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

## TL;DR

This paper studies the entanglement entropy of free fermions with a random Hamiltonian, showing it follows a volume law and proving the universality of Page's formula.

## Contribution

The paper introduces a new asymptotic regime and proves the volume law for entanglement entropy in free fermion systems with random Hamiltonians.

## Key findings

- Entanglement entropy follows a volume law in the asymptotic regime with random Hamiltonians.
- Page’s formula is universally valid for typical ground states of free fermions.
- The results demonstrate the typicality and universality of