Free energy density for mean field perturbation of states of a one-dimensional spin chain
aa r X i v : . [ m a t h - ph ] J a n Free energy density formean field perturbation of statesof a one-dimensional spin chain
Dedicated to Professor Walter Thirring on his 80th birthday
Fumio Hiai , Mil´an Mosonyi , Hiromichi Ohno and D´enes Petz , Graduate School of Information Sciences, Tohoku UniversityAoba-ku, Sendai 980-8579, Japan Graduate School of Mathematics, Kyushu University,1-10-6 Hakozaki, Fukuoka 812-8581, Japan Alfr´ed R´enyi Institute of Mathematics,H-1364 Budapest, POB 127, Hungary
Abstract
Motivated by recent developments on large deviations in states of the spin chain, wereconsider the work of Petz, Raggio and Verbeure in 1989 on the variational expressionof free energy density in the presence of a mean field type perturbation. We extend theirresults from the product state case to the Gibbs state case in the setting of translation-invariant interactions of finite range. In the special case of a locally faithful quantumMarkov state, we clarify the relation between two different kinds of free energy densities(or pressure functions).
AMS subject classification : 82B10, 82B20
Key words and phrases : free energy density, mean relative entropy, interactions, Gibbsstates, KMS states, finitely correlated states, quantum Markov states, Legendre transform E-mail: [email protected]; Partially supported by Grant-in-Aid for Scientific Research (B)17340043. E-mail: [email protected]; Partially supported by Grant-in-Aid for JSPS Fellows 18 · E-mail: [email protected]; Partially supported by Grant-in-Aid for JSPS Fellows 19 · E-mail: [email protected]; Partially supported by the Hungarian Research Grant OTKA T068258. Introduction
The theoretical description of the statistical mechanics of quantum spin chains was the firstsuccess of the operator algebraic approach to quantum physics. A one-dimensional spin chainis described by a quasi-local C*-algebra A := N k ∈ Z A k which is the infinite tensor product offull matrix algebras A k = M d ( C ) and the limit of the local algebras A Λ := N k ∈ Λ A k , whereΛ ⊂ Z is finite. A state ϕ of the spin chain is uniquely specified by its local restrictions ϕ Λ := ϕ | A Λ . A local state ω of A Λ can equivalently be given by its density matrix D ( ω )satisfying ω ( A ) = Tr D ( ω ) A , A ∈ A Λ .A translation-invariant interaction Φ of the spins determines a local Hamiltonian H Λ (Φ) := X X ⊂ Λ Φ( X ) (1.1)with corresponding local Gibbs state D ( ϕ G Λ ) := e − H Λ (Φ) Tr e − H Λ (Φ) (1.2)for all finite Λ ⊂ Z . The local Gibbs state is the unique maximizer of the functional ω ω ( H Λ (Φ)) + S ( ω ), where ω is an arbitrary state of A Λ and S ( ω ) is the von Neumann entropy S ( ω ) := − Tr D ( ω ) log D ( ω ). Furthermore,log Tr Λ e − H Λ (Φ) = max {− ω ( H Λ (Φ)) + S ( ω ) : ω state of A Λ } . (1.3)One of the main problems in the statistical mechanics of the spin chain is the determination ofthe global equilibrium states of A for a given interaction. When Φ is of relatively short range,it is well known [11, 22] that the variational formula (1.3) holds in the asymptotic limit: P (Φ) = max {− ω ( A Φ ) + s ( ω ) : ω translation-invariant state of A} , (1.4)where P (Φ) := lim Λ → Z | Λ | log Tr e − H Λ (Φ) , (1.5) s ( ω ) := lim Λ → Z | Λ | S ( ω | A Λ ) , (1.6) A Φ := X X ∋ Φ( X ) | X | (1.7)are the pressure (or free energy density ) of Φ, the mean entropy of ω and the mean energy ofΦ, respectively. (Here note that the term “free energy” should be used with minus sign in theexact sense of physics.) Maximizers of the right-hand side of (1.4) are the equilibrium statesfor the interaction Φ. If Φ is of finite range, then the equilibrium state is unique.One of the main subjects of the present paper is an extension of the free energy density(1.5) when the interaction is perturbed by a mean field term. Let γ be the right-translationautomorphism of A and set s n ( A ) := 1 n X Λ+ k ⊂ [1 ,n ] γ k ( A ) ∈ A [1 ,n ] A ∈ A saΛ with a finite Λ ⊂ Z . We will study the limitlim n →∞ n log Tr exp (cid:0) − H [1 ,n ] (Φ) − nf ( s n ( A )) (cid:1) , (1.8)where f is a real continuous function. This kind of problem was initiated by Petz, Raggio andVerbeure [33] in the particular case when there is no interaction between the spins. The moti-vation came from mean field models and the extension of large deviation theory for quantumchains [32]. An important tool was Størmer’s quantum version of the de Finetti theorem forsymmetric states. The subject was treated in details in the monograph [31] under the name“perturbational limits” by using the concept of approximately symmetric sequences [36]. Sincethe interaction Φ in the general situation is not invariant under the permutation of the spins,our method in the general case is the extremal decomposition theory for translation-invariantstates that is standard in quantum statistical mechanics, see [10]. In the present paper we willshow that the limit is expressed by a variational formula generalizing (1.4).The limit (1.8) has a direct physical meaning in the case when f ( x ) = x and A = A ∈ A .Then − H [1 ,n ] (Φ) − n n X i,j =1 A i A j is a mean field perturbation of the interaction Φ, where A j := γ j ( A ). The limit is the freeenergy density for the mean field model and the variational formula has an important physicalinterpretation.The limit density (1.8) can be considered in a different way as well. Given a translation-invariant state ϕ , we can study the limit p ϕ ( A, f ) := lim n →∞ n log Tr exp (cid:0) log D ( ϕ | A [1 ,n ] ) − nf ( s n ( A )) (cid:1) (1.9)and its variational expression under the duality between the observable space A sa and thetranslation-invariant state space S γ ( A ). In particular, when f ( x ) = x , the limit (1.9) becomesa simply perturbed free energy density function (or pressure function) p ϕ ( A ) := lim n →∞ n log Tr exp (cid:0) log D ( ϕ | A [1 ,n ] ) − ns n ( A ) (cid:1) for local observables A in A sa (if the limit exists). The dual function of the function p ϕ ( A ) isthe mean relative entropy S M ( ω, ϕ ) := lim n →∞ n S ( ω | A [1 ,n ] , ϕ | A [1 ,n ] ) (1.10)with respect to ϕ defined for ω ∈ S γ ( A ). The existence of the mean relative entropy and itsproperties were worked out in [18, 20, 21].When Φ is a translation-invariant interaction of finite range and ϕ is the equilibrium statefor Φ, the limits (1.8) and (1.9) are the same (up to an additive term P (Φ)), but (1.9) can alsobe studied for a wider class of translation-invariant states, for example, for finitely correlatedstates which were introduced by Fannes, Nachtergaele and Werner [14]. A slightly differentconcept of quantum Markov states was formerly introduced by Accardi and Frigerio [3]. A3ranslation-invariant and locally faithful quantum Markov state in the sense of Accardi andFrigerio is known to be a finitely correlated state as well as the equilibrium state for a nearest-neighbor interaction [4, 30]. Remarkably, a Markovian structure similar to the special quantumMarkov state just mentioned appears in the recent characterization [15, 28] of the quantumstates which saturate the strong subadditivity of von Neumann entropy.A similar but different version of the free energy density function p ϕ ( A ) is˜ p ϕ ( A ) := lim n →∞ n log ϕ (cid:0) e ns n ( A ) (cid:1) = lim n →∞ n log ϕ exp n X k =1 γ k ( A ) !! , which gives the logarithmic moment generating function for a sequence of compactly supportedprobability measures on the real line. Large deviations governed by this generating functionhave recently been studied in [29, 26, 17] for example. In fact, our first motivation of the presentpaper came from large deviation results in [29, 26] with respect to Gibbs-KMS states. It isnot known in general for p ϕ to have the interpretation as the logarithmic moment generatingfunction as ˜ p ϕ does. Indeed, this question is nothing more than the so-called BMV-conjecture[9]. On the other hand, since ˜ p ϕ is not a convex function in general, it is impossible for ˜ p ϕ toenjoy such a variational expression as p ϕ does.The paper is organized as follows. Section 2 is a preliminary on translation-invariant inter-actions and Gibbs-KMS equilibrium states of the one-dimensional spin chain. In Section 3 theexistence of the functional free energy density (1.9) and its variational expression are obtainedwhen ϕ is the Gibbs state for a translation-invariant interaction of finite range. In Section 4the existence of the density p ϕ ( A ) is proven for a general finitely correlated state ϕ , and theexact relation between the functionals p ϕ and ˜ p ϕ introduced above is clarified in the specialcase when ϕ is a locally faithful quantum Markov state. Section 5 is a brief guide to how ourresults for a Gibbs state ϕ can be extended to the case of arbitrary dimension. A one-dimensional spin chain is described by the infinite tensor product C ∗ -algebra A := N k ∈ Z A k of full matrix algebras A k := M d ( C ) over Z . The right-translation automorphism of A is denoted by γ . We denote by S γ ( A ) the set of all γ -invariant states of A . The C ∗ -subalgebraof A corresponding to a subset X of Z is A X := N k ∈ X A k with convention A ∅ := C , where is the identity of A . If X ⊂ Y ⊂ Z , then A X ⊂ A Y by a natural inclusion. The local algebrais the dense ∗ -subalgebra A loc := S ∞ n =1 A [ − n,n ] of A . The self-adjoint parts of A loc and A aredenoted by A saloc and A sa , respectively. The usual trace on A X for each finite X ⊂ Z is denotedby Tr without referring to X since it causes no confusion.An interaction Φ in A is a mapping from the nonempty finite subsets of Z into A such thatΦ( X ) = Φ( X ) ∗ ∈ A X for each finite X ⊂ Z . Given an interaction Φ and a finite subset Λ ⊂ Z ,we have the local Hamiltonian H Λ (Φ) given in (1.1) and the surface energy W Λ (Φ) W Λ (Φ) := X { Φ( X ) : X ∩ Λ = ∅ , X ∩ Λ c = ∅} whenever the sum converges in norm. We always assume that Φ is γ -invariant, i.e., γ (Φ( X )) =Φ( X + 1) for every finite X ⊂ Z , where X + 1 := { k + 1 : k ∈ X } . We denote by B ( A ) the set4f all γ -invariant interactions Φ in A such that k Φ k := X X ∋ k Φ( X ) k + sup n ≥ k W [1 ,n ] (Φ) k < + ∞ . It is easy to see that B ( A ) is a real Banach space with the usual linear operations and the norm k Φ k . Associated with Φ ∈ B ( A ) we have a strongly continuous one-parameter automorphismgroup α Φ of A given by α Φ t ( A ) = lim m →−∞ ,n →∞ e itH [ m,n ] (Φ) Ae − itH [ m,n ] (Φ) ( A ∈ A ) . Then it is known [6, 24] that there exists a unique α Φ -KMS state (at β = − ϕ of A , whichis automatically faithful and ergodic (i.e., an extremal point of S γ ( A )). The KMS state ϕ ischaracterized by the Gibbs condition and so it is also called the (global) Gibbs state for Φ. Thestate ϕ is also characterized by the variational principle s ( ϕ ) = ϕ ( A Φ ) + P (Φ), the equalitycase of the expression (1.4), where P (Φ), s ( ϕ ) and A Φ are given in (1.5)–(1.7). See [11, 22] fordetails on these equivalent characterizations of equilibrium states.In the rest of this section, assume that Φ is a γ -invariant interaction of finite range, i.e.,there is an N ∈ N such that Φ( X ) = 0 whenever the diameter of X is greater than N . Ofcourse, Φ ∈ B ( A ). Let ϕ be the α Φ -KMS state (at β = −
1) of A . The next lemma will playan essential role in our discussions below; the proof can be found in [5, 7, 8]. Lemma 2.1.
