  The generalized Gibbs ensemble (GGE), which involves multiple conserved
quantities other than the Hamiltonian, has served as the statistical-mechanical
description of the long-time behavior for several isolated integrable quantum
systems. The GGE may involve a noncommutative set of conserved quantities in
view of the maximum entropy principle, and show that the GGE thus generalized
(noncommutative GGE, NCGGE) gives a more qualitatively accurate description of
the long-time behaviors than that of the conventional GGE. Providing a clear
understanding of why the (NC)GGE well describes the long-time behaviors, we
construct, for noninteracting models, the exact NCGGE that describes the
long-time behaviors without an error even at finite system size. It is
noteworthy that the NCGGE involves nonlocal conserved quantities, which can be
necessary for describing long-time behaviors of local observables. We also give
some extensions of the NCGGE and demonstrate how accurately they describe the
long-time behaviors of few-body observables.
