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

TL;DR

This paper analyzes the entanglement entropy of large free fermion systems with a random matrix Hamiltonian, revealing a volume law and establishing the universality of Page's formula for black hole radiation.

Contribution

It introduces a simultaneous limit regime for entanglement entropy and proves the volume law for systems with random matrix Hamiltonians, also confirming the universality of Page's formula.

Findings

Entanglement entropy obeys a volume law in the large system limit.

The universality of Page's formula is established for a broad class of states.

The analysis applies to systems with long-range hopping modeled by random matrices.

Abstract

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 heuristic 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 heuristic 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 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.