There is a constant λ ≥ independent of n ) such that λ − ϕ n ≤ ϕ Gn ≤ λϕ n for all n ∈ N , where ϕ Gn is the local Gibbs state (1.2) with Λ = [1 , n ] . For ω ∈ S γ ( A ) and Ψ ∈ B ( A ) we write for short ω n and H n (Ψ) for ω | A [1 ,n ] and H [1 ,n ] (Ψ),respectively. Lemma 2.1 gives (cid:12)(cid:12)(cid:12)(cid:12) n log Tr exp (cid:0) log D ( ϕ n ) − H n (Ψ) (cid:1) − n log Tr exp (cid:0) log D ( ϕ Gn ) − H n (Ψ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ log λn . Since Tr exp (cid:0) log D ( ϕ Gn ) − H n (Ψ) (cid:1) = Tr e − H n (Φ+Ψ) Tr e − H n (Φ) , we have Lemma 2.2.
For every Ψ ∈ B ( A ) the limit P ϕ (Ψ) := lim n →∞ n log Tr exp (cid:0) log D ( ϕ n ) − H n (Ψ) (cid:1) exists and P ϕ (Ψ) = P (Φ + Ψ) − P (Φ) . For every ω ∈ S γ ( A ) the mean relative entropy (1.10) exists and S M ( ω, ϕ ) = lim n →∞ n S ( ω n , ϕ n ) = lim n →∞ n S ( ω n , ϕ Gn ) , (2.1)5ee [20, p. 710]. In fact, since S ( ω n , ϕ Gn ) = − S ( ω n ) + ω ( H n (Φ)) + log Tr e − H n (Φ) and lim n →∞ ω ( H n (Φ)) n = ω ( A Φ ) , we have Lemma 2.3.
For every ω ∈ S γ ( A ) , S M ( ω, ϕ ) = − s ( ω ) + ω ( A Φ ) + P (Φ) . Hence, the function ω S M ( ω, ϕ ) is affine and lower semicontinuous in the weak* topologyon S γ ( A ) . Theorem 2.4. (a)
For every Ψ ∈ B ( A ) , P ϕ (Ψ) = max {− ω ( A Ψ ) − S M ( ω, ϕ ) : ω ∈ S γ ( A ) } . (b) For every ω ∈ S γ ( A ) , S M ( ω, ϕ ) = sup {− ω ( A Ψ ) − P ϕ (Ψ) : Ψ ∈ B ( A ) } . (c) The function P ϕ on B ( A ) is Gˆateaux-differentiable at any Ψ ∈ B ( A ) , i.e., the limit ∂ ( P ϕ ) Ψ (Ψ ′ ) := lim t → P ϕ (Ψ + t Ψ ′ ) − P ϕ (Ψ) t exists for every Ψ ′ ∈ B ( A ) . Moreover, when ϕ Ψ is the unique α Φ+Ψ -KMS state, ∂ ( P ϕ ) Ψ (Ψ ′ ) = − ϕ Ψ ( A Ψ ′ ) . Proof.
The variational expressions in (a) and (b) are just rewriting of (1.4) and s ( ω ) = inf { ω ( A Ψ ) + P (Ψ) : Ψ ∈ B ( A ) } thanks to Lemmas 2.2 and 2.3 (see [22, § II.3] for the above expression of s ( ω ) complementaryto (1.4)). Note also that the maximum in (a) is attained by the unique Gibbs state for Φ + Ψ.The differentiability of P ϕ in (c) was essentially shown in [26, Corollary 3.5]; we give theproof for completeness. Let B ( A ) ∗ be the dual Banach space of B ( A ). For each ω ∈ S γ ( A )define f ω ∈ B ( A ) ∗ by f ω (Ψ) := − ω ( A Ψ ). Then ω f ω is an injective and continuous (inthe weak* topologies) affine map [22, Lemma II.1.1]; hence Γ := { f ω : ω ∈ S γ ( A ) } is a weak*compact convex subset of B ( A ) ∗ and F ( f ) := ( S M ( ω, ϕ ) if f = f ω with ω ∈ S γ ( A ) , + ∞ otherwise6s a well-defined function on B ( A ) ∗ which is convex and weakly* lower semicontinuous. Theassertion (a) means that P ϕ is the conjugate function of F , which in turn implies that theconjugate function of P ϕ on B ( A ) is F . By the general theory of conjugate functions (see[13, Proposition I.5.3] for example), P ϕ is Gˆateaux-differentiable at Ψ ∈ B ( A ) if and only ifthere is a unique f ∈ B ( A ) ∗ such that ( P ϕ ) ∗ ( f ) = f (Ψ) − P ϕ (Ψ), that is, there is a unique ϕ Ψ ∈ S γ ( A ) such that S M ( ϕ Ψ , ϕ ) = − ϕ Ψ ( A Ψ ) − P ϕ (Ψ) . (2.2)By Lemmas 2.2 and 2.3 the above equality is equivalent to the variational principle s ( ϕ Ψ ) = ϕ Ψ ( A Φ+Ψ ) + P (Φ + Ψ) , which is equivalent to ϕ Ψ being the α Φ+Ψ -KMS state. Hence the differentiability assertion of P ϕ follows. Moreover, by (a) we get P ϕ (Ψ + t Ψ ′ ) ≥ − ϕ Ψ ( A Ψ+ t Ψ ′ ) − S M ( ω, ϕ )for any Ψ ′ ∈ B ( A ) and t ∈ R . This together with equality (2.2) for t = 0 gives the formula ∂ ( P ϕ ) Ψ (Ψ ′ ) = − ϕ Ψ ( A Ψ ′ ). Corollary 2.5. (1)
For every A ∈ A saloc so that A ∈ A saΛ with a finite Λ ⊂ Z , the free energy density p ϕ ( A ) := lim n →∞ n log Tr exp log D ( ϕ n ) − X Λ+ k ⊂ [1 ,n ] γ k ( A ) ! (2.3) exists ( independently of the choice of Λ) . (2) The function p ϕ on A saloc is Gˆateaux-differentiable at any A ∈ A saloc in the sense that thelimit lim t → p ϕ ( A + tB ) − p ϕ ( A ) t exists for every B ∈ A saloc . In particular, the function t ∈ R p ϕ ( tA ) is differentiable forevery A ∈ A saloc . (3) The above function p ϕ on A saloc uniquely extends to a function ( denoted by the same p ϕ ) on A sa which is convex and Lipschitz continuous with | p ϕ ( A ) − p ϕ ( B ) | ≤ k A − B k , A, B ∈ A sa . (4) For every A ∈ A sa , p ϕ ( A ) = max {− ω ( A ) − S M ( ω, ϕ ) : ω ∈ S γ ( A ) } . (5) For every ω ∈ S γ ( A ) , S M ( ω, ϕ ) = sup {− ω ( A ) − p ϕ ( A ) : A ∈ A saloc } = sup {− ω ( A ) − p ϕ ( A ) : A ∈ A sa } . roof. To show (1), we may assume A ∈ A sa[1 ,ℓ ( A )] with some ℓ ( A ) ∈ N , and set a γ -invariantinteraction Ψ A of finite range (hence Ψ A ∈ B ( A )) byΨ A ( X ) := ( γ k ( A ) if X = [ k + 1 , k + ℓ ( A )], k ∈ Z , . Since P n − ℓ ( A ) k =0 γ k ( A ) = H n (Ψ A ), the limit (2.3) exists by Lemma 2.2 and its independenceof the choice of Λ is obvious. The differentiability in (2) immediately follows from Theorem2.4 (c). (In fact, the derivative of p ϕ at A is ∂ ( p ϕ ) A ( B ) = − ϕ A ( B ) for every B ∈ A saloc , where ϕ A is the unique α Φ+Ψ A -KMS state.) Moreover, since A Ψ A = 1 ℓ ( A ) ℓ ( A ) X k =1 γ − k ( A )so that ω ( A Ψ A ) = ω ( A ) for all ω ∈ S γ ( A ), Theorem 2.4 (a) implies the variational expressionin (4) for any A ∈ A saloc . The Lipschitz inequality in (3) for every A, B ∈ A saloc is immediatelyseen from the formula (2.3). Hence p ϕ uniquely extends to a Lipschitz continuous function on A sa , and the convexity of p ϕ on A sa is obvious. To prove (4) for general A ∈ A sa let { A n } be a sequence in A saloc such that k A n − A k →
0. It is clear by convergence that p ϕ ( A ) ≥− ω ( A ) − S M ( ω, ϕ ) for all ω ∈ S γ ( A ). Let ω n be the maximizer of the right-hand side of (4) for A n ; here it may be assumed that { ω n } converges to ω ∈ S γ ( A ) in the weak* topology. Thenwe get p ϕ ( A ) = lim n →∞ {− ω n ( A n ) − S M ( ω n , ϕ ) } ≤ ω ( A ) − S M ( ω, ϕ )by Lemma 2.3 (the weak* lower semicontinuity), which proves (4). Finally, (5) follows fromLemma 2.3 and the duality theorem for conjugate functions [13, Proposition I.4.1].For each A ∈ A sa we have the convex and continuous function t p ϕ ( tA ) on R by Corollary2.5 (3). We now introduce the function I A ( x ) := inf { S M ( ω, ϕ ) : ω ∈ S γ ( A ) , ω ( A ) = x } ( x ∈ R ) . (2.4)Obviously, I A ( x ) = + ∞ for x [ λ min ( A ) , λ max ( A )], where λ min ( A ) and λ max ( A ) are theminimum and the maximum of the spectrum of A . The next proposition says that p ϕ ( tA ) and I A ( x ) are the Legendre transforms of each other, which are the contractions of the expressionsin the above (5) and (4) into the real line via ω ω ( A ). Proposition 2.6.
