Generalized deep thermalization for free fermions

In noninteracting isolated quantum systems out of equilibrium, local subsystems typically relax to nonthermal 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.

Automated summary

In noninteracting isolated quantum systems, local subsystems relax to nonthermal stationary states, described by a generalized Gibbs ensemble. This study shows that a recently introduced projected ensemble, which involves projective measurements on the rest of the system, can be completely characterized by the generalized Gibbs ensemble. A random ensemble called deep GGE is proposed and shown to coincide with a universal Haar random ensemble for infinite-temperature initial states. Numerical tests confirm the predictions of the deep GGE and its agreement with the projected ensemble for both infinite and finite temperatures. This work contributes to the systematic characterization of projected ensembles beyond chaotic systems and infinite temperatures.