Generalized deep thermalization for free fermions

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