For every A ∈ A sa , I A ( x ) = sup {− tx − p ϕ ( tA ) : t ∈ R } , x ∈ R ,p ϕ ( tA ) = max {− tx − I A ( x ) : x ∈ [ λ min ( A ) , λ max ( A )] } , t ∈ R . Proof.
We have I A ( x ) = min ω ∈S γ ( A ) sup t ∈ R { t ( − x + ω ( A )) + S M ( ω, ϕ ) } = sup t ∈ R min ω ∈S γ ( A ) { t ( − x + ω ( A )) + S M ( ω, ϕ ) } = sup t ∈ R {− tx − p ϕ ( tA ) }
8y Corollary 2.5 (4). In the above, the second equality follows from Sion’s minimax theorem[35] thanks to Lemma 2.3. (The elementary proof in [25] for real-valued functions can alsowork for functions with values in ( −∞ , + ∞ ].) The second formula follows from the first byduality. Remark 2.7.
An alternative notion of free energy density˜ p ϕ ( A ) := lim n →∞ n log ϕ exp n − X k =0 γ k ( A ) !! (2.5)was recently studied in [29, 26, 17] in relation with large deviation problems on the spin chain.The function t ∈ R ˜ p ϕ ( tA ) is the so-called logarithmic moment generating function [12] ofa sequence of probability measures and existence of the limit guarantees large deviation upperbound to hold, while if the limit is even differentiable that provides full large deviation principle.The existence of the limit was proven for any A ∈ A saloc when ϕ is the unique Gibbs state of atranslation-invariant interaction of finite range [26] and when ϕ is a finitely correlated state [17].Differentiability was shown in [29] and [17] for certain special cases. The Golden-Thompsoninequality shows that p ϕ ( A ) ≤ ˜ p ϕ ( A ) (2.6)holds for any A ∈ A saloc . For instance, for a product state ϕ = N Z ρ with D ( ρ ) = e − H and a one-site observable A , since ˜ p ϕ ( A ) = log Tr ( e − H e − A ) while p ϕ ( A ) = log Tr ( e − H − A ), the equality p ϕ ( A ) = ˜ p ϕ ( A ) occurs only when A commutes with H (see [16]). Although the Lipschitzcontinuity of ˜ p ϕ on A saloc and its variational expression as in the above (4) are impossible, itmight be possible to get the variational expression as in (5) with ˜ p ϕ in place of p ϕ . This isequivalent to saying that p ϕ on A sa is the lower semicontinuous convex envelope of ˜ p ϕ on A saloc ,as will be shown in a special case in Section 4 (see Corollary 4.10). Remark 2.8.
An equivalent formulation of the celebrated conjecture due to Bessis, Moussaand Villani [9] (the so-called BMV-conjecture) is stated as follows [27]: If H and H are N × N Hermitian matrices with H ≥
0, then there exists a positive measure µ on [0 , ∞ ) such thatTr e H − tH = Z ∞ e − ts dµ ( s ) , t > e H − tH on t > H := log D ( ϕ n ) , H := 1 n X Λ+ k ⊂ [1 ,n ] γ k ( A ) , where A ∈ A saΛ with a finite Λ ⊂ Z , we would have a probability measure µ n supported in[ λ min ( A ) , λ max ( A )] such thatTr exp log D ( ϕ n ) − X Λ+ k ⊂ [1 ,n ] γ k ( tA ) ! = Z ∞−∞ e − nts dµ n ( s ) , t ∈ R . (The restriction on the support of µ n easily follows from the Paley-Wiener theorem.) In thissituation, the free energy density p ϕ ( tA ) is the logarithmic moment generating function of thesequence of measures ( µ n ), and Corollary 2.5 and Proposition 2.6 combined with the G¨artner-Ellis theorem [12, Theorem 2.3.6] yield that ( µ n ) satisfies the large deviation principle with thegood rate function I A ( x ) given in (2.4). 9 Perturbation of Gibbs states
When the reference state ϕ is a product state and A is a one-site observable, the variationalexpression of functional free energy densitylim n →∞ n log Tr exp (cid:0) log D ( ϕ n ) − nf ( s n ( A )) (cid:1) = sup ω n − lim n →∞ ω ( f ( s n ( A ))) − S M ( ω, ϕ ) o was obtained in [33], where ω runs over the symmetric (or permutation-invariant) states. Acomprehensive exposition on the subject is also found in [31, § ϕ is still a product state. In this section we consider thecase when the reference state ϕ is the Gibbs state for a translation-invariant interaction Φ offinite range.Let A ∈ A saloc . We may assume without loss of generality that A ∈ A sa[1 ,ℓ ( A )] with some ℓ ( A ) ∈ N , and set s n ( A ) := 1 n n − ℓ ( A ) X k =0 γ k ( A ) ∈ A [1 ,n ] . Given A and a continuous function f : [ λ min ( A ) , λ max ( A )] → R the functional free energydensity is defined as the limit lim n →∞ n log Z ϕ ( n, A, f )for Z ϕ ( n, A, f ) := Tr exp (cid:0) log D ( ϕ n ) − nf ( s n ( A )) (cid:1) as n → ∞ . We will show the existence of the limit in Theorem 3.4.The extreme boundary ex S γ ( A ) of the set S γ ( A ) consists of the ergodic states. It is knownthat ex S γ ( A ) is a G δ -subset of S γ ( A ) (see [34, Proposition 1.3]). Since ( A , γ ) is asymptoticallyAbelian in the norm sense, S γ ( A ) is a so-called Choquet simplex (see [10, Corollary 4.3.11]) sothat each ω ∈ S γ ( A ) has a unique extremal decomposition ω = Z ex S γ ( A ) ψ dν ω ( ψ )with a probability Borel measure ν ω on ex S γ ( A ) (see [34, p. 66], [10, Theorem 4.1.15]). Lemma 3.1.
For every continuous f : [ λ min ( A ) , λ max ( A )] → R and for every ω ∈ S γ ( A ) thelimit E A,f ( ω ) := lim n →∞ ω ( f ( s n ( A ))) exists and E A,f ( ω ) = Z ex S γ ( A ) f ( ψ ( A )) dν ω ( ψ ) for the extremal decomposition ω = R ex S γ ( A ) ψ dν ω ( ψ ) . roof. The first assertion is contained in [31, Proposition 13.2]. However, we use a differentmethod to prove the two statements together.First let ψ ∈ ex S γ ( A ) and ( π ψ , H ψ , U ψ , Ω ψ ) be the GNS construction associated with ψ ,i.e., π ψ is a representation of A on H ψ with a cyclic vector Ω ψ and U ψ is a unitary on H ψ such that ψ ( A ) = h π ψ ( A )Ω ψ , Ω ψ i and π ψ ( γ ( A )) = U ψ π ψ ( A ) U ∗ ψ for all A ∈ A . Thanks to theasymptotic Abelianness, the extremality of ψ means (see [10, Theorem 4.3.17]) that the setof U ψ -invariant vectors in H ψ is the one-dimensional subspace C Ω ψ . Hence the mean ergodictheorem implies that π ψ ( s n ( A ))Ω ψ = 1 n n − ℓ ( A ) X k =0 U kψ π ψ ( A )Ω ψ converges in norm to ψ ( A )Ω ψ as n → ∞ . The case f ( x ) = x m easily follows from this, and byapproximating f by polynomials, we getlim n →∞ k π ψ ( f ( s n ( A )))Ω ψ − f ( ψ ( A ))Ω ψ k = 0so that lim n →∞ ψ ( f ( s n ( A ))) = f ( ψ ( A )) . Finally, for a general ω ∈ S γ ( A ) with the extremal decomposition ω = R ex S γ ( A ) ψ dν ω ( ψ ), theLebesgue convergence theorem giveslim n →∞ ω ( f ( s n ( A ))) = lim n →∞ Z ex S γ ( A ) ψ ( f ( s n ( A ))) dν ω ( ψ ) = Z ex S γ ( A ) f ( ψ ( A )) dν ω ( ψ ) , as required.In the following proofs we will often use a state perturbation technique. For the convenienceof the reader, we here summarize some basic properties of state perturbation restricted to thesimple case of matrix algebras. See [11, 31] for the general theory of the subject matter. Let ρ be a faithful state of B := M N ( C ) with density matrix e − H . For each h ∈ B sa define theperturbed functional ρ h by ρ h ( A ) := Tr e − H − h A ( A ∈ B )and the normalized version[ ρ h ]( A ) := ρ h ( A ) ρ h ( ) = Tr e − H − h A Tr e − H − h ( A ∈ B ) . The state [ ρ h ] is characterized as the unique minimizer of the functional ω S ( ω, ρ ) + ω ( h )on the states of B . It is plain to see the chain rule: [[ ρ h ] k ] = [ ρ h + k ] for all h, k ∈ B sa . For eachstate ω of B , from the equality S ( ω, [ ρ h ]) = S ( ω, ρ ) + ω ( h ) + log ρ h ( )and the Golden-Thompson inequality ρ h ( ) ≤ ρ ( e − h ), the following are readily seen:log ρ h ( ) ≥ − ω ( h ) − S ( ω, ρ ) , (3.1) | S ( ω, ρ ) − S ( ω, [ ρ h ]) | ≤ k h k . (3.2)11 emma 3.2. For every continuous f : [ λ min ( A ) , λ max ( A )] → R and for every ω ∈ S γ ( A ) , lim inf n →∞ n log Z ϕ ( n, A, f ) ≥ sup {− E A,f ( ω ) − S M ( ω, ϕ ) : ω ∈ S γ ( A ) } holds.Proof. For n ∈ N write h n := nf ( s n ( A )) for simplicity. The perturbed functional ϕ h n n of ϕ n on A [1 ,n ] has the density exp(log D ( ϕ n ) − h n ) and so Z ϕ ( n, A, f ) = ϕ h n n ( ). Hence it follows from(3.1) that log Z ϕ ( n, A, f ) ≥ − ω n ( h n ) − S ( ω n , ϕ n ) , ω ∈ S γ ( A ) . By Lemma 3.1 and (2.1) we havelim inf n →∞ n log Z ϕ ( n, A, f ) ≥ − E A,f ( ω ) − S M ( ω, ϕ )for all ω ∈ S γ ( A ). Lemma 3.3.
