DLR–KMS correspondence on lattice spin systems

Abstract

The Dobrushin–Lanford–Ruelle condition (Dobrushin in Theory Prob Appl 17:582–600, 1970. https://doi.org/10.1137/1115049; Lanford and Ruelle in Commun Math Phys 13:194–215, 1969. https://doi.org/10.1007/BF01645487) and the classical Kubo–Martin–Schwinger (KMS) condition (Gallavotti and Verboven in Nuov Cim B 28:274–286, 1975. https://doi.org/10.1007/BF02722820) 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.

1 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. Eq. (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 Sect. 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 ${\mathbb {R}}^d$ [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. Sect. 3 for the precise definition. Thus, it requires a Poisson structure on the system under investigation.

At this juncture, a natural question arises, namely whether 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 ${\mathbb {R}}^d$ 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 $\Omega =M^{{\mathbb {Z}}^d}$ is the space of functions ${\mathbb {Z}}^d\rightarrow M$ 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 ${\mathcal {G}}(\Phi )$ be the convex set of infinite-volume Gibbs states on $\Omega =M^{{\mathbb {Z}}^d}$—cf. Sect. 3 and Eq. (6). Let ${\mathcal {K}}(\Phi )$ be the convex set of $X^\Phi$-KMS states defined as per Eq. (14), cf. Sect. 3. Then, ${\mathcal {G}}(\Phi )={\mathcal {K}}(\Phi )$.$\lozenge$

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 ${\mathcal {G}}(\Phi )\subseteq {\mathcal {K}}(\Phi )$ is a direct computation, cf. Proposition 5. Instead, the converse inclusion requires a more refined analysis of the conditional probability $\varphi _\Lambda (\;|\eta )$ of a given KMS state $\varphi$ with respect to the complement $\Lambda ^c$ of a finite region $\Lambda \Subset {\mathbb {Z}}^d$. This is where the results of [8, 15] apply, ensuring that such conditional probabilities $\varphi _\Lambda (\;|\eta )$ coincide with the local Gibbs state $\varphi ^\Phi _\Lambda (\;|\eta )$.

The paper is organized as follows: Sect. 2 is devoted to a brief introduction to the lattice spin system of interest together with the precise definition of the DLR condition. Similarly, Sect. 3 considers the classical KMS condition in the same setting. Finally, Sect. 4 proves Theorem 1, cf. Propositions 5–6.

2 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 ${\mathbb {Z}}^d$ with values in a symplectic manifold M. The differential structure of M is needed for the formulation of the classical KMS condition, cf. Sect. 3.

Let $(M,\varsigma )$ be a compact, connected symplectic manifold with symplectic form $\varsigma$. 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 $\mu _M{:}{=}c_M\varsigma ^m/m!$ the induced volume form on M, where $\dim M=2m$, while the normalization constant $c_M>0$ is chosen so that $\int _M\mu _M=1$. For $f,g\in C^\infty (M)$, we denote by $\{f,g\}$ the induced Poisson bracket between f and g and set $X_f{:}{=}\{,f\}$ the Hamiltonian vector field associated with f.

The configuration space of interest is $\Omega {:}{=}M^{{\mathbb {Z}}^d}$ which is compact in the product topology. Moreover, $\Omega$ is also metrizable once we set

$$\begin{aligned} d_\Omega (\omega ,\eta ){:}{=}\sum _{i\in {\mathbb {Z}}^d}2^{-\Vert i\Vert _\infty }\frac{d_M(\omega _i,\eta _i)}{1+d_M(\omega _i,\eta _i)}\,, \end{aligned}$$

where $\Vert i\Vert _\infty {:}{=}\sup \nolimits _{k\in \{1,\ldots ,d\}}|i_k|$.

If $\Lambda \subset {\mathbb {Z}}^d$, we set $\Omega _\Lambda {:}{=}M^\Lambda$ and denote by

$$\begin{aligned} \pi _\Lambda :\Omega \rightarrow \Omega _\Lambda \qquad \pi _\Lambda (\omega ){:}{=}\omega _\Lambda {:}{=}\omega |_\Lambda \,, \end{aligned}$$

the projection over $\Omega _\Lambda$. With a standard slight abuse of notation, we will set, given $\omega _\Lambda \in \Omega _\Lambda$ and $\omega _{\Lambda ^c}\in \Omega _{\Lambda ^c}$, $\Lambda \subset {\mathbb {Z}}^d$, $\omega _\Lambda \omega _{\Lambda ^c}{:}{=}\omega _\Lambda \otimes \omega _{\Lambda ^c}\in \Omega$. For a given $\Lambda \subset {\mathbb {Z}}^d$, we will write $\Lambda \Subset {\mathbb {Z}}^d$ if $|\Lambda |<+\infty$: Notice that in this latter case $C(\Omega _\Lambda )\simeq C(M^{|\Lambda |})$ where $M^{|\Lambda |}$ is again a connected compact symplectic manifold with symplectic form $\varsigma _\Lambda {:}{=}\oplus _{i\in \Lambda }\varsigma$.

