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