For every continuous f : [ λ min ( A ) , λ max ( A )] → R , sup {− E A,f ( ω ) − S M ( ω, ϕ ) : ω ∈ S γ ( A ) } = sup {− f ( ψ ( A )) − S M ( ψ, ϕ ) : ψ ∈ ex S γ ( A ) } = max {− f ( ω ( A )) − S M ( ω, ϕ ) : ω ∈ S γ ( A ) } = max {− f ( x ) − I A ( x ) : x ∈ [ λ min ( A ) , λ max ( A )] } . Proof.
For every ω ∈ S γ ( A ) let ω = R ex S γ ( A ) ψ dν ω ( ψ ) be the extremal decomposition of ω . ByLemma 2.3 it follows from [34, Lemma 9.7] that S M ( ω, ϕ ) = Z ex S γ ( A ) S M ( ψ, ϕ ) dν ω ( ψ ) . This together with Lemma 3.1 shows that − E A,f ( ω ) − S M ( ω, ϕ ) = Z ex S γ ( A ) ( − f ( ψ ( A )) − S M ( ψ, ϕ )) dν ω ( ψ ) ≤ sup {− f ( ψ ( A )) − S M ( ψ, ϕ ) : ψ ∈ ex S γ ( A ) } . Therefore, sup {− E A,f ( ω ) − S M ( ω, ϕ ) : ω ∈ S γ ( A ) }≤ sup {− f ( ψ ( A )) − S M ( ψ, ϕ ) : ψ ∈ ex S γ ( A ) } , and the converse inequality is obvious. Hence the first equality follows. The last equalityimmediately follows from the definition (2.4).To prove the second equality, let ˜ ω be a maximizer of ω
7→ − f ( ω ( A )) − S M ( ω, ϕ ) on S γ ( A ).For each m ∈ N with m > ℓ ( A ) we introduce a product state ψ := O Z ˜ ω m
12f the re-localized spin chain N i ∈ Z A [ im +1 , ( i +1) m ] and define ¯ ψ ∈ S γ ( A ) to be the average¯ ψ := 1 m m − X k =0 ψ ◦ γ k . First we prove that ¯ ψ is γ -ergodic. For every B , B ∈ A loc choose an i ∈ N such that B , B ∈ A [ − i m, ( i − m ] . Let n ∈ N be given so that n = jm + r with j ∈ N , j > i and0 ≤ r < m . When i ≥ i , 1 ≤ t ≤ m and 0 ≤ k ≤ m −
1, we have ψ ( γ k ( B ) γ k + im + t ( B )) = ψ ( γ k ( B )) ψ ( γ k + im + t ( B )) = ψ ( γ k ( B )) ψ ( γ k + t ( B )) , because γ k ( B ) ∈ A ( −∞ ,i m ] and γ k + im + t ( B ) ∈ A [( i − i ) m +1 , ∞ ) with i ≤ i − i . Hence forevery i ≥ i we get m X t =1 ¯ ψ ( B γ im + t ( B )) = 1 m m X t =1 m − X k =0 ψ ( γ k ( B )) ψ ( γ k + im + t ( B ))= m − X k =0 ψ ( γ k ( B )) m m X t =1 ψ ( γ k + t ( B )) ! = m − X k =0 ψ ( γ k ( B )) ¯ ψ ( B ) = m ¯ ψ ( B ) ¯ ψ ( B ) . Therefore,1 n n X t =1 ¯ ψ ( B γ t ( B )) = 1 n i m X t =1 + jm + r X t = jm +1 ! ¯ ψ ( B γ t ( B )) + 1 n j − X i =2 i m X t =1 ¯ ψ ( B γ im + t ( B ))= 1 n i m X t =1 + jm + r X t = jm +1 ! ¯ ψ ( B γ t ( B )) + ( j − i ) mn ¯ ψ ( B ) ¯ ψ ( B ) , which obviously implies thatlim n →∞ n n X t =1 ¯ ψ ( B γ t ( B )) = ¯ ψ ( B ) ¯ ψ ( B ) . By [10, Theorems 4.3.17 and 4.3.22] this is equivalent to ¯ ψ ∈ ex S γ ( A ). Furthermore, since¯ ψ ( A ) = m − ℓ ( A ) + 1 m ˜ ω ( A ) + 1 m m − X k = m − ℓ ( A )+1 ψ ( γ k ( A )) , we get | ¯ ψ ( A ) − ˜ ω ( A ) | ≤ ℓ ( A ) k A k m . (3.3)Now for m greater than both the range of Φ and ℓ ( A ), we set a product state φ ( m ) := O Z ϕ Gm (3.4)13f the re-localized N i ∈ Z A [ im +1 , ( i +1) m ] , where ϕ Gm is the local Gibbs state of A [1 ,m ] for Φ. Wealso set W := X { Φ( X ) : X ∩ ( −∞ , = ∅ , X ∩ [1 , ∞ ) = ∅} , (3.5) K := X {k Φ( X ) k : X ∩ ( −∞ , = ∅ , X ∩ [1 , ∞ ) = ∅} ( ≥ k W k ) . (3.6)For each j ∈ N , since H jm (Φ) = j − X i =0 γ im ( H m (Φ)) + j − X i =1 γ im ( W ) , it is clear that φ ( m ) | A [1 ,jm ] = N j ϕ Gm is the perturbed state of ϕ Gjm as follows: N j ϕ Gm = (cid:2) ( ϕ Gjm ) − W ( m ) ] , (3.7)where W ( m ) := P j − i =1 γ im ( W ). Hence by Lemma 2.1 and (3.2) we get S ( ψ jm , ϕ jm ) ≤ S ( ψ jm , ϕ Gjm ) + log λ ≤ S ( N j ˜ ω m , N j ϕ Gm ) + 2( j − K + log λ (3.8)= jS (˜ ω m , ϕ Gm ) + 2( j − K + log λ ≤ jS (˜ ω m , ϕ m ) + 2( j − K + ( j + 1) log λ . Since ϕ can be considered as the Gibbs state for an interaction of finite range in the re-localized N i ∈ Z A [ im +1 , ( i +1) m ] , Lemma 2.3 (the affine property) implies that S M ( ¯ ψ, ϕ ) = 1 m lim j →∞ j S ( ¯ ψ | A [1 ,jm ] , ϕ | A [1 ,jm ] )= 1 m m − X k =0 lim j →∞ j S ( ψ ◦ γ k | A [1 ,jm ] , ϕ | A [1 ,jm ] )= 1 m lim j →∞ j S ( ψ | A [1 ,jm ] , ϕ | A [1 ,jm ] ) (3.9)similarly to [31, (13.29)]. Therefore, S M ( ¯ ψ, ϕ ) ≤ m S (˜ ω m , ϕ m ) + 2 K + log λm . (3.10)From (3.3) and (3.10) together with (2.1), for any ε > − f ( ¯ ψ ( A )) − S M ( ¯ ψ, ϕ ) ≥ − f (˜ ω ( A )) − S M (˜ ω, ϕ ) − ε , whenever m is sufficiently large. With ¯ ψ ∈ ex S γ ( A ) this proves the second equality.The next theorem showing the variational expression of the functional free energy densitywith respect to the state ϕ is a generalization of [33, Theorem 12] as well as [31, Theorem13.11]. In fact, when ϕ is a product state N Z ρ and A is a one-site observable in A , one caneasily see that max {− f ( ω ( A )) − S M ( ω, ϕ ) : ω ∈ S γ ( A ) } = max {− f ( σ ( A )) − S ( σ, ρ ) : σ state of A } I A ( x ) = min { S ( σ, ρ ) : σ state of A , σ ( A ) = x } = sup {− tx − log ρ tA ( I ) : t ∈ R } so that Theorem 3.4, together with Lemma 3.3, exactly becomes [33, Theorem 12]. A typicalcase is the quadratic function f ( x ) = x , which is familiar in quantum models of mean fieldtype as remarked in [33] (also in Introduction).The proof below is a modification of that of [31, Theorem 13.11]. Here it should be notedthat the quantities c ( ϕ, nf ( s n ( A )) in [31, §
13] and Z ϕ ( n, A, f ) here are in the relation c ( ϕ, nf ( s n ( A ))) = − log Z ϕ ( n, A, f )as long as ϕ is a product state. Theorem 3.4.