The algebra of observables over the configuration space $\Omega$ is identified with the commutative $C^*$-algebra $C(\Omega )$ equipped with the supremum norm $\Vert f\Vert _\infty {:}{=}\sup _{\omega \in \Omega }|f(\omega )|$. This space is best described in terms of (continuous) local functions. In particular, $f\in C(\Omega )$ is called (continuous) local function, denoted $f\in C_{\textsc {loc}}(\Omega )$, if there exists $\Lambda \Subset {\mathbb {Z}}^d$ and $f_\Lambda \in C(\Omega _\Lambda )\simeq C(M^{|\Lambda |})$ such that $f(\omega )=f_\Lambda (\omega _\Lambda )$ for all $\omega \in \Omega$. Then, for all $f\in C(\Omega )$ there exists a sequence $(f_n)_{n\in {\mathbb {N}}}$ of local functions such that $\lim _{n\rightarrow \infty }\Vert f-f_n\Vert _\infty =0$ [16, Lem. 6.21] (e.g., one considers $\eta \in \Omega$ and a sequence $(\Lambda _n)_{n\in {\mathbb {N}}}$ of increasing subsets $\Lambda _n\subset \Lambda _{n+1}\Subset {\mathbb {Z}}^d$ such that $\cup _n\Lambda _n=\Omega$ and set $f_n(\omega ){:}{=}f(\omega _{\Lambda _n}\eta _{\Lambda _n^c})$ for all $n\in {\mathbb {N}}$). For this reason, $C(\Omega )$ is usually referred to as the $C^*$-algebra of (continuous) quasi-local functions.

Per definition, a state on $C(\Omega )$ is a normalized linear, positive functional $\varphi :C(\Omega )\rightarrow {\mathbb {C}}$. By the Riesz–Markov–Kakutani theorem, every state $\varphi$ on $C(\Omega )$ is completely described by a Radon probability measure $\mu \in {\mathcal {P}}(\Omega ,{\mathcal {F}})$ on $(\Omega ,{\mathcal {F}})$—here ${\mathcal {F}}$ denotes the Borel $\sigma$-algebra over $\Omega$. In particular, $\varphi \equiv \varphi _\mu$ and

$$\begin{aligned} \varphi _\mu (f)=\int _\Omega f\textrm{d}\mu \qquad \forall f\in C(\Omega )\,. \end{aligned}$$

(1)

For later convenience, we recall the definition of the $\sigma$-algebra ${\mathcal {F}}_\Lambda$ over $\Omega$ of events occurring in $\Lambda \subset {\mathbb {Z}}^d$:

$$\begin{aligned} {\mathcal {F}}_\Lambda {:}{=} \sigma \left( \bigcup _{\begin{array}{c} \Lambda '\Subset {\mathbb {Z}}^d\\ \Lambda '\subset \Lambda \end{array}}\{\pi _{\Lambda '}^{-1}(E)\,|\,E\in {\mathcal {B}}_{\Lambda '}\}\right) \,, \end{aligned}$$

where $\sigma (A)$ is the $\sigma$-algebra generated by the collection A, while, if $\Lambda \Subset {\mathbb {Z}}^d$, ${\mathcal {B}}_\Lambda$ denotes the Borel $\sigma$-algebra over $\otimes _{i\in \Lambda } M\simeq M^{|\Lambda |}$. Notice that

$$\begin{aligned} {\mathcal {F}}_\Lambda =\{\pi _{\Lambda }^{-1}(E)\,|\,E\in {\mathcal {B}}_{\Lambda }\} \qquad \forall \Lambda \Subset {\mathbb {Z}}^d\,, \end{aligned}$$

while ${\mathcal {F}}_{{\mathbb {Z}}^d}$ coincides with the Borel $\sigma$-algebra ${\mathcal {F}}$ over $\Omega$. A function $f:\Omega \rightarrow {\mathbb {C}}$ is ${\mathcal {F}}_\Lambda$-measurable if and only if f is $\Lambda {\textbf {-local}}$, namely there exists a measurable function $f_\Lambda :\Omega _\Lambda \rightarrow {\mathbb {C}}$ such that $f(\omega )=f_\Lambda (\omega _\Lambda )$ for all $\omega \in \Omega$ [16, Lem. 6.3].

Among all possible states $\varphi _\mu$ on $C(\Omega )$, we will be interested in the infinite-volume Gibbs states. The definition of the latter requires to introduce a potential, namely a collection $\Phi =\{\Phi _\Lambda \}_{\Lambda \Subset {\mathbb {Z}}^d}$, where $\Phi _\Lambda \in C(\Omega )$ is ${\mathcal {F}}_\Lambda$-measurable for all $\Lambda \Subset {\mathbb {Z}}^d$. In what follows, we will assume the following technical assumptions on $\Phi$:

• (I) For all $\Lambda \Subset {\mathbb {Z}}^d$, $\Phi _\Lambda \in C^1(\Omega _\Lambda )\simeq C^1(M^{|\Lambda |})$.

