Thermalization and pseudolocality in extended quantum systems

TL;DR

This paper rigorously explores pseudolocal charges in extended quantum systems, demonstrating their role in thermalization and the formation of generalized Gibbs ensembles, applicable to non-integrable, translation-invariant lattice models.

Contribution

It introduces a Hilbert space structure for pseudolocal charge densities and defines pseudolocal states, showing their preservation under time evolution and their role in thermalization.

Findings

Pseudolocal charge densities form a Hilbert space based on thermodynamic susceptibilities.

Pseudolocal states include thermal and generalized Gibbs states.

Under certain conditions, stationary states are thermal Gibbs states, confirming Gibbs thermalization.

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 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.