On the KMS states for the Bernoulli shift

TL;DR

This paper characterizes the types of extremal KMS states for a flow on a crossed product algebra associated with the Bernoulli shift, revealing conditions for states of types II_infinity and III_lambda based on potential dependencies.

Contribution

It provides a detailed classification of extremal KMS states for the Bernoulli shift crossed product, including explicit conditions for types II_infinity and III_lambda states.

Findings

Existence of extremal β-KMS states of type II_infinity for all β ≠ 0.

Determination of λ values for which extremal β-KMS states of type III_λ exist when potentials are rationally dependent.

Explicit characterization of KMS states based on potential properties.

Abstract

Let $\Omega:=\{0,1\}^{\mathbb{Z}}$ be the Cantor space, and let $\tau:\Omega \to \Omega$ be the Bernoulli shift. For the flow on the crossed product $C(\Omega)\rtimes_\tau \mathbb{Z}$ determined by a potential that depends on only one coordinate, we show that for every $\beta \neq 0$, there is an extremal $\beta$-KMS state on $C(\Omega)\rtimes_\tau \mathbb{Z}$ of type $II_\infty$. Also, when the potential takes values that are rationally dependent, we determine the values of $\lambda \in (0,1)$ for which there is a an extremal $\beta$-KMS state of type $III_\lambda$.