• (II) For all $i\in {\mathbb {Z}}^d$

  $$\begin{aligned} \sum _{\begin{array}{c} \Lambda \Subset {\mathbb {Z}}^d\\ i\in \Lambda \end{array}}\Vert \Phi _\Lambda \Vert _{C^1(\Omega _\Lambda )}<+\infty \,, \end{aligned}$$

  (2)

  where $\Vert \Phi _\Lambda \Vert _{C^1(M_\Lambda )}$ denotes the supremum of $\Phi _\Lambda$ and $\textrm{d}\Phi _\Lambda$ over $\Omega _\Lambda$; thus, it coincides with the norm of the $C^*$-algebra $C^1(M^{|\Lambda |})$.

Within these assumptions, we can define

$$\begin{aligned} H_\Lambda ^\Phi {:}{=}\sum _{\begin{array}{c} \Lambda '\Subset {\mathbb {Z}}^d\\ \Lambda '\cap \Lambda \ne \emptyset \end{array}}\Phi _{\Lambda '}\,, \end{aligned}$$

(3)

which is a well-defined element in $C(\Omega )$ on account of assumption (II). For a fixed $\eta \in \Omega$, one then introduces the probability measure $\mu ^\Phi _\Lambda (\;|\eta )$ on $(\Omega ,{\mathcal {F}})$ defined by

$$\begin{aligned} \mu ^\Phi _\Lambda (A|\eta ) {:}{=}\frac{1}{Z^\Phi _{\eta ,\Lambda }}\int _{\Omega _\Lambda } 1_A(\omega _\Lambda \eta _{\Lambda ^c}) e^{-H^\Phi _\Lambda (\omega _\Lambda \eta _{\Lambda ^c})}\textrm{d}\mu _{M^{|\Lambda |}}(\omega _\Lambda ) \quad \forall A\in {\mathcal {F}}\,, \end{aligned}$$

(4)

where $\mu _{M^{|\Lambda |}}{:}{=}\otimes _{i\in \Lambda }\mu _M$ denotes the (normalized) Liouville volume form over $M^{|\Lambda |}$ while $Z^\Phi _{\eta ,\Lambda }$ is a normalization constant—we have implicitly identified $\Omega _\Lambda \simeq M^{|\Lambda |}$. We will refer to $\mu ^\Phi _\Lambda (\;|\eta )$ as the $\Lambda$-Gibbs measure associated with the potential $\Phi$ and with boundary condition $\eta$ (at fixed inverse temperature $\beta =1$). We will denote by $\varphi ^\Phi _\Lambda (\;|\eta )$ the associated state on $C(\Omega )$ which will be referred to as the $\Lambda$-Gibbs state associated with $\Phi$ and $\eta$.

Remark 2

The assumptions on the potential $\Phi$ are slightly stronger than those usually imposed [16, §6]. As a matter of fact, one usually requires

$$\begin{aligned} \sum _{\begin{array}{c} \Lambda \Subset {\mathbb {Z}}^d\\ i\in \Lambda \end{array}} \Vert \Phi _\Lambda \Vert _{C(\Omega _\Lambda )}<+\infty \,, \end{aligned}$$

which ensures the well-definiteness of $H^\Phi _\Lambda$ as per Eq. (3). The stronger assumptions (I)–(II) are necessary for discussing the Poisson bracket on $\Omega$—cf. Sect. 3. $\lozenge$

The collection $\mu ^\Phi {:}{=}\{\mu ^\Phi _\Lambda \}_{\Lambda \Subset {\mathbb {Z}}^d}$ is called Gibbs specification and enjoys the following properties:

• (a) For all $\Lambda \Subset {\mathbb {Z}}^d$ and $\eta \in \Omega$, $\mu ^\Phi _\Lambda (\;|\eta )$ is a probability measure over $(\Omega ,{\mathcal {F}})$;

• (b) For all $\Lambda \Subset {\mathbb {Z}}^d$ and $A\in {\mathcal {F}}$, the function $\Omega \ni \eta \mapsto \mu ^\Phi _\Lambda (A|\eta )$ is ${\mathcal {F}}_{\Lambda ^c}$-measurable. Properties (a)-(b) can be summarized by saying that $\mu ^\Phi _\Lambda$ is a probability kernel from ${\mathcal {F}}_{\Lambda ^c}$ to ${\mathcal {F}}$ [21, §5] (also known as transition measures [7, Def. 10.7.1]).

• (c) For all $\Lambda \Subset {\mathbb {Z}}^d$ and $A\in {\mathcal {F}}_{\Lambda ^c}$, it holds $\mu ^\Phi _\Lambda (A|\eta )=1_A(\eta )$. Probability kernels abiding by this assumption are called proper.

