DLR-KMS correspondence on lattice spin systems

The Dobrushin-Lanford-Ruelle (DLR) condition [13, 22] and the classical Kubo-Martin-Schwinger (KMS) condition [18] are considered in the context of classical lattice systems. In particular, we prove that these conditions are equivalent for the case of a lattice spin system with values in a compact symplectic manifold by showing that inﬁnite volume Gibbs states are in bijection with KMS states.


Introduction
Identifying the proper notion of thermal equilibrium in an infinite system is of paramount importance in statistical mechanics [26]. A common approach to this problem is to consider a finite size approximation of the system under consideration, where the notion of thermal equilibrium is captured by the renown Gibbs state [16], cf. Equation (4). A subsequent limit procedure, where the size of the system diverges, leads to states which are identified (a posteriori) as those describing thermal equilibrium in the infinite system. While being a convenient and fruitful approach, this method leaves open the problem of identifying thermal equilibrium states directly on the infinite system. Such possibility is essential because it leads to a proper notion of phase transitions, independently on the chosen order parameters [16, §3].
For lattice spin systems, the Dobrushin-Lanford-Ruelle (DLR) condition provides a satisfactory solution to the stated problem [10,11,12,13,22,25]. Therein, thermal equilibrium states are identified with probability measures which reduce to the Gibbs state when conditioned on the complement of a finite region, see Section 2 for the precise definition. States abiding by the DLR condition are usually called infinite-volume Gibbs states. The main ingredients of this approach are the lattice formulation of the system of interest together with the notion of Gibbs states on a local region of such lattice [19].
Another conceptually clear yet different in spirit condition to pinpoint thermal equilibrium states is the renown Kubo-Martin-Schwinger (KMS) condition. The quantum version of this condition, usually formulated in the C * -algebraic setting, identifies thermal equilibrium in terms of an analytic condition on the correlations of the state [9,20]. A classical version of this condition is at disposal and has been investigated in different contexts ranging from systems of infinitely many particles on R d [1,2,17,18] to Poisson geometry [5,8,15]. The classical KMS condition fixes the expectation value of the state of interest on Poisson brackets, cf. Section 3 for the precise definition. Thus, it requires a Poisson structure on the system under investigation.
At this juncture a natural question arises, namely if the DLR and the (classical) KMS conditions agree whenever the system under investigation is described in terms of a lattice structure endowed with a Poisson bracket. A positive answer for systems of particles on R d has been found in [2] and similar positive results have been obtained on the quantum side [9,Thm. 6.2.18].
In this paper we deal with the DLR-KMS correspondence for the case of a lattice system whose configuration space Ω = M Z d is the space of functions Z d → M with values in a compact symplectic manifold M . This provides a natural setting where both the DLR and KMS approaches apply, leading to the natural question of whether the DLR-KMS correspondence holds true also in this setting. The main result of this paper is the following theorem, which provides a positive answer to this question.
Theorem 1: Let G(Φ) be the convex set of infinite volume Gibbs states on Ω = M Z d -cf. Section 3 and Equation (6). Let K(Φ) be the convex set of X Φ -KMS states defined as per Equation (14), cf. Section 3. Then ♦ Aside from its relevance in the discussion on the notion of thermal equilibrium, Theorem 1 is relevant for its connection with strict deformation quantization (SDQ) [3,4,6]. The latter provides a mathematically sound description of the classical limit of quantum theories and it is particularly suitable for the analysis of the semi-classical limit of quantum states. In view of the recent results in this framework [14,23,24,27,28], Theorem 1 stands as a technical result which may lead to a deeper understanding of the quantum-to-classical limit of thermal equilibrium states. We plan to address this question in a future investigation.
From a technical point of view this paper profits from the strategy applied in [2] together with the results obtained in [8,15]. As a matter of fact the proof of G(Φ) ⊆ K(Φ) is a direct computation, cf. Proposition 5. Instead, the converse inclusion requires a more refined analysis of the conditional probability ϕ Λ ( |η) of a given KMS state ϕ with respect to the complement Λ c of a finite region Λ ⋐ Z d . This is where the results of [8,15] apply, ensuring that such conditional probabilities ϕ Λ ( |η) coincide with the local Gibbs state ϕ Φ Λ ( |η). The paper is organized as follows: Section 2 is devoted to a brief introduction to the lattice spin system of interest together with the precise definition of the DLR condition. Similarly, Section 3 considers the classical KMS condition in the same setting. Finally Section 4 proves Theorem 1, cf. Proposition 5-6.

