  Recently, it was understood that modified concepts of locality played an
important role in the study of extended quantum systems out of equilibrium, in
particular in so-called generalized Gibbs ensembles. In this paper, we
rigorously study pseudolocal charges and their involvement in time evolutions
and in the thermalization process of arbitrary states with strong enough
clustering properties. We show that the densities of pseudolocal charges form a
Hilbert space, with inner product determined by thermodynamic susceptibilities.
Using this, we define the family of pseudolocal states, which are determined by
pseudolocal charges. This family includes thermal Gibbs states at high enough
temperatures, as well as (a precise definition of) generalized Gibbs ensembles.
We prove that the family of pseudolocal states is preserved by finite time
evolution, and that, under certain conditions, the stationary state emerging at
infinite time is a generalized Gibbs ensemble with respect to the evolution
dynamics. If the evolution dynamics does not admit any conserved pseudolocal
charges other than the evolution Hamiltonian, we show that any stationary
pseudolocal state with respect to this dynamics is a thermal Gibbs state, and
that Gibbs thermalization occurs. The framework is that of
translation-invariant states on hypercubic quantum lattices of any
dimensionality (including quantum chains) and finite-range Hamiltonians, and
does not involve integrability.