• (d) The family of probability kernels $\mu ^\Phi$ is compatible, namely, for all $\Lambda _1\subset \Lambda _2\Subset {\mathbb {Z}}^d$ it holds $\mu ^\Phi _{\Lambda _2}\mu ^\Phi _{\Lambda _1}=\mu ^\Phi _{\Lambda _2}$, where

  $$\begin{aligned} \mu ^\Phi _{\Lambda _2}\mu ^\Phi _{\Lambda _1}:{\mathcal {F}}\times \Omega \ni (A,\eta )\mapsto \int _\Omega \mu ^\Phi _{\Lambda _1}(A|\omega )\textrm{d}\mu ^\Phi _{\Lambda _2}(\omega |\eta )\,. \end{aligned}$$

For any probability measure $\mu \in {\mathcal {P}}(\Omega ,{\mathcal {F}})$ and $\Lambda \Subset {\mathbb {Z}}^d$, one may introduce a new probability measure $\mu \mu ^\Phi _\Lambda \in {\mathcal {P}}(\Omega ,{\mathcal {F}})$ defined by

$$\begin{aligned} \mu \mu ^\Phi _\Lambda (A) {:}{=}\int _\Omega \mu ^\Phi _\Lambda (A|\eta )\textrm{d}\mu (\eta )\,. \end{aligned}$$

(5)

A state $\varphi _\mu$ is called infinite-volume Gibbs state, denoted $\varphi _\mu \in {\mathcal {G}}(\Phi )$, if the associated Radon probability measure $\mu$ fulfills

$$\begin{aligned} \mu \mu ^\Phi _\Lambda =\mu \qquad \forall \Lambda \Subset {\mathbb {Z}}^d\,. \end{aligned}$$

(6)

Any $\varphi _\mu \in {\mathcal {G}}(\Phi )$ is interpreted as a state in thermal equilibrium (at fixed inverse temperature $\beta =1$) with respect to the formal Hamiltonian $H_{{\mathbb {Z}}^d}{:}{=}\sum _{\Lambda \Subset {\mathbb {Z}}^d}\Phi _\Lambda$. This condition allows to characterize thermal equilibrium states on the whole configuration space $\Omega$, without resorting to a finite configuration space $\Omega _\Lambda$, $\Lambda \Subset {\mathbb {Z}}^d$. This approach provides a rather neat definition of phase transition: A phase transition occurs whenever ${\mathcal {G}}(\Phi )$ 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 reflects the property that locally, i.e., on $\Omega _\Lambda$ for $\Lambda \Subset {\mathbb {Z}}^d$ with fixed boundary conditions on $\Lambda ^c$, 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.

Remark 3

Condition (6) can be interpreted by saying that, for all $\Lambda \Subset {\mathbb {Z}}^d$, $\mu ^\Phi _\Lambda (\;|\eta )$ is a regular conditional measure for $\mu$ with respect to ${\mathcal {F}}_{\Lambda ^c}$ [7, Def. 10.4.1]. For later convenience, we stress that any probability measure $\mu \in {\mathcal {P}}(\Omega ,{\mathcal {F}})$ arising from a state $\varphi _\mu$ on $C(\Omega )$ has a regular conditional measure $\mu _\Lambda$ with respect to ${\mathcal {F}}_{\Lambda ^c}$ for all $\Lambda \Subset {\mathbb {Z}}^d$. This follows from [7, Cor. 10.4.7], together with the observation that: (a) since M is metrizable, $\Omega$ is second-countable (and also a Polish space because it is metrizable); therefore, ${\mathcal {F}}$ is countably generated; (b) since $\Omega$ is compact, the Riesz–Markov–Kakutani theorem ensures that the probability measure $\mu$ associated with any state $\varphi _\mu$ is Radon; thus, it has an approximating compact class (made by compact subsets of $\Omega$). Thus, for all $\Lambda \Subset {\mathbb {Z}}^d$ there exists

$$\begin{aligned} \mu _\Lambda :{\mathcal {F}}\times \Omega \ni (A,\eta )\mapsto \mu _\Lambda (A|\eta )\in [0,1]\,, \end{aligned}$$

such that $\mu _\Lambda (\;|\eta )\in {\mathcal {P}}(\Omega ,{\mathcal {F}})$ for $\mu$-almost all $\eta \in \Omega$ and $\mu _\Lambda (A|\;)$ is ${\mathcal {F}}_{\Lambda ^c}$-measurable, moreover,

$$\begin{aligned} \mu (A\cap B)=\int _B \mu _\Lambda (A|\eta )\textrm{d}\mu (\eta ) \qquad \forall A\in {\mathcal {F}}\,,\, \forall B\in {\mathcal {F}}_{\Lambda ^c}\,. \end{aligned}$$

(7)

Thus, $\varphi _\mu \in {\mathcal {G}}(\Phi )$ if and only if $\mu _\Lambda (\;|\eta )=\mu ^\Phi _\Lambda (\;|\eta )$ for $\mu$-almost all $\eta \in \Omega$ and for all $\Lambda \Subset {\mathbb {Z}}^d$. It is worth observing that since ${\mathcal {F}}$ is countably generated, $\mu _\Lambda (\;|\eta )$ is unique for all $\eta \in \Omega$ up to a $\mu$-null set [7, Lem. 10.4.3].

