Random free fermions: An analytical example of eigenstate thermalization

TL;DR

The paper provides an analytic realization of the Eigenstate Thermalization Hypothesis (ETH) using Gaussian random free fermions with GOE couplings. It shows that in the multiparticle sector, eigenstate expectation values reproduce thermal predictions with an effective temperature set by the particle density N_p/N, and that the correlation matrix decomposes into a thermal part plus GOE fluctuations. Entanglement entropies, computable via the Peschel method, are thermal for subsystems up to size O(N_p) in high-N_p sectors, while single-particle sectors remain non-extensive; for N_p ≈ N/2, entanglement becomes fully thermal for all subsystems. A comparison with fully random Hamiltonians yields quantitative randomness measures through variance analyses, linking ETH in Gaussian models to broader random-matrix perspectives and black-hole physics implications.

Abstract

Having analytical instances of the Eigenstate Thermalization Hypothesis (ETH) is of obvious interest, both for fundamental and applied reasons. This is generically a hard task, due to the belief that non-linear interactions are basic ingredients of the thermalization mechanism. In this article we proof that random gaussian free fermions satisfy ETH in the multiparticle sector, by analytically computing the correlations and entanglement entropies of the theory. With the explicit construction at hand, we finally comment on the differences between fully random Hamiltonians and random Gaussian systems, and on the connection between chaotic energy spectra and ETH.