Random Free Fermions: An Analytical Example of Eigenstate Thermalization

Having analytical instances of the eigenstate thermalization hypothesis (ETH) is of obvious interest, both for fundamental and applied reasons. This is generally a hard task, due to the belief that nonlinear interactions are basic ingredients of the thermalization mechanism. In this article we prove 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, providing a physically motivated notion of randomness of the microscopic quantum state.