Moreover, since $\mu _\Lambda (\;|\eta )$ is $\mu$-integrable, a density argument shows that, for all $f\in C(\Omega )$,

$$\begin{aligned} \varphi _\mu (f)=\int _\Omega \varphi _\Lambda (f|\eta )\textrm{d}\mu (\eta )\,, \end{aligned}$$

where $\varphi _\Lambda (\;|\eta )$ denotes the state on $C(\Omega )$ associated with $\mu _\Lambda (\;|\eta )$. In particular, if $f,g\in C(\Omega )$ and g is ${\mathcal {F}}_{\Lambda ^c}$-measurable, $\Lambda \Subset {\mathbb {Z}}_+^d$, then

$$\begin{aligned} \varphi _\mu (fg)=\int _\Omega \varphi _\Lambda (f|\eta )g(\eta )\textrm{d}\mu (\eta )\,. \end{aligned}$$

(8)

Finally, it is worth observing that as a consequence of [7, Cor. 10.4.10] $\varphi _\Lambda (\;|\eta )$ is concentrated on $\pi _{\Lambda ^c}^{-1}(\eta _{\Lambda ^c})$ for all $\Lambda \Subset {\mathbb {Z}}^d$ and $\mu$-almost all $\eta \in \Omega$. Notice that $\pi _{\Lambda ^c}^{-1}(\eta _{\Lambda ^c})\in {\mathcal {F}}_{\Lambda ^c}$ because $\pi _{\Lambda ^c}^{-1}(\eta _{\Lambda ^c})=\bigcap _{n\in {\mathbb {N}}}\pi _{\Lambda _n\setminus \Lambda }^{-1}(\eta _{\Lambda _n\setminus \Lambda ^c})$ where $(\Lambda _n)_{n\in {\mathbb {N}}}$ is an increasing sequence of finite subsets $\Lambda _n\Subset {\mathbb {Z}}^d$ such that $\bigcup _{n\in {\mathbb {N}}}\Lambda _n={\mathbb {Z}}^d$. $\lozenge$

3 Classical KMS condition on lattice systems

This section is devoted to describe the classical KMS condition on the lattice system $\Omega =M^{{\mathbb {Z}}^d}$. This condition is used to select a particular class of states $\varphi _\mu$ 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 $\Lambda$-local Gibbs measure (4).

To set the stage, we recall that for all $\Lambda \Subset {\mathbb {Z}}^d$, $\Omega _\Lambda \simeq M^{|\Lambda |}$ is a compact, connected symplectic manifold with $\varsigma _\Lambda {:}{=}\oplus _{i\in \Lambda }\varsigma$. For $f,g\in C^\infty (\Omega _\Lambda )$, we denote by $\{f,g\}_\Lambda$ the associated Poisson bracket.

Remark 4

For later convenience, we recall the following property of the state $\varphi _{\Omega _\Lambda }$ on $C(\Omega _\Lambda )\simeq C(M^{|\Lambda |})$ induced by the normalized Liouville measure $\mu _{M^{|\Lambda |}}$ on $M^{|\Lambda |}$. Actually, $\varphi _{\Omega _\Lambda }$ is the unique state invariant under all Hamiltonian vector fields. In fact, if $\psi _\Lambda$ is a state on $C(\Omega _\Lambda )$ such that

$$\begin{aligned} \psi _\Lambda (\{f,g\}_\Lambda )=0 \qquad \forall f,g\in C^\infty (\Omega _\Lambda )\,, \end{aligned}$$

then $\psi _\Lambda =\varphi _{\Omega _\Lambda }$, i.e.,

$$\begin{aligned} \psi _\Lambda (f)=\int _{\Omega _\Lambda }f(\omega _\Lambda )\textrm{d}\mu _{M^{|\Lambda |}}(\omega _\Lambda )\,. \end{aligned}$$

The proof of this result can be found in [2, Lem. 1] as well as in [8, Cor. 2.7]. It is based on a localization process in coordinated charts equipped with Darboux coordinates—so that one can write $\partial _a f=\{f,x^a\}_\Lambda$—together with the observation that the Lebesgue measure on ${\mathbb {R}}^d$ is the unique (up to multiplicative constants) measure vanishing on derivatives of compactly supported functions [8, Lem. 2.6]. $\lozenge$

The KMS condition requires to endow $C(\Omega )$ with a Poisson bracket. We recall that a Poisson bracket $\{\,,\;\}$ on $C(\Omega )$ is a skew-symmetric bilinear map $\{\;,\;\}:D\times D\rightarrow D$ defined on a dense sub-algebra D of $C(\Omega )$ such that $\{\;,f\}$ is a derivation on D for all $f\in D$ and fulfilling the Jacobi identity

$$\begin{aligned} \{\{f,g\},h\} =\{\{f,h\},g\} +\{f,\{g,h\}\} \qquad \forall f,g,h\in D\,. \end{aligned}$$

In what follows, we will consider $D=C^\infty _{\textsc {loc}}(\Omega )$ where $C^\infty _{\textsc {loc}}(\Omega )$ is the algebra of smooth local functions: $f\in C^\infty _{\textsc {loc}}(\Omega$) if there exists $\Lambda \Subset {\mathbb {Z}}^d$ and $f_\Lambda \in C^\infty (\Omega _\Lambda )$ such that $f(\omega )=f_\Lambda (\omega _\Lambda )$.

