Classical ergodicity and quantum eigenstate thermalization: Analysis in fully connected Ising ferromagnets

TL;DR

This paper explores the connection between classical ergodicity and quantum thermalization in fully connected Ising ferromagnets, revealing how classical dynamics influence quantum eigenstate properties and system equilibration.

Contribution

It demonstrates that in fully connected Ising models, quantum eigenstate expectation values relate to classical long-time averages, even when classical dynamics are non-ergodic, highlighting a general semiclassical property.

Findings

Quantum expectation values match classical long-time averages in spin-1/2 systems.

In spin-1 systems, eigenstate statistics align with classical averages from random initial states.

The results inform understanding of thermalization and dynamical transitions in semiclassical systems.

Abstract

We investigate the relation between the classical ergodicity and the quantum eigenstate thermalization in the fully connected Ising ferromagnets. In the case of spin-1/2, an expectation value of an observable in a single energy eigenstate coincides with the long-time average in the underlying classical dynamics, which is a consequence of the Wentzel-Kramers-Brillouin approximation. In the case of spin-1, the underlying classical dynamics is not necessarily ergodic. In that case, it turns out that, in the thermodynamic limit, the statistics of the expectation values of an observable in the energy eigenstates coincides with the statistics of the long-time averages in the underlying classical dynamics starting from random initial states sampled uniformly from the classical phase space. This feature seems to be a general property in semiclassical systems, and the result presented here is crucial in discussing equilibration, thermalization, and dynamical transitions of such systems.