For every continuous f : [ λ min ( A ) , λ max ( A )] → R the limit p ϕ ( A, f ) := lim n →∞ n log Z ϕ ( n, A, f ) exists and p ϕ ( A, f ) = sup {− E A,f ( ω ) − S M ( ω, ϕ ) : ω ∈ S γ ( A ) } = max {− f ( x ) − I A ( x ) : x ∈ [ λ min ( A ) , λ max ( A )] } . Proof.
By Lemmas 3.2 and 3.3 we only have to show thatlim sup n →∞ n log Z ϕ ( n, A, f ) ≤ sup {− E A,f ( ω ) − S M ( ω, ϕ ) : ω ∈ S γ ( A ) } . To prove this, we may assume by approximation that f is a polynomial. For each m ∈ N greater than both ℓ ( A ) and the range of Φ, let φ ( m ) := N Z ϕ Gm , a product state of the re-localized N i ∈ Z A [ im +1 , ( i +1) m ] as in (3.4). Furthermore, we set A ( m ) := 1 m m − ℓ ( A ) X k =0 γ k ( A ) ∈ A [1 ,m ] . According to [33, Theorem 1] (or [31, Proposition 13.8]), for any ε > γ m -invariant) state ψ of N i ∈ Z A [ im +1 , ( i +1) m ] such thatlim j →∞ j log Z ( m ) φ ( m ) ( j, A ( m ) , mf ) < − E ( m ) A ( m ) ,mf ( ψ ) − S ( m ) M ( ψ, φ ( m ) ) + ε , (3.11)where Z ( m ) φ ( m ) ( j, A ( m ) , mf ) := Tr exp log (cid:0)N j D ( ϕ Gm ) (cid:1) − jmf j j − X i =0 γ im ( A ( m ) ) !! ,E ( m ) A ( m ) ,mf ( ψ ) := lim j →∞ ψ mf j j − X i =0 γ im ( A ( m ) ) !! , ( m ) M ( ψ, φ ( m ) ) := lim j →∞ j S (cid:0) ψ | A [1 ,jm ] , φ ( m ) | A [1 ,jm ] (cid:1) . Then one can define an ω ∈ S γ ( A ) by ω := m P m − k =0 ψ ◦ γ k . Since we assumed that f isa polynomial, there is a constant M > k A k ) such that k f ( B ) − f ( B ) k ≤ M k B − B k for all B , B ∈ A sa with k B k , k B k ≤ k A k .For each n ∈ N with n ≥ m , write n = jm + r where j ∈ N and 0 ≤ r < m . Since m isgreater than the range of Φ, one can write H n (Φ) = j − X i =0 γ im ( H m (Φ)) + j − X i =1 γ im ( W ) + W j , where W is given in (3.5) and W j := X { Φ( X ) : X ⊂ [1 , n ] , X ∩ [ jm + 1 , jm + r ] = ∅} . We have by Lemma 2.1log D ( ϕ n ) ≤ log D ( ϕ Gn ) + log λ = − j − X i =0 γ im ( H m (Φ)) − j − X i =1 γ im ( W ) − W j − log Tr exp − j − X i =0 γ im ( H m (Φ)) − j − X i =1 γ im ( W ) − W j ! + log λ ≤ log (cid:0)N j D ( ϕ Gm ) (cid:1) + 2 jK + 2 k W j k + log λ with K given in (3.6). Here it is clear that k W j k ≤ mL with L := P X ∋ k Φ( X ) k . Furthermore,it is readily seen that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s n ( A ) − j j − X i =0 γ im ( A ( m ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:18) j + ℓ ( A ) m (cid:19) k A k and hence (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ( s n ( A )) − f j j − X i =0 γ im ( A ( m ) ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:18) j + ℓ ( A ) m (cid:19) M k A k . (3.12)Therefore, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) nf ( s n ( A )) − jmf j j − X i =0 γ im ( A ( m ) ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ m k f k ∞ + (2 m + jℓ ( A )) M k A k , where k f k ∞ is the sup-norm of f on [ λ min ( A ) , λ max ( A )]. From the above estimates we get1 n log Z ϕ ( n, A, f ) ≤ n log Z ( m ) φ ( m ) ( j, A ( m ) , mf )+ 1 n (cid:8) jK + 2 mL + log λ + m k f k ∞ + (2 m + jℓ ( A )) M k A k (cid:9)
16o thatlim sup n →∞ n log Z ϕ ( n, A, f ) ≤ m lim j →∞ j log Z ( m ) φ ( m ) ( j, A ( m ) , mf ) + 2 Km + ℓ ( A ) M k A k m . (3.13)Next, thanks to (3.12) we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω ( f ( s n ( A ))) − ψ f j j − X i =0 γ im ( A ( m ) ) !!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m m − X k =0 k f ( γ k ( s n ( A ))) − f ( s n ( A )) k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ( s n ( A )) − f j j − X i =0 γ im ( A ( m ) ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ Mm m − X k =0 k γ k ( s n ( A )) − s n ( A ) k + (cid:18) j + ℓ ( A ) m (cid:19) M k A k≤ (cid:18) j + ℓ ( A ) m (cid:19) M k A k . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) E A,f ( ω ) − m E A ( m ) ,mf ( ψ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ ( A ) M k A k m . (3.14)Furthermore, we get S ( ψ jm , ϕ jm ) ≤ S ( ψ jm , ϕ Gjm ) + log λ ≤ S ( ψ | A [1 ,jm ] , φ ( m ) | A [1 ,jm ] ) + 2( j − K + log λ similarly to (3.8) using the state perturbation (3.7). Since S M ( ω, ϕ ) = 1 m lim j →∞ j S ( ψ | A [1 ,jm ] , ϕ | A [1 ,jm ] )in the same way as (3.9), it follows that S M ( ω, ϕ ) ≤ m S ( m ) M ( ψ, φ ( m ) ) + 2 Km . (3.15)Inserting (3.13)–(3.15) into (3.11) giveslim sup n →∞ n log Z ϕ ( n, A, f ) ≤ − E A,f ( ω ) − S M ( ω, ϕ ) + 1 m ( ε + 4 K + 2 ℓ ( A ) M k A k ) , implying the required inequality because m and ε are arbitrary.The following is a straightforward consequence of Theorem 3.4. Corollary 3.5.
For every continuous f : R → R , the function p ϕ ( · , f ) on A saloc uniquely extendsto a continuous function ( denoted by the same p ϕ ( · , f )) on A sa satisfying p ϕ ( A, f ) = max {− f ( x ) − I A ( x ) : x ∈ [ λ min ( A ) , λ max ( A )] } for all A ∈ A sa . Moreover, for every continuous f, g : R → R and every A ∈ A sa , | p ϕ ( A, f ) − p ϕ ( A, g ) | ≤ max {| f ( x ) − g ( x ) | : x ∈ [ λ min ( A ) , λ max ( A )] } . emark 3.6. Suppose the “semi-classical” case where the observable A ∈ A saloc commutes withall Φ( X ). Since α Φ t ( A ) = A for all t ∈ R , A belongs to the centralizer of ϕ , i.e., ϕ ( AB ) = ϕ ( BA )for all B ∈ A . (To see this, apply [23, p. 617] in the GNS von Neumann algebra π ϕ ( A ) ′′ havingthe modular automorphism group which extends α Φ t .) This implies that s n ( A ) commutes with D ( ϕ n ) for every n ∈ N . As stated in Remark 2.8, p ϕ ( tA ) becomes the logarithmic momentgenerating function of ( µ n ) satisfying the large deviation principle with the good rate function I A ( x ) in (2.4). For any continuous f : R → R we have Z ϕ ( n, A, f ) = ϕ n (exp( − nf ( s n ( A )))) = Z λ max ( A ) λ min ( A ) e − nf ( s ) dµ n ( s ) . Now Varadhan’s integral lemma [12, Theorem 4.3.1] can be applied to obtainlim n →∞ n log Z ϕ ( n, A, f ) = max {− f ( x ) − I A ( x ) : x ∈ [ λ min ( A ) , λ max ( A )] } . The exact large deviation principle is not formulated in our noncommutative setting as longas the BMV-conjecture remains unsolved (see Remark 2.8); nevertheless Varadhan’s formulais valid as stated in Theorem 3.4.
The notion of ( C ∗ -)finitely correlated states was introduced by Fannes, Nachtergaele andWerner in [14]. Let B be a finite-dimensional C ∗ -algebra, E : A ⊗ B → B ( A = M d ( C ))a completely positive unital map and ρ a state of B such that ρ ( E ( I ⊗ b )) = ρ ( b ) for all b ∈ B .For each A ∈ A define a map E A : B → B by E A ( b ) := E ( A ⊗ b ), b ∈ B . Then the finitelycorrelated state ϕ determined by the triple ( B , E , ρ ) is the γ -invariant state of A given by ϕ ( A ⊗ A ⊗ · · · ⊗ A n ) := ρ ( E A ◦ E A ◦ · · · ◦ E A n ( B )) ( A i ∈ A i , ≤ i ≤ n ) . As was shown in the proof of [17, Proposition 4.4], a finitely correlated state has the followingupper factorization property, which will be useful in our discussions below.
Lemma 4.1. If ϕ is a finitely correlated state of A , then there exists a constant α ≥ suchthat ϕ ≤ α (cid:0) ϕ | A ( −∞ , (cid:1) ⊗ (cid:0) ϕ | A [1 , ∞ ) (cid:1) . The next proposition is a generalization of [31, Proposition 11.2].
Proposition 4.2.