To define a Poisson bracket on $C(\Omega )$, we notice that $\{f,g\}_\Lambda$ makes sense also for $f,g\in C^\infty (\Omega _{\Lambda '})$ for all $\Lambda \subset \Lambda '\Subset {\mathbb {Z}}^d$. As a matter of fact, if $\Lambda _1\subset \Lambda _2\Subset {\mathbb {Z}}^d$, denoting $\pi ^2_1:\Omega _{\Lambda _2}\rightarrow \Omega _{\Lambda _1}$ the projection $\omega _{\Lambda _2}=\omega _{\Lambda _1}\omega _{\Lambda _2\setminus \Lambda _1}\mapsto \omega _{\Lambda _1}$, we find

$$\begin{aligned} \{(\pi ^2_1)^*f,g\}_{\Lambda _2} =\{(\pi ^2_1)^*f,g\}_{\Lambda _1}\,, \end{aligned}$$

(9)

for all $f\in C^\infty (\Omega _{\Lambda _1})$ and $g\in C^\infty (\Omega _{\Lambda _2})$ where $(\pi ^2_1)^*:C(\Omega _{\Lambda _1})\rightarrow C(\Omega _{\Lambda _2})$—in what follows we will not write $\pi ^2_1$ since its use will be clear from the context. The Poisson bracket $\{,\;\}$ on $C(\Omega )$ is defined by

$$\begin{aligned} \{f,g\}{:}{=}\{f,g\}_{\Lambda }\,, \end{aligned}$$

(10)

where $f,g\in C^\infty _{\textsc {loc}}(\Omega )$ while $\Lambda \Subset {\mathbb {Z}}^d$ is such that $f,g\in C^\infty (\Omega _{\Lambda })$. On account of Eq. (9), the value of $\{f,g\}_{\Lambda }$ does not change if we enlarge $\Lambda$. By direct inspection, $\{,\;\}$ defines a Poisson bracket on $C(\Omega )$.

The next ingredient for stating the KMS condition is the choice of a vector field X on $C(\Omega )$ which plays the role of the infinitesimal generator of the dynamics on $\Omega$. 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

$$\begin{aligned} X^\Phi _{\Lambda }:C^\infty _{\textsc {loc}}(\Omega )\rightarrow C(\Omega ) \qquad X^\Phi _\Lambda (f) {:}{=}\{f,H^\Phi _\Lambda \} =\sum _{\begin{array}{c} \Lambda '\Subset {\mathbb {Z}}^d\\ \Lambda '\cap \Lambda \ne \emptyset \end{array}} \{f,\Phi _{\Lambda '}\}_{\Lambda '}\,, \end{aligned}$$

(11)

where the series converges on account of assumption (II). Notice that $\{f,H^\Phi _\Lambda \}$ is only continuous on $\Omega$; moreover, it is not a local function.

By direct inspection, we also have, for all $\Lambda ''\subseteq \Lambda \Subset {\mathbb {Z}}^d$ and $f\in C^\infty (\Omega _{\Lambda ''})$,

$$\begin{aligned} \{f,H^\Phi _\Lambda \} =\sum _{\begin{array}{c} \Lambda '\Subset {\mathbb {Z}}^d\\ \Lambda '\cap \Lambda \ne \emptyset \end{array}} \{f,\Phi _{\Lambda '}\}_{\Lambda '\cap \Lambda ''} =\sum _{\begin{array}{c} \Lambda '\Subset {\mathbb {Z}}^d\\ \Lambda '\cap \Lambda ''\ne \emptyset \end{array}} \{f,\Phi _{\Lambda '}\}_{\Lambda '\cap \Lambda ''} =\{f,H^\Phi _{\Lambda ''}\}\,. \end{aligned}$$

(12)

Equation (12) implies that if f is $\Lambda ''$-local and $\Lambda ''\subseteq \Lambda \Subset {\mathbb {Z}}^d$, then $X^\Phi _\Lambda (f)$ does not change if we enlarge $\Lambda$. This allows to introduce a global vector field $X^\Phi$ defined by

$$\begin{aligned} X^\Phi :C^\infty _{\textsc {loc}}(\Omega )\rightarrow C(\Omega ) \qquad X^\Phi (f)=\{f,H^\Phi _\Lambda \} \qquad \forall f\in C^\infty (\Omega _\Lambda )\,. \end{aligned}$$

(13)

Loosely speaking, one may think of $X^\Phi$ as the Hamiltonian vector field associated to the formal Hamiltonian $H=\sum _{\Lambda \Subset {\mathbb {Z}}^d}\Phi _\Lambda$.

