# Clustering of correlations in extremal thermal states

## Background

Consider a quantum system described by an algebra $$\mathcal{A}$$ of local observables, which are supported on subsets of the lattice $$\Lambda = \mathbb{Z}^{d}$$. Given an observable $$A$$, let $$A(t) = e^{i H t} A e^{-i H t}$$ denote its time-evolution by the Hamiltonian $$H$$. Let us restrict in this question to short-range Hamiltonians, in the sense that $$H$$ can be written as $$H = \sum_{X} h_{X}$$, where the local term $$h_{X}$$ is supported on $$X \subset \Lambda$$, and $$\parallel h_{X} \parallel$$ is bounded and decays sufficiently rapidly with $$\mathrm{diam}(X)$$.

A state $$\rho$$ of this system is a linear functional on $$\mathcal{A}$$, whose action on an observable $$A$$ we will write as $$\langle A \rangle_{\rho}$$. We say that $$\rho$$ is a $$\beta$$-KMS state if for any local observables $$A,B\in \mathcal{A}$$ we have

$$\langle A(t) B \rangle_{\rho} = \langle B A(t + i \beta) \rangle_{\rho}.$$

We further say that $$\rho$$ is an extremal $$\beta$$-KMS state if it cannot be written as a convex combination

$$\rho = \lambda \rho_{1} + (1-\lambda) \rho_{2},$$

with $$0 < \lambda < 1$$, where $$\rho_{1}$$ and $$\rho_{2}$$ are both $$\beta$$-KMS states.

Finally, we say that a state $$\rho$$ satisfies cluster decomposition if, for local observables $$A_{X}$$ and $$B_{Y}$$ respectively supported on regions $$X,Y \subset \Lambda$$, we have the decay to zero of the connected correlation function,

$$\left| \langle A_{X} B_{Y} \rangle_{\rho} - \langle A_{X} \rangle_{\rho} \langle B_{Y} \rangle_{\rho} \right| \to 0,$$

as the distance between their supports $$d(X,Y) \to \infty$$.

## Question

Is it the case that all extremal $$\beta$$-KMS states of short-range Hamiltonians satisfy cluster decomposition? Does the answer depend on $$d$$ or $$\beta$$?

If this has not been proved, I would still be interested in knowing whether it is expected to be true, or if there are known counterexamples.

I am particularly interested in the low-temperature limit $$\beta \to \infty$$, where e.g. symmetry-breaking might be relevant. I also do not mind how rapidly the correlations go to zero.

• You should look at Simon's The Statistical Mechanics of Lattice Gases (see, for instance, Corollary IV.1.8). Jun 4, 2021 at 5:56

Definition: Let $$\mathcal{P}_{f}(\mathbb{Z}^{d})$$ denote the set of all finite subsets of $$\mathbb{Z}^{d}$$.
Definition: Given the algebra $$\mathcal{A}$$ of quasilocal observables and a subset $$X \subset \mathbb{Z}^{d}$$, let $$\mathcal{A}_{X}$$ denote the subset of $$\mathcal{A}$$ consisting of observables supported on $$X$$.
Below will be made reference to a "strongly continuous one-parameter automorphism group" $$\alpha$$. One can have in mind time-evolution by a Hamiltonian, i.e. $$\alpha = \{\alpha_{t}\}$$, where $$\alpha_{t}$$ acts on an observable $$A$$ as $$\alpha_{t}(A) = e^{i H t} A e^{-i H t}$$.
Definition: Given a strongly continuous one-parameter automorphism group $$\alpha$$, let $$K_{\beta}(\alpha)$$ denote the set of $$\beta$$-KMS states with respect to $$\alpha$$.
Corollary IV.4.17: Let $$\alpha$$ be a strongly continuous one-parameter automorphism group, and let $$\beta \neq 0$$. Then $$\rho \in K_{\beta}(\alpha)$$ is an extreme point of that set if and only if, for all [quasilocal observables] $$A \in \mathcal{A}$$ and $$\epsilon > 0$$, there is a $$\Lambda \in \mathcal{P}_{f}(\mathbb{Z}^{d})$$ with $$| \langle A B \rangle_{\rho} - \langle A \rangle_{\rho} \langle B \rangle_{\rho} | \leq \epsilon \parallel \!\! B \parallel$$ for all $$B \in \mathcal{A}_{\mathbb{Z}^{d} \setminus \Lambda}$$.