Equilibration towards generalized Gibbs ensembles in non-interacting theories

TL;DR

This paper rigorously analyzes how non-interacting lattice fermion models equilibrate towards generalized Gibbs ensembles, highlighting the role of translation invariance and initial state correlations, with numerical validation and implications for quantum simulators.

Contribution

It provides a mathematically rigorous proof of equilibration to generalized Gibbs ensembles in non-interacting systems with specific initial conditions, using Kusmin-Landau bounds.

Findings

Systems equilibrate rapidly following a power-law in time.

Translation invariance and finite correlation length are sufficient for equilibration.

Numerical simulations confirm analytical predictions with Anderson insulator initial states.

Abstract

Even after almost a century, the foundations of quantum statistical mechanics are still not completely understood. In this work, we provide a precise account on these foundations for a class of systems of paradigmatic importance that appear frequently as mean-field models in condensed matter physics, namely non-interacting lattice models of fermions (with straightforward extension to bosons). We demonstrate that already the translation invariance of the Hamiltonian governing the dynamics and a finite correlation length of the possibly non-Gaussian initial state provide sufficient structure to make mathematically precise statements about the equilibration of the system towards a generalized Gibbs ensemble, even for highly non-translation invariant initial states far from ground states of non-interacting models. Whenever these are given, the system will equilibrate rapidly according to a power-law in time as long as there are no long-wavelength dislocations in the initial second moments that would render the system resilient to relaxation. Our proof technique is rooted in the machinery of Kusmin-Landau bounds. Subsequently, we numerically illustrate our analytical findings by discussing quench scenarios with an initial state corresponding to an Anderson insulator observing power-law equilibration. We discuss the implications of the results for the understanding of current quantum simulators, both in how one can understand the behaviour of equilibration in time, as well as concerning perspectives for realizing distinct instances of generalized Gibbs ensembles in optical lattice-based architectures.