We can finally introduce the KMS condition associated with the vector field $X^\Phi$ and the Poisson bracket $\{,\;\}$. The latter condition selects a particular subclass of states $\varphi _\mu$ on $C(\Omega )$ with a constraint on the expectation value of Poisson brackets.

In more details, a state $\varphi _\mu$ on $C(\Omega )$ is called $X^\Phi$-KMS state if

$$\begin{aligned} \varphi _\mu (\{f,g\})=\varphi _\mu (gX^\Phi (f)) \qquad \forall f,g\in C^\infty _{\textsc {loc}}(\Omega )\,. \end{aligned}$$

(14)

We denote by ${\mathcal {K}}(\Phi )$ the convex set of KMS states.

4 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 into two propositions.

Proposition 5

It holds ${\mathcal {G}}(\Phi )\subseteq {\mathcal {K}}(\Phi )$, namely every infinite-volume Gibbs state is also a $X^\Phi$-KMS state. $\lozenge$

Proof

Let $\varphi _\mu \in {\mathcal {G}}(\Phi )$, $\Lambda \Subset {\mathbb {Z}}^d$ and $f,g\in C^\infty (\Omega _\Lambda )$. Equation (6) implies

$$\begin{aligned} \varphi _\mu (\{f,g\}) =\int \varphi ^\Phi _\Lambda (\{f,g\}_\Lambda |\eta )\textrm{d}\mu (\eta )\,. \end{aligned}$$

By direct inspection, we also have

$$\begin{aligned} \varphi ^\Phi _\Lambda (\{f,g\}_\Lambda |\eta )&{:}{=}\frac{1}{Z^\Phi _{\eta ,\Lambda }}\int _{\Omega _\Lambda } \{f,g\}_\Lambda (\omega _\Lambda \eta _{\Lambda ^c}) e^{-H^\Phi _\Lambda (\omega _\Lambda \eta _{\Lambda ^c})}\textrm{d}\mu _{M^{|\Lambda |}}(\omega _\Lambda ) \\&=\frac{1}{Z^\Phi _{\eta ,\Lambda }}\int _{\Omega _\Lambda } \{f,g e^{-H^\Phi _\Lambda }\}_\Lambda (\omega _\Lambda \eta _{\Lambda ^c})\textrm{d}\mu _{M^{|\Lambda |}}(\omega _\Lambda ) \\&\quad +\frac{1}{Z^\Phi _{\eta ,\Lambda }}\int _{\Omega _\Lambda } g\{f,H_\Lambda ^\Phi \}_\Lambda (\omega _\Lambda \eta _{\Lambda ^c}) e^{-H^\Phi _\Lambda (\omega _\Lambda \eta _{\Lambda ^c})}\textrm{d}\mu _{M^{|\Lambda |}}(\omega _\Lambda ) \\&=\varphi ^\Phi _\Lambda (g\{f,H^\Phi _\Lambda \}_\Lambda |\eta ) =\varphi ^\Phi _\Lambda (gX^\Phi (f)|\eta )\,, \end{aligned}$$

where in the last line we used that $\varphi _{\Omega _\Lambda }(\{h_1,h_2\}_\Lambda )=0$ for all $h_1,h_2\in C^1(\Omega _\Lambda )$. Overall, we have

$$\begin{aligned} \varphi _\mu (\{f,g\}) =\int _\Omega \varphi ^\Phi _\Lambda (gX^\Phi (f)|\eta )\textrm{d}\mu (\eta ) =\varphi _\mu (gX^\Phi (f))\,, \end{aligned}$$

and therefore, $\varphi _\mu \in {\mathcal {K}}(\Phi )$. $\square$

Proposition 6

It holds ${\mathcal {K}}(\Phi )\subseteq {\mathcal {G}}(\Phi )$, namely every $X^\Phi$-KMS state is an infinite-volume Gibbs state. $\lozenge$

Proof

Using Eq. (7), it is enough to prove that $\varphi _\Lambda (\;|\eta )=\varphi ^\Phi _\Lambda (\;|\eta )$ for $\mu$-almost all $\eta \in \Omega$ and all $\Lambda \Subset {\mathbb {Z}}^d$. We recall that for each $f\in C(\Omega )$, the function $\eta \mapsto \varphi _\Lambda (f|\eta )$ is ${\mathcal {F}}_{\Lambda ^c}$-measurable; thus, it only depends on $\eta _{\Lambda ^c}$. Moreover, $\varphi _\Lambda (\;|\eta )$ is concentrated on $\pi _{\Lambda ^c}^{-1}(\eta _{\Lambda ^c})$ $\mu$-almost all $\eta \in \Omega$.

