Entanglement Entropy of Free Fermions with a Random Matrix as a One-Body Hamiltonian

We consider a quantum system of large size N and its subsystem of size L assuming that N is much larger than L, which can also be sufficiently large, i. e. , 1 L N. A widely accepted mathematical version of this heuristic inequality is the asymptotic regime of successive limits: first the macroscopic limit N, then an asymptotic analysis of the entanglement entropy as L. In this paper, we consider another version of the above heuristic inequality: the regime of asymptotically proportional L and N, i. e. , the simultaneous limits L, \; N, L/N >0. Specifically, we consider the system of free fermions which is in its ground state and such that its one-body Hamiltonian is a large random matrix, that is often used to model the long-range hopping. By using random matrix theory, we show that in this case, the entanglement entropy obeys the volume law known for systems with short-ranged hopping but described either by a mixed state or a pure strongly excited state of the Hamiltonian. We also give a streamlined proof of Page's formula for the entanglement entropy of the black hole radiation for a wide class of typical ground states, thereby proving the universality of the formula.