Generalized Gibbs ensemble in integrable lattice models

Lev Vidmar, Marcos Rigol

Published 2016-04-13Version 1

The generalized Gibbs ensemble (GGE) was introduced ten years ago to describe observables in isolated integrable quantum systems after equilibration. Since then, the GGE has been demonstrated to be a powerful tool to predict the outcome of the relaxation dynamics of few-body observables in a variety of integrable models, a process we call generalized thermalization. This review discusses several fundamental aspects of the GGE and generalized thermalization in integrable systems. In particular, we focus on questions such as: which observables equilibrate to the GGE predictions and who should play the role of the bath; what conserved quantities can be used to construct the GGE; what are the differences between generalized thermalization in noninteracting systems and in interacting systems mappable to noninteracting ones; why is it that the GGE works when traditional ensembles of statistical mechanics fail. While there has been a lot of interest in these issues in recent years, no definite answers have been given to these questions. Here, we review results for the XX model (hard-core bosons) and report original ones for the transverse field Ising model. For the latter, we show that the GGE describes spin-spin correlations over the entire system. This shows that there is no need to trace out any part of the system in real space for equilibration to occur and the GGE to apply. In addition, an analysis of the spectral decomposition of the weights of various ensembles reveals that generalized eigenstate thermalization occurs in the transverse field Ising model, namely, that eigenstates of the Hamiltonian with similar distributions of conserved quantities have similar expectation values of few-spin observables.