DLR approach on continuous lattice system
In this section we briefly recollect a few crucial results and definitions from continuous lattice systems, see [16]. In what follows we will consider a configuration space over the lattice Z d with values in a symplectic manifold M . The differential structure of M is needed for the formulation of the classical KMS condition, cf. Section 3.
Let (M, ς) be a compact, connected symplectic manifold with symplectic form ς. For technical convenience we shall assume that M is metrizable and denote with d M a complete metric on M whose induced topology coincides with the one of M . We will denote by The configuration space of interest is Ω := M Z d which is compact in the product topology. Moreover, Ω is also metrizable once we set If Λ ⊂ Z d we set Ω Λ := M Λ and denote by the projection over Ω Λ . With a standard slight abuse of notation we will set, given where M |Λ| is again a connected compact symplectic manifold with symplectic form ς Λ := ⊕ i∈Λ ς. The algebra of observables over the configuration space Ω is identified with the commutative C * -algebra C(Ω) equipped with the supremum norm f ∞ := sup ω∈Ω |f (ω)|. This space is best described in terms of (continuous) local functions. In particular, Then, for all f ∈ C(Ω) there exists a sequence (f n ) n∈N of local functions such that lim n→∞ f − f n ∞ = 0 [16, Lem. 6.21] (e.g. one considers η ∈ Ω and a se- . For this reason, C(Ω) is usually referred to as the C * -algebra of (continuous) quasi-local functions.
Per definition, a state on C(Ω) is a normalized linear, positive functional ϕ : C(Ω) → C. By the Riesz-Markov-Kakutani theorem every state ϕ on C(Ω) is completely described by a Radon probability measure µ ∈ P(Ω, F) on (Ω, F) -here F denotes the Borel σ-algebra over Ω. In particular ϕ ≡ ϕ µ and For later convenience we recall the definition of the σ-algebra F Λ over Ω of events occurring in Λ ⊂ Z d : Among all possible states ϕ µ on C(Ω) we will be interested in the infinite volume Gibbs states. The definition of the latter requires to introduce a potential, namely a collection In what follows we will assume the following technical assumptions on Φ: where Φ Λ C 1 (M Λ ) denotes the supremum of Φ Λ and dΦ Λ over Ω Λ , thus it coincides with the norm of the C * -algebra C 1 (M |Λ| ).
Within these assumptions we can define which is a well-defined element in C(Ω) on account of assumption (II). For a fixed η ∈ Ω one then introduces the probability measure µ Φ Λ ( |η) on (Ω, F) defined by where µ M |Λ| := ⊗ i∈Λ µ M denotes the (normalized) Liouville volume form over M |Λ| while Z Φ η,Λ is a normalization constant -we have implicitly identified Ω Λ ≃ M |Λ| . We will refer to µ Φ Λ ( |η) as the Λ-Gibbs measure associated with the potential Φ and with boundary condition η (at fixed inverse temperature β = 1). We will denote by ϕ Φ Λ ( |η) the associated state on C(Ω) which will be referred to as the Λ-Gibbs state associated with Φ and η.
Remark 2: The assumption on the potential Φ are slightly stronger than those usually imposed [16, §6]. As a matter of fact, one usually requires which ensures the well-definiteness of H Φ Λ as per Equation (3). The stronger assumptions (I)-(II) are necessary for discussing the Poisson bracket on Ω -cf. Section 3.
♦ The collection µ Φ := {µ Φ Λ } Λ⋐Z d is called Gibbs specification and enjoys the following properties: (a) For all Λ ⋐ Z d and η ∈ Ω, µ Φ Λ ( |η) is a probability measure over (Ω, F); Properties (a)-(b) can be summarized by saying that µ Φ Λ is a probability kernel from F Λ c to F (c) For all Λ ⋐ Z d and A ∈ F Λ c , it holds µ Φ Λ (A|η) = 1 A (η). Probability kernels abiding by this assumption are called proper.
(d) The family of probability kernels µ Φ is compatible, namely, for all For any probability measure µ ∈ P(Ω, F) and Λ ⋐ Z d one may introduce a new probability measure µµ Φ Λ ∈ P(Ω, F) defined by A state ϕ µ is called infinite volume Gibbs state, denoted ϕ µ ∈ G(Φ), if the associated Radon probability measure µ fulfils Any ϕ µ ∈ G(Φ) is interpreted as a state in thermal equilibrium (at fixed inverse temperature β = 1) with respect to the formal Hamiltonian H Z d := Λ⋐Z d Φ Λ . This condition allows to characterize thermal equilibrium states on the whole configuration space Ω, without resorting to a finite configuration space Ω Λ , Λ ⋐ Z d . This approach provides a rather neat definition of phase transition: A phase transition occurs whenever G(Φ) is not a singleton. In the physical jargon this is equivalent to the existence of different equilibrium states in the infinite volume system.
The compatibility condition (6) with the Gibbs specification reflect the property that locally, i.e. on Ω Λ for Λ ⋐ Z d with fixed boundary conditions on Λ c , all equilibrium states coincide with the Gibbs measure (4). In physical terms, there are no phase transitions in a finite system.
As we shall see in the next section, in the current situation the symplectic structure on M allows for a different characterization of thermal equilibrium states by means of the (classical) KMS condition [17,18]. The goal of this paper is to prove that these two characterizations of thermal states coincide.

