  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. Despite a
lot of interest in these questions in recent years, no definite answers have
been given. We review results for the XX model and for the transverse field
Ising model. For the latter model, we also report original results and show
that the GGE describes spin-spin correlations over the entire system. This
makes apparent that there is no need to trace out a part of the system in real
space for equilibration to occur and for the GGE to apply. In the past, a
spectral decomposition of the weights of various statistical ensembles revealed
that generalized eigenstate thermalization occurs in the XX model (hard-core
bosons). Namely, eigenstates of the Hamiltonian with similar distributions of
conserved quantities have similar expectation values of few-spin observables.
Here we show that generalized eigenstate thermalization also occurs in the
transverse field Ising model.