Let ϕ be a finitely correlated state of A . For every ω ∈ S γ ( A ) the meanrelative entropy S M ( ω, ϕ ) = lim n →∞ n S ( ω n , ϕ n ) exists. Moreover, the function ω ∈ S γ ( A ) S M ( ω, ϕ ) is affine and weakly* lower semicontin-uous on S γ ( A ) . roof. The proof of the first assertion is a slight modification of that of [20, Theorem 2.1] whileit will be repeated below for the convenience of the remaining proof. For each n, m ∈ N with n ≥ m , write n = jm + r with j ∈ N and 0 ≤ r < m . Lemma 4.1 implies that ϕ n ≤ α j j − O i =0 (cid:0) ϕ | A [ im +1 , ( i +1) m ] (cid:1)! ⊗ (cid:0) ϕ | A [ jm +1 ,jm + r ] (cid:1) . (4.1)Consider the product state φ ( m ) := N Z ϕ m of the re-localized spin chain N i ∈ Z A [ im +1 , ( i +1) m ] .For every ω ∈ S γ ( A ) we have S ( ω n , ϕ n ) ≥ S ( ω jm , ϕ jm ) ≥ S ( ω jm , N j ϕ m ) − j log α (4.2)due to the monotonicity of relative entropy and (4.1). Dividing (4.2) by n and letting n → ∞ with m fixed we get lim inf n →∞ n S ( ω n , ϕ n ) ≥ m S ( m ) M ( ω, φ ( m ) ) − log αm , where S ( m ) M ( ω, φ ( m ) ) denotes the mean relative entropy in the re-localized N i ∈ Z A [ im +1 , ( i +1) m ] as in (3.11). Since S ( m ) M ( ω, φ ( m ) ) ≥ S ( ω m , ϕ m ) by [18, (2.1)], we further getlim inf n →∞ n S ( ω n , ϕ n ) ≥ m S ( ω m , ϕ m ) − log αm . Since m ∈ N is arbitrary, this shows the existence of S M ( ω, ϕ ) and the above inequalitiesbecome S M ( ω, ϕ ) ≥ m S ( ω m , ϕ m ) − log αm . (4.3)The affinity of ω S M ( ω, ϕ ) is a consequence of the general property [31, Proposition5.24]. Assume that ω, ω ( k ) ∈ S γ ( A ) and ω ( k ) → ω weakly*. Then from (4.3) we havelim inf k →∞ S M ( ω ( k ) , ϕ ) ≥ m lim inf k →∞ S ( ω ( k ) m , ϕ m ) − log αm ≥ m S ( ω m , ϕ m ) − log αm thanks to the lower semicontinuity of relative entropy (in fact, ω S ( ω m , ϕ m ) is continuous dueto finite dimensionality). Letting m → ∞ shows the lower semicontinuity of ω S M ( ω, ϕ ).Next we show the existence of the free energy density with respect to a finitely correlatedstate ϕ . Since ϕ is not assumed to be locally faithful in the sense that D ( ϕ n ) is strictly positivefor every n ∈ N , we need to be careful in defining Tr exp (cid:0) log D ( ϕ n ) − B (cid:1) for B ∈ A sa[1 ,n ] . Let D be a nonzero positive semidefinite matrix and B a Hermitian matrix in M N ( C ). It is known[19, Lemma 4.1] that lim ε ց e log( D + εI ) − B = P ( e P (log D ) P − P BP ) P , where P is the support projection of D . Hence one can define Tr e log D − B byTr e log D − B := lim ε ց Tr e log( D + εI ) − B = Tr P e P (log D ) P − P BP . (4.4)19 roposition 4.3. Let ϕ be a finitely correlated state of A . For every A ∈ A saloc so that A ∈ A saΛ with a finite Λ ⊂ Z , the free energy density p ϕ ( A ) := lim n →∞ n log Tr exp log D ( ϕ n ) − X Λ+ k ⊂ [1 ,n ] γ k ( A ) ! exists ( independently of the choice of Λ) . Moreover, p ϕ is convex and Lipschitz continuous with | p ϕ ( A ) − p ϕ ( B ) | ≤ k A − B k , and therefore it uniquely extends to a function on A sa with thesame properties.Proof. To prove the first assertion we may assume that A ∈ A sa[1 ,ℓ ( A )] with some ℓ ( A ) ∈ N . Foreach n, m ∈ N with n ≥ m > ℓ ( A ), write n = jm + r with 0 ≤ r < m . From (4.1) we get D ( ϕ n ) ≤ α j j − Y i =0 γ im ( D ( ϕ m ))with a constant α ≥ n, m . For any ε > D ( ϕ n ) + ε j I ≤ α j j − Y i =0 γ im ( D ( ϕ m ) + εI ) . Furthermore, it is immediately seen that n − ℓ ( A ) X k =0 γ k ( A ) ≥ j − X i =0 γ im m − ℓ ( A ) X k =0 γ k ( A ) ! − ( j ( ℓ ( A ) −
1) + r ) k A k . Set h n := P n − ℓ ( A ) k =0 γ k ( A ). From the above two inequalities we getTr exp (cid:0) log( D ( ϕ n ) + ε j I ) − h n (cid:1) ≤ (cid:8) Tr exp (cid:0) log( D ( ϕ m ) + εI ) − h m (cid:1)(cid:9) j exp (cid:0) j log α + ( j ( ℓ ( A ) −
1) + r ) k A k (cid:1) . In view of the definition (4.4), letting ε ց (cid:0) log D ( ϕ n ) − h n (cid:1) ≤ (cid:8) Tr exp (cid:0) log D ( ϕ m ) − h m (cid:1)(cid:9) j exp (cid:0) j log α + ( j ( ℓ ( A ) −
1) + r ) k A k (cid:1) so that lim sup n →∞ n log Tr exp (cid:0) log D ( ϕ n ) − h n (cid:1) ≤ m log Tr exp (cid:0) log D ( ϕ m ) − h m (cid:1) + log αm + ℓ ( A ) − m k A k . Since m ( > ℓ ( A )) is arbitrary, this shows the existence of the limit p ϕ ( A ). It is obvious that p ϕ ( A ) is independent of the choice of Λ. It is also clear that p ϕ ( A ) on A saloc is convex andsatisfies | p ϕ ( A ) − p ϕ ( B ) | ≤ k A − B k for all A, B ∈ A saloc , from which the second part of theproposition follows. 20 emark 4.4.
The limit ˜ p ϕ ( A ) similar to p ϕ ( A ) was referred to in Remark 2.7 from the view-point of large deviations. In [17] the limit ˜ p ϕ ( A ) was shown to exist for any A ∈ A saloc when ϕ is a finitely correlated state (as well as when ϕ is a Gibbs state). The proof for ˜ p ϕ ( A ) is moreinvolved than the above for p ϕ ( A ) and relies on the estimate in [26, Theorem 3.7] related toGibbs state perturbation.Once we had Propositions 4.2 and 4.3, it is natural to expect that S M ( ω, ϕ ) and p ϕ ( A )enjoy the same Legendre transform formulas as (4) and (5) of Corollary 2.5 in the Gibbs statecase. But this is still unsolved while the following inequality is easy as Lemma 3.2. For theproof use [20, (4.2)] or [31, Proposition 1.11], the extended version of (3.1). Proposition 4.5.
Let ϕ be a finitely correlated state of A . For every A ∈ A sa , p ϕ ( A ) ≥ max {− ω ( A ) − S M ( ω, ϕ ) : ω ∈ S γ ( A ) } . Remark 4.6.
Suppose that ϕ satisfies the lower factorization property ϕ ≥ β (cid:0) ϕ | A ( −∞ , (cid:1) ⊗ (cid:0) ϕ | A [1 , ∞ ) (cid:1) for some β > ϕ . The proofs can be carried out similarly to those in Section 3; in fact, theyare even easier without the state perturbation technique. However, the lower factorizationproperty for finitely correlated states is quite strong; for example, one can easily see thata classical irreducible Markov chain has this property if and only if its transition stochasticmatrix is strictly positive (i.e., all entries are strictly positive), which is stronger than the strongmixing property. More details are in [17].In the rest of this section, we assume that ϕ is a γ -invariant quantum Markov state ofAccardi and Frigerio type [3], and further assume that ϕ is locally faithful. According to[4, 30], there exists a conditional expectation E from M d ( C ) ⊗ M d ( C ) into M d ( C ) such that ϕ ◦ E ( I ⊗ A ) = ϕ ( A ) for all A ∈ M d ( C ) and ϕ ( A ⊗ A ⊗ · · · ⊗ A n ) = ϕ ( E ( A ⊗ E ( A ⊗ · · · ⊗ E ( A n − ⊗ A n ) · · · )))for all A , A , . . . , A n ∈ M d ( C ), where ϕ := ϕ | A . Set B := E ( M d ( C ) ⊗ M d ( C )), a subalgebraof M d ( C ), E := E | M d ( C ) ⊗B and ρ := ϕ | B . Then ϕ is a finitely correlated state with the triple( B , E , ρ ). Let q , . . . , q k be the minimal central projections of B ; then B q i ∼ = M d i ( C ) and B isdecomposed as B = k M i =1 B q i = k M i =1 (cid:0) M d i ( C ) ⊗ I m i (cid:1) , where m i is the multiplicity of M d i ( C ) in M d ( C ). Let B ′ be the relative commutant of B in M d ( C ) so that B ′ = L ki =1 I d i ⊗ M m i ( C ). For each m, n ∈ Z , m ≤ n , set e A [ m,n ] := B ′ ⊗ A [ m +1 ,n − ⊗ B ( ⊂ A [ m,n ] )with convention e A [ n,n ] := C I ( ⊂ A n ). Let C := L ki =1 M d i ( C ) ⊗ M m i ( C ) ( ⊂ M d ( C )) and E C bethe pinching A ∈ M d ( C ) P ki =1 q i Aq i ∈ C (or the conditional expectation onto C with respectto the trace). The following properties were shown in [4, 30]:21i) There exist positive linear functionals ρ ij on M m i ( C ) ⊗ M d j ( C ), 1 ≤ i, j ≤ k , such that E = k M i,j =1 id M di ( C ) ⊗ ρ ij ! ◦ ( E C ⊗ id B ) . (ii) Let T ij be the density matrices of ρ ij for 1 ≤ i, j ≤ k . Then the density matrix of ϕ | e A [ m,n ] is e D [ m,n ] := M i m ,i m +1 ,...,i n ρ ( q i m ) T i m i m +1 ⊗ T i m +1 i m +2 ⊗ · · · ⊗ T i n − i n . (4.5)The density matrices e D [ m,n ] have a simple form of product type. Since T ij is strictly positivein M m i ( C ) ⊗ M d j ( C ) for each i, j due to the local faithfulness of ϕ , a γ -invariant nearest-neighborinteraction Φ can be defined byΦ([0 , − k X i,j =1 log T ij ∈ B ′ ⊗ B ⊂ A [0 , , Φ([ n, n + 1]) := γ n (Φ([0 , . Then the density of the local Gibbs state of A [ m,n ] for Φ is M i m ,...,i n T i m i m +1 ⊗ · · · ⊗ T i n − i n , and the automorphism group α Φ t is given by α Φ t ( A ) = lim m →−∞ ,n →∞ e D − it [ m,n ] A e D it [ m,n ] ( A ∈ A ) . (4.6)Hence ϕ is the α Φ -KMS state (or the Gibbs state for Φ) and so all the results in Sections 2 and3 hold for ϕ . Below let us further investigate the relation between p ϕ ( A ) in (2.3) and ˜ p ϕ ( A ) in(2.5).The centralizer of ϕ is given by A ϕ := { A ∈ A : ϕ ( AB ) = ϕ ( BA ) for all B ∈ A} , which is a γ -invariant C ∗ -subalgebra of A . For each m, n ∈ Z with m ≤ n , we also define( e A [ m,n ] ) ϕ := { A ∈ e A [ m,n ] : ϕ ( AB ) = ϕ ( BA ) for all B ∈ e A [ m,n ] } . Lemma 4.7. If m ′ ≤ m ≤ n ≤ n ′ in Z , then ( e A [ m,n ] ) ϕ ⊂ ( e A [ m ′ ,n ′ ] ) ϕ ⊂ A ϕ . Moreover, e A ϕ, loc := S ∞ n =1 ( e A [ − n,n ] ) ϕ is a dense ∗ -subalgebra of A ϕ .Proof. Since ( e A [ m,n ] ) ϕ is the relative commutant of { e D [ m,n ] } in e A [ m,n ] , the first assertion isimmediately seen from the form (4.5) of e D [ m,n ] . Furthermore, it is also obvious from (4.6) that α Φ t ( e A [ m,n ] ) = e A [ m,n ] , t ∈ R , for any m ≤ n . By [37] applied in the GNS von Neumann algebra π ϕ ( A ) ′′ with the modular automorphism group extending α Φ t , there exists the conditionalexpectation E [ m,n ] : A → e A [ m,n ] with ϕ ◦ E [ m,n ] = ϕ . Then it is clear that k E [ m,n ] ( A ) − A k → m → −∞ and n → ∞ for any A ∈ A . Now let A ∈ A ϕ . Since ϕ ( E [ m,n ] ( A ) B ) = ϕ ( AB ) = ϕ ( BA ) = ϕ ( BE [ m,n ] ( A )) , B ∈ e A [ m,n ] , we have E [ m,n ] ( A ) ∈ ( e A [ m,n ] ) ϕ for any m ≤ n , implying the latter assertion.22 emma 4.8. For every ω ∈ S γ ( A ) , S M ( ω, ϕ ) = lim n →∞ n S (cid:0) ω | ( e A [1 ,n ] ) ϕ , ϕ | ( e A [1 ,n ] ) ϕ (cid:1) and hence S M ( ω, ϕ ) is determined by ω | A ϕ . Moreover, if ω, ω ( i ) ∈ S γ ( A ) , i ∈ N , and ω ( i ) | A ϕ → ω | A ϕ in the weak* topology, then S M ( ω, ϕ ) ≤ lim inf i →∞ S M ( ω ( i ) , ϕ ) . Proof.
The proof of the first assertion is essentially the same as that of [18, Theorem 2.1] as willbe sketched below. Let T ij = P L ij ℓ =1 λ ijℓ e ijℓ be the spectral decomposition of T ij for 1 ≤ i, j ≤ k ,and Θ be the set of all ( i, j, ℓ ) with 1 ≤ i, j ≤ k and 1 ≤ ℓ ≤ L ij . For each n ∈ N let K n be theset of all tuples ( n θ ) θ ∈ Θ of nonnegative integers such that P θ ∈ Θ n θ = n −
1. For each 1 ≤ i ≤ k and ( n θ ) ∈ K n we denote by I i, ( n θ ) the set of all ( i , i , . . . , i n ; ℓ , ℓ , . . . , ℓ n − ) such that i = i and { r ∈ [1 , n −
1] : ( i r , i r +1 , ℓ r ) = θ } = n θ for all θ ∈ Θ, and define the projection P i, ( n θ ) in e A [1 ,n ] and λ i, ( n θ ) ∈ R by P i, ( n θ ) := X ( i ,...,i n ; ℓ ,...,ℓ n − ) ∈ I i, ( nθ ) e i i ℓ ⊗ e i i ℓ ⊗ · · · ⊗ e i n − i n ℓ n − ,λ i, ( n θ ) := ρ ( q i ) Y θ ∈ Θ λ n θ θ where λ θ := λ ijℓ for θ = ( i, j, ℓ ) . Then P ki =1 P ( n θ ) ∈ K n P i, ( n θ ) = I and e D [1 ,n ] is written as e D [1 ,n ] = k X i =1 X ( n θ ) ∈ K n λ i, ( n θ ) P i, ( n θ ) . Now, for each ω ∈ S γ ( A ), the proof of [18, Theorem 2.1] implies that S ( ω n − , ϕ n − ) ≤ S (cid:0) ω | e A [1 ,n ] , ϕ | e A [1 ,n ] (cid:1) ≤ S (cid:0) ω | ( e A [1 ,n ] ) ϕ , ϕ | ( e A [1 ,n ] ) ϕ (cid:1) + log k + log K n for every n ≥
3. Since K n ≤ n , we get S M ( ω, ϕ ) ≤ lim inf n →∞ n S (cid:0) ω | ( e A [1 ,n ] ) ϕ , ϕ | ( e A [1 ,n ] ) ϕ (cid:1) , which proves the first assertion.Set γ := 1 / min ≤ i ≤ k ρ ( q i ). For each m, m ′ ∈ N , since it follows from (4.5) that ϕ | ( e A [1 ,m ] ) ϕ ⊗ ( e A [ m +1 ,m + m ′ ] ) ϕ ≤ γ (cid:0) ϕ | ( e A [1 ,m ] ) ϕ (cid:1) ⊗ (cid:0) ϕ | ( e A [ m +1 ,m + m ′ ] ) ϕ (cid:1) , we get S (cid:0) ω | ( e A [1 ,m + m ′ ] ) ϕ , ϕ | ( e A [1 ,m + m ′ ] ) ϕ (cid:1) ≥ S (cid:0) ω | ( e A [1 ,m ] ) ϕ ⊗ ( e A [ m +1 ,m + m ′ ] ) ϕ , ϕ | ( e A [1 ,m ] ) ϕ ⊗ ( e A [ m +1 ,m + m ′ ] ) ϕ (cid:1) − log γ ≥ S (cid:0) ω | ( e A [1 ,m ] ) ϕ , ϕ | ( e A [1 ,m ] ) ϕ (cid:1) + S (cid:0) ω | ( e A [1 ,m ′ ] ) ϕ , ϕ | ( e A [1 ,m ′ ] ) ϕ (cid:1) − log γ ω and ω ( i ) be given as stated in the lemma. For any m ∈ N and n = jm + r with j ∈ N and0 ≤ r < m , the above inequality gives S (cid:0) ω ( i ) | ( e A [1 ,n ] ) ϕ , ϕ | ( e A [1 ,n ] ) ϕ (cid:1) ≥ jS (cid:0) ω ( i ) | ( e A [1 ,m ] ) ϕ , ϕ | ( e A [1 ,m ] ) ϕ (cid:1) − j log γ . Dividing this by n and letting n → ∞ with m fixed we get S M ( ω ( i ) , ϕ ) ≥ m S (cid:0) ω ( i ) | ( e A [1 ,m ] ) ϕ , ϕ | ( e A [1 ,m ] ) ϕ (cid:1) − log γm and hence lim inf i →∞ S M ( ω ( i ) , ϕ ) ≥ m S (cid:0) ω | ( e A [1 ,m ] ) ϕ , ϕ | ( e A [1 ,m ] ) ϕ (cid:1) − log γm . Letting m → ∞ gives the latter assertion.In addition to the variational expression in Corollary 2.5 (5) we have Theorem 4.9.
