Abstract
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 these 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.
Highlights
The physics of quantum systems out of equilibrium has received a large amount of attention recently
We showed that pseudolocal charges are in bijection with the Hilbert space associated to susceptibilities, which can be interpreted as the space of their densities
We further defined a pseudolocal state as a state lying at the end-point of a flow whose tangents are determined by pseudolocal charges
Summary
The physics of quantum systems out of equilibrium has received a large amount of attention recently. We consider general quantum states satisfying strong enough clustering properties that guarantee the existence and finiteness of thermodynamic susceptibilities Within this context, we prove that pseudolocal charges (a refinement, and generalization to arbitrary dimensions, of the concepts developed in [57,58,64,65,66]) are in bijection with the countable-dimensional Hilbert space induced by Cauchy completing the positive semidefinite sesquilinear form built out of susceptibilities. Under certain conditions on the existence of the long-time limit, any pseudolocal state reaches a GGE with respect to the evolution dynamics. The conditions include the existence of the long-time limit of dynamical susceptibilities This generalized thermalization result holds independently of integrability, the latter only affecting the family of conserved pseudolocal charges available and the manifold of final states.
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