Classical KMS condition on lattice systems
This section is devoted to describe the classical KMS condition on the lattice system Ω = M Z d . This condition is used to select a particular class of states ϕ µ by means of the Poisson structure carried by M [1,2,17,18]. This is parallel to the notion of infinite volume Gibbs states (6) which is instead based on the Λ-local Gibbs measure (4).
Remark 4: For later convenience we recall the following property of the state ϕ Ω Λ on C(Ω Λ ) ≃ C(M |Λ| ) induced by the normalized Liouville measure µ M |Λ| on M |Λ| . Actually, ϕ Ω Λ is the unique state invariant under all Hamiltonian vector fields. In fact, if ψ Λ is a state on C(Ω Λ ) such that In what follows we will consider D = C ∞ loc (Ω) where C ∞ loc (Ω) is the algebra of smooth local functions: To define a Poisson bracket on C(Ω) we notice that, {f, g} Λ makes sense also for f, for all f ∈ C ∞ (Ω Λ 1 ) and g ∈ C ∞ (Ω Λ 2 ) where (π 2 1 ) * : C(Ω Λ 1 ) → C(Ω Λ 2 ) -in what follows we will not write π 2 1 since its use will be clear from the context. The Poisson bracket { , } on C(Ω) is defined by where f, g ∈ C ∞ loc (Ω) while Λ ⋐ Z d is such that f, g ∈ C ∞ (Ω Λ ). On account of Equation (9)  The next ingredient for stating the KMS condition is the choice of a vector field X on C(Ω) which plays the role of the infinitesimal generator of the dynamics on Ω. Notice that, despite this interpretation, the (classical) KMS condition does not rely on the existence of a dynamics integrating X. We now consider the vector field where the series converges on account of assumption (II). Notice that, {f, H Φ Λ } is only continuous on Ω, moreover, it is not a local function. By direct inspection we also have, for all Λ ′′ ⊆ Λ ⋐ Z d and f ∈ C ∞ (Ω Λ ′′ ), Equation (12) implies that, if f is Λ ′′ -local and Λ ′′ ⊆ Λ ⋐ Z d , then X Φ Λ (f ) does not change if we enlarge Λ. This allows to introduce a global vector field X Φ defined by Loosely speaking, one may think of X Φ as the Hamiltonian vector field associated to the formal Hamiltonian H = Λ⋐Z d Φ Λ . We can finally introduce the KMS condition associated with the vector field X Φ and the Poisson bracket { , }. The latter condition select a particular subclass of states ϕ µ on C(Ω) with a constraint on the expectation value of Poisson brackets.
In more details, a state ϕ µ on C(Ω) is called We denote by K(Φ) the convex set of KMS states.

Proof of the classical DLR-KMS correspondence
This section is devoted to the proof of Theorem 1 which asserts the equivalence between the DLR condition (6) and the KMS condition (14). The argument profits of the one presented in [2,Lem. 2].
To lighten the presentation, the proof of Theorem 1 is divided in two propositions.