For every ω ∈ S γ ( A ) , S M ( ω, ϕ ) = sup {− ω ( A ) − p ϕ ( A ) : A ∈ A sa ϕ } = sup {− ω ( A ) − ˜ p ϕ ( A ) : A ∈ A saloc } , where ˜ p ϕ ( A ) is given in (2.5) .Proof. The proof of the first equality is a simple duality argument. Set Γ := { ω | A sa ϕ : ω ∈S γ ( A ) } , which is a weakly* compact and convex subset of ( A sa ϕ ) ∗ , the dual Banach space ofthe real Banach space A sa ϕ . From Lemma 4.8 one can define F : ( A sa ϕ ) ∗ → [0 , + ∞ ] by F ( f ) := ( S M ( ω, ϕ ) if f = ω | A sa ϕ with some ω ∈ S γ ( A ) , + ∞ otherwise , which is affine and weakly* lower semicontinuous on ( A sa ϕ ) ∗ by Proposition 4.2 and Lemma 4.8.Corollary 2.5 (4) says that p ϕ ( A ) = max {− f ( A ) − F ( f ) : f ∈ ( A sa ϕ ) ∗ } , A ∈ A sa ϕ . Hence it follows by duality [13, Proposition I.4.1] that F ( f ) = sup {− f ( A ) − p ϕ ( A ) : A ∈ A sa ϕ } , f ∈ ( A sa ϕ ) ∗ . For every ω ∈ S γ ( A ) this means the first equality, which also gives S M ( ω, ϕ ) = sup {− ω ( A ) − p ϕ ( A ) : A ∈ e A sa ϕ, loc } (4.7)thanks to Lemma 4.7.To prove the second equality, we show that p ϕ ( A ) = ˜ p ϕ ( A ) for all A ∈ e A sa ϕ, loc . Thanks toLemma 4.7 and the γ -invariance of p ϕ and ˜ p ϕ , we may assume that A ∈ ( e A [1 ,m ] ) sa ϕ for some m ∈ N . For each n ∈ N and 0 ≤ k ≤ n − m , we have γ k ( A ) ∈ ( e A [1+ k,m + k ] ) ϕ ⊂ ( e A [1 ,n ] ) ϕ so that24xp (cid:0) − P n − mk =0 γ k ( A ) (cid:1) ∈ ( e A [1 ,n ] ) ϕ . Furthermore, since e A [1 ,n ] ⊂ A [1 ,n ] ⊂ e A [0 ,n +1] , it is easy to seeby Lemma 4.7 that ( e A [1 ,n ] ) ϕ ⊂ ( A [1 ,n ] ) ϕ . Hence we get exp (cid:0) − P n − mk =0 γ k ( A ) (cid:1) ∈ ( A [1 ,n ] ) ϕ , whichimplies that exp (cid:0) − P n − mk =0 γ k ( A ) (cid:1) commutes with the density D ( ϕ n ) so that ϕ exp − n − m X k =0 γ k ( A ) !! = Tr exp log D ( ϕ n ) − n − m X k =0 γ k ( A ) ! , showing p ϕ ( A ) = ˜ p ϕ ( A ) by definitions (2.3) and (2.5). From this and (4.7) we get S M ( ω, ϕ ) ≤ sup {− ω ( A ) − ˜ p ϕ ( A ) : A ∈ A saloc }≤ sup {− ω ( A ) − p ϕ ( A ) : A ∈ A saloc } = S M ( ω, ϕ )thanks to (2.6) and Corollary 2.5 (5), implying the second equality. Corollary 4.10.
The function p ϕ on A sa is the lower semicontinuous convex envelope of ˜ p ϕ on A saloc in the sense that p ϕ is the largest among lower semicontinuous and convex functions q on A sa satisfying q ≤ ˜ p ϕ on A saloc .Proof. Let q be as stated in the corollary. Define Q : ( A sa ) ∗ → ( −∞ , + ∞ ] by Q ( f ) := sup {− f ( A ) − q ( A ) : A ∈ A sa } ( f ∈ ( A sa ) ∗ ) . Let us prove that ( Q ( ω ) ≥ S M ( ω, ϕ ) if ω ∈ S γ ( A ) ,Q ( f ) = + ∞ if f ∈ ( A sa ) ∗ \ S γ ( A ) . (4.8)For ω ∈ S γ ( A ) Theorem 4.9 gives Q ( ω ) ≥ sup {− ω ( A ) − ˜ p ϕ ( A ) : A ∈ A saloc } = S M ( ω, ϕ ) . For f ∈ ( A sa ) ∗ \ S γ ( A ) we may consider the following three cases:(a) f ( A ) < A ∈ A loc ,(b) f ( ) = 1,(c) f ( A ) = f ( γ ( A )) for some A ∈ A sa .In case (a), since q ( αA ) ≤ ˜ p ϕ ( αA ) ≤ α >
0, we get − f ( αA ) − q ( αA ) ≥ − αf ( A ) → + ∞ as α → + ∞ . In case (b), since q ( α ) ≤ ˜ p ϕ ( α ) = − α , we get − f ( α ) − q ( α ) ≥ − α ( f ( ) − → + ∞ as α → + ∞ or −∞ according as f ( ) < f ( ) >
1. Finally in case (c), since q ( α ( A − γ ( A ))) ≤ ˜ p ϕ ( α ( A − γ ( A ))) = lim n →∞ n log ϕ ( e − α ( A − γ n ( A )) ) = 0 , we get − f ( α ( A − γ ( A ))) − q ( α ( A − γ ( A ))) ≥ − αf ( A − γ ( A )) → + ∞ as α → + ∞ or −∞ according as f ( A ) < f ( γ ( A )) or f ( A ) > f ( γ ( A )). Hence (4.8) follows. By duality this impliesthat q ≤ p ϕ on A sa .In particular, when ϕ is the product state N Z ρ of a not necessarily faithful ρ , all thevariational expressions in Corollary 2.5 and Theorem 4.9 are valid for ϕ , and so Corollary 4.10holds for ϕ . Although we have no strong evidence, it might be conjectured that Corollary 4.10is true generally for the Gibbs-KMS state ϕ treated in Sections 2 and 3.25 Concluding remarks: guide to the case of arbitrary dimen-sion
In this paper we confined ourselves to the one-dimensional spin chain case for the follow-ing reasons. First, our main motivation came from recent developments on large deviationsin spin chains, where the differentiability of logarithmic moment generating functions is cru-cial. The corresponding functions in our setting are free energy density functions so thatwe wanted to provide their differentiability (see Theorem 2.4 (c) and Corollary 2.5 (2)), andthe one-dimensionality is essential for this. Secondly, finitely correlated states treated in thelatter half are defined only in a one-dimensional spin chain though some attempts to multi-dimensional extension were made for similar states of quantum Markov type (see [1, 2] forexample). However, all the discussions (except the differentiability assertions) presented for aGibbs state of one-dimension in Sections 2 and 3 can also work well in the setting of arbitrarydimension but in high temperature regime, which we outline below.Consider a ν -dimensional spin chain A := N k ∈ Z ν A k , A k = M d ( C ), with the translationautomorphism group γ k , k ∈ Z ν , and local algebras A Λ := N k ∈ Λ A k for finite Λ ⊂ Z ν . Wedenote by B ( A ) the set of all translation-invariant interactions Φ in A of relatively short range,i.e., ||| Ψ ||| := P X ∋ k Ψ( X ) k / | X | < + ∞ , which is a real Banach space with the norm ||| Ψ ||| . LetΦ ∈ B ( A ) and assume further that Φ is of finite body, i.e., N (Φ) := sup {| X | : Φ( X ) = 0 } < + ∞ (weaker than the assumption of finite range). Then Φ is automatically of short range, i.e., k Φ k := P X ∋ k Φ( X ) k < + ∞ . It is well known [11, 22] that the one-parameter automorphismgroup α Φ t of A is defined and all of the α Φ -KMS condition, the Gibbs condition and thevariational principle for states ϕ ∈ S γ ( A ) are equivalent. The pressure (1.5) and the meanentropy (1.6), the main ingredients in the variational principle, can be defined in the van Hovelimit of Λ → ∞ (see [22, p. 12] or [11, p. 287]), but in our further discussions we may simplyrestrict to the parallelepipeds Λ = { ( k , . . . , k ν ) : 1 ≤ k i ≤ n i , ≤ i ≤ ν } with Λ → ∞ meaning n i → ∞ for 1 ≤ i ≤ ν .A crucial point in the arbitrary dimensional setting is the following generalization of Lemma2.1 given in [8] in high temperature regime with an inverse temperature β . Lemma 5.1.
Let Φ be given as above and r (Φ) := { k Φ k ( N (Φ) − } − ( meant + ∞ if N (Φ) ≤ . Assume that < β < r (Φ) and ϕ ∈ S γ ( A ) satisfies the Gibbs condition for β Φ ( equivalently, the α Φ -KMS condition at − β ) . Then there are constants λ Λ such that λ − ϕ Λ ≤ ϕ β,G Λ ≤ λ Λ ϕ Λ and lim Λ →∞ log λ Λ | Λ | = 0 , (5.1) where ϕ β,G Λ is the local Gibbs state of A Λ for β Φ . Even though a Gibbs state ϕ ∈ S γ ( A ) for β Φ is not necessarily unique and constants λ Λ are depending on Λ, property (5.1) is enough for us to show all the results in Section 2 (exceptthe differentiability assertions mentioned above) in the same way under the situation where Φis replaced by β Φ with β as in Lemma 5.1 and B ( A ) is replaced by B ( A ). In particular, itwas formerly observed in [20, p. 710–711] that for every ω ∈ S γ ( A ) the mean relative entropy262.1) exists and furthermore S M ( ω, ϕ ) = 0 if and only if ω is a Gibbs state for β Φ too. In fact,the latter assertion is immediate from the formula in Lemma 2.3 due to the equivalence of theGibbs condition and the variational principle.Next let A ∈ A saloc so that we may assume that A ∈ A saΛ with some parallelepiped Λ ⊂ Z ν of the form mentioned above. Let f be a real continuous function on [ λ min ( A ) , λ max ( A )]. Foreach parallelepiped Λ of the same form, we set s Λ ( A ) := 1 | Λ | X Λ + k ⊂ Λ γ k ( A )and Z ϕ (Λ , A, f ) := Tr exp (cid:0) log D ( ϕ Λ ) − | Λ | f ( s Λ ( A )) (cid:1) . Then Lemmas 3.1 and 3.2 hold true in the same way as before. Moreover, the proof of Lemma3.3 can easily be carried out in the present framework with slight modifications, for example,with replacing the uniform boundedness of surface energies by the asymptotic property1 | Λ | X {k Φ( X ) k : X ∩ Λ = ∅ , X ∩ Λ c = ∅} −→ → ∞ of parallelepipeds Λ. This property holds in general for translation-invariant inter-actions of short range.Finally, we can prove the existence of the functional free energy density p ϕ ( A, f ) := lim Λ →∞ | Λ | log Z ϕ (Λ , A, f )and its variational expressions in the same way as in Theorem 3.4. A key point in proving thisis that the result for the product state case in [33] (or [31]) used in the proof of Theorem 3.4can be applied as well since the dimension of the integer lattice is irrelevant in the situation ofproduct/symmetric states. In this way, all the proofs in Section 3 of one dimension can easilybe adapted to the present framework by using Lemma 5.1 and the property of short range forΦ, and the condition of finite range is not necessary. Acknowledgements
The authors are grateful to an anonymous referee for his comments that are very helpful inimproving the final version of the paper.
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