Generalized Deep Thermalization for Free Fermions

TL;DR

This paper introduces the deep GGE, a new ensemble framework for describing local subsystems in non-interacting quantum systems, showing it matches the projected ensemble and captures thermalization behavior.

Contribution

It proposes the deep GGE as a novel ensemble that characterizes the projected ensemble in free fermion systems, extending understanding beyond chaotic systems and infinite temperatures.

Findings

Deep GGE coincides with a universal Haar random ensemble at infinite temperature.

Numerical tests show excellent agreement between deep GGE predictions and the projected ensemble.

The approach advances systematic characterization of projected ensembles in non-chaotic systems.

Abstract

In non-interacting isolated quantum systems out of equilibrium, local subsystems typically relax to non-thermal stationary states. In the standard framework, information on the rest of the system is discarded, and such states are described by a Generalized Gibbs Ensemble (GGE), maximizing the entropy while respecting the constraints imposed by the local conservation laws. Here we show that the latter also completely characterize a recently introduced projected ensemble (PE), constructed by performing projective measurements on the rest of the system and recording the outcomes. By focusing on the time evolution of fermionic Gaussian states in a tight-binding chain, we put forward a random ensemble constructed out of the local conservation laws, which we call deep GGE (dGGE). For infinite-temperature initial states, we show that the dGGE coincides with a universal Haar random ensemble on the manifold of Gaussian states. For both infinite and finite temperatures, we use a Monte Carlo approach to test numerically the predictions of the dGGE against the PE. We study in particular the $k$-moments of the state covariance matrix and the entanglement entropy, finding excellent agreement. Our work provides a first step towards a systematic characterization of projected ensembles beyond the case of chaotic systems and infinite temperatures.