Ergodic property of Markovian semigroups on standard forms of von Neumann algebras

TL;DR

This paper establishes conditions under which Markovian semigroups on von Neumann algebras are ergodic, with applications to quantum spin systems at high temperatures ensuring unique equilibrium states.

Contribution

It provides new sufficient conditions for ergodicity of Markovian semigroups on standard forms of von Neumann algebras, extending previous methods.

Findings

Ergodicity conditions for Markovian semigroups on von Neumann algebras.

Application to quantum spin systems showing ergodicity at high temperatures.

Demonstration of ergodicity linked to the uniqueness of KMS-states.

Abstract

We give sufficient conditions for ergodicity of the Markovian semigroups associated to Dirichlet forms on standard forms of von Neumann algebras constructed by the method proposed in Refs. [Par1,Par2]. We apply our result to show that the diffusion type Markovian semigroups for quantum spin systems are ergodic in the region of high temperatures where the uniqueness of the KMS-state holds.