Let $\Lambda \Subset {\mathbb {Z}}^d$, $\Lambda '\Subset \Lambda ^c$ and consider $f\in C^\infty (\Omega _\Lambda )$, $h\in C^\infty (\Omega _{\Lambda '})$, $g\in C^\infty _{\textsc {loc}}(\Omega )$. Then, Eq. (9) implies that

$$\begin{aligned} h\{f,g\}=\{f,gh\}\,, \end{aligned}$$

since $\{f,h\}=0$. Moreover, Eq. (8) and the KMS condition (14) imply

$$\begin{aligned}{} & {} \int _\Omega \varphi _\Lambda (\{f,g\}|\eta )h(\eta )\textrm{d}\mu (\eta ) =\varphi _\mu (h\{f,g\}) =\varphi _\mu (\{f,gh\}) \nonumber \\{} & {} =\varphi _\mu (ghX^\Phi (f)) =\int _\Omega \varphi _\Lambda (g\{f,H^\Phi _\Lambda \}|\eta )h(\eta )\textrm{d}\mu (\eta ). \end{aligned}$$

(15)

The arbitrariness of $h\in C^\infty (\Omega _{\Lambda '})\subset C(\Omega _{\Lambda ^c})$ and of $\Lambda '\Subset \Lambda ^c$ entails that, by an approximation argument,

$$\begin{aligned} \int _\Omega \big [\varphi _\Lambda (\{f,g\}_\Lambda |\eta )-\varphi _\Lambda (g\{f,H^\Phi _\Lambda \}_\Lambda |\eta )\big ]h(\eta )\textrm{d}\mu (\eta ) =0 \qquad \forall h\in C(\Omega _{\Lambda ^c})\,. \end{aligned}$$

Since $\eta \mapsto \varphi _\Lambda (f|\eta )$ in ${\mathcal {F}}_{\Lambda ^c}$-measurable, it follows that there exists $N_{f,g}\in {\mathcal {F}}_{\Lambda ^c}\subset {\mathcal {F}}$ with $\mu (N_{f,g})=0$ such that

$$\begin{aligned} \varphi _\Lambda (\{f,g\}_\Lambda |\eta ) =\varphi _\Lambda (g\{f,H^\Phi _\Lambda \}_\Lambda |\eta ) \qquad \forall \eta \in \Omega \setminus N_{f,g}\,. \end{aligned}$$

(16)

To proceed, we need to cope with the f, g-dependence of $N_{f,g}$. However, since $C(\Omega _\Lambda )$ is separable, we may choose $N_{f,g}$ independently on f, g; indeed, it suffices to consider Eq. (16) for f, g on a countable dense set of $C^\infty (\Omega _\Lambda )$. This leads to countably many $\mu$-null sets whose union leads to the $\mu$-null set N of interest.

We now prove that condition (16) implies that $\varphi _\Lambda (\;|\eta )=\varphi ^\Phi _\Lambda (\;|\eta )$ for all $\eta \in \Omega \setminus N$. For each $\eta \in \Omega \setminus N$, we consider the linear positive functional $\psi _\Lambda$ on $C(\Omega _\Lambda )$ defined by

$$\begin{aligned} \psi _\Lambda (f) {:}{=}\varphi _\Lambda (fe^{H^\Phi _\Lambda }|\eta ) \qquad \forall f\in C(\Omega _\Lambda )\,. \end{aligned}$$

By direct inspection, we have, for all $f,g\in C^\infty (\Omega _\Lambda )$,

$$\begin{aligned} \psi _\Lambda (\{f,g\}_\Lambda ){} & {} =\varphi _\Lambda (\{f,g\}_\Lambda e^{H^\Phi _\Lambda }) \\{} & {} =\varphi _\Lambda (\{f,ge^{H^\Phi _\Lambda }\}_\Lambda ) -\varphi _\Lambda (g\{f,H^\Phi _\Lambda \}_\Lambda e^{H^\Phi _\Lambda }) =0, \end{aligned}$$

where we applied Eq. (16).

Thus, $\psi _\Lambda$ is a linear, positive functional on $C(\Omega _\Lambda )\simeq C(M^{|\Lambda |})$ which is invariant under all Hamiltonian vector fields. Remark 4 implies that $\widehat{\psi }_\Lambda =\varphi _{\Omega _\Lambda }$ where $\widehat{\psi }_\Lambda (f)=\psi _\Lambda (f)/\psi _\Lambda (1)$. Finally, since $\varphi _\Lambda (\;|\eta )$ is concentrated on $\pi _{\Lambda ^c}^{-1}(\eta _{\Lambda ^c})$, we have

$$\begin{aligned} \varphi _\Lambda (f|\eta ) =\widehat{\psi }_\Lambda (f_\eta e^{-H^\Phi _{\Lambda ,\eta }}) =\varphi ^\Phi _\Lambda (f|\eta )\,, \end{aligned}$$

where $f_\eta e^{-H^\Phi _{\Lambda ,\eta }}\in C(\Omega _\Lambda )$ is defined by $(f_\eta e^{-H^\Phi _{\Lambda ,\eta }})(\omega _\Lambda ){:}{=}(fe^{-H^\Phi _\Lambda })(\omega _\Lambda \eta _{\Lambda ^c})$. $\square$