W L,M E σ y ;σ z b,3 σ x b,3 (2 σ x ;σ z b,3 σ y b,3 2 σ x b,3 σ y b,3 σ z b,3 + σ x b,3 σ y ;σ z b,3 ) β2 b E σ y ;σ z b,3, 2 V L M x,y,z W L,M = 2βb c E µ3 L b,3, (52) This bound and the boundedness of E µ 3 L b,3 enable us to prove the second equality (47) as well as the first one (46). The second equality (47) for i = 1 is proved by showing the bound b E ρ1 1,2 2 b,1 5βb 8 c E µ3 L b,1. This bound and the boundedness of E µ 3 L b,1 enable us to prove the second equality (47) for i = 1, and this completes the proof. Proof of Theorem 2.1 Since S 2 x E is orthogonal to E for an arbitrary eigenstate E of the Hamiltonian, we obtain S 2 x = 0 and S2 x S2 y = δ x,y/4. These imply E R1,2 2 = 0, E R2 2 1,2 = 1 16 V L, then Theorem 2.1 is valid trivially for R1,2 2. Therefore, we consider Ri α,β for i = 1,3. Since M E ρi 1,2 b,i exists by Lemma 3.9, this it exists also for b = 0 by Lemma Ri 1,2 = M E ρi 1,2 0,i = First, we use the Ghirlanda-Guerra identities for b 0. For n = 2 and f = ρ i 1,2, the identity in Lemma 3.10 b 0 M E ρi 1,2 b,i. M [2E ρi 1,2 ρi 1,3 b,i (E ρ i 1,2 b,i) 2 E ρ i 2 1,2 b,i ] = 0. (53) For n = 3 and f = ρ i 2,3, the identity in Lemma 3.10 gives M [3E ρi 2,3 ρi 1,4 b,i (E ρ i 1,2 b,i) 2 E ρ i 2,3 ρi 1,2 b,i E ρ i 2,3 ρi 1,3 b,i] = 0. (54) These two identities and ρ i 2,3 ρi 1,4 b,i = ρ i 1,22 b,i and ρ i 1,2 ρi 1,3 b,i = ρ 2,3 ρ i 1,2 b,i = ρ i 2,3 ρi 1,3 b,i in the replica symmetric Gibbs state imply 2 M [E ρi 1,2 2 b,i (E ρi 1,2 b,i) 2 ] = M [E ρi 2 1,2 b,i E ρ i 1,2 2 b,i ] 14

15 Since the right hand side vanishes in the above for any b because of Lemma 3.6, the left hand side vanishes for almost all b 0. This fact and Lemma 3.11 imply that the left hand side vanishes also for b = 0.Then, (31) yields [E(Ri 1,2,Ri 1,2 ) D (E R1,2 i )2 ] = [E(R1,2 i,ri 1,2 ) D E R1,2 i 2 +E R1,2 i 2 (E R1,2 i )2 ] = M [E ρi 1,2 2 0,i (E ρ i 1,2 0,i ) 2 ] = 0. Harris inequality of the Bogolyubov type between the Duhamel product and the Gibbs expectation of the square of arbitrary self-adjoint operator O [17] enables us to obtain Therefore (O,O) D O 2 (O,O) D + β [O,[H,O]], (55) 12 E(Ri 1,2,Ri 1,2 ) D = E R i 2 1,2. This completes the proof of Theorem 2.1. Acknowledgments [E Ri 2 1,2 (E R i 1,2 ) 2 ] = 0. (56) It is pleasure to thank R. M. Woloshyn for reading the manuscript and helpful suggestions. I am grateful to M. Aoyagi for discussions in early stage of this work. I would like to thank the anonymous referees for essential comments. References [1] Aizenman, M., Contucci, P. : On the stability of quenched state in mean-field spin glass models. J. Stat. Phys. 92, (1997) [2] Aizenman, M., Greenblatt,R.L., Lebowitz, J. L. :Proof of rounding by quenched disorder of first order transitions in low-dimensional quantum systems J. Math. Phys , (2012) [3] Chatterjee, S. : Absence of replica symmetry breaking in the random field Ising model. Commun. Math.Phys. 337, (2015) [4] Chatterjee,S.: The Ghirlanda-Guerra identities without averaging. preprint, arxiv: (2009). [5] Chatterjee, S. : Disorder chaos and multiple valleys in spin glasses. preprint, arxiv: (2009). [6] Contucci, P., Giardinà, C. : The Ghirlanda-Guerra identities. J. Stat. Phys. 126, ,(2007) 15

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