Thermalization and its Breakdown for a Large Nonlinear Spin

TL;DR

This paper investigates how a large nonlinear spin thermalizes over time, revealing that most eigenstates satisfy thermalization conditions, but certain unstable classical fixed points can cause a breakdown, leading to long-term memory effects.

Contribution

It introduces a semi-classical analysis based on the Eigenstate Thermalization Hypothesis to explain thermalization and its breakdown in a large nonlinear spin system.

Findings

Eigenstate Thermalization Hypothesis holds for most eigenstates.

Thermalization is generally observed despite classical integrability.

Unstable fixed points can cause breakdown of thermalization and memory effects.

Abstract

By developing a semi-classical analysis based on the Eigenstate Thermalization Hypothesis, we determine the long time behavior of a large spin evolving with a nonlinear Hamiltonian. Despite integrable classical dynamics, we find the Eigenstate Thermalization Hypothesis for the diagonal matrix elements of observables is satisfied in the majority of eigenstates, and thermalization of long time averaged observables is generic. The exception is a novel mechanism for the breakdown of thermalization based on an unstable fixed point in the classical dynamics. Using the semi-classical analysis we derive how the equilibrium values of observables encode properties of the initial state. This analysis shows an unusual memory effect in which the remembered initial state property is not conserved in the integrable classical dynamics. We conclude with a discussion of relevant experiments and the potential generality of this mechanism for long time memory and the breakdown of thermalization.