Eigenstate Gibbs Ensemble in Integrable Quantum Systems

TL;DR

This paper demonstrates that most eigenstates in integrable quantum systems are locally Gibbs-like, requiring only energy density for their description, and introduces a method to sample and analyze rare eigenstates with generalized ensembles.

Contribution

The authors show that the majority of eigenstates in integrable systems are locally Gibbs-like and develop an unbiased sampling method for rare eigenstates requiring generalized ensembles.

Findings

Most eigenstates are locally Gibbs-like, needing only energy density.

Rare eigenstates are described by truncated Generalized Gibbs Ensembles.

Presence of rare eigenstates leads to long-time GGE description after quenches.

Abstract

The Eigenstate Thermalization Hypothesis implies that for a thermodynamically large system in one of its eigenstates, the reduced density matrix describing any finite subsystem is determined solely by a set of {\it relevant} conserved quantities. In a generic system, only the energy plays that role and hence eigenstates appear locally thermal. Integrable systems, on the other hand, possess an extensive number of such conserved quantities and hence the reduced density matrix requires specification of an infinite number of parameters (Generalized Gibbs Ensemble). However, here we show by unbiased statistical sampling of the individual eigenstates with a given finite energy density, that the local description of an overwhelming majority of these states of even such an integrable system is actually Gibbs-like, i.e. requires only the energy density of the eigenstate. Rare eigenstates that cannot be represented by the Gibbs ensemble can also be sampled efficiently by our method and their local properties are then shown to be described by appropriately {\it truncated} Generalized Gibbs Ensembles. We further show that the presence of these rare eigenstates differentiates the model from the generic (non-integrable) case and leads to the system being described by a Generalized Gibbs Ensemble at long time under a unitary dynamics following a sudden quench, even when the initial state is a Gibbs-like eigenstate of the pre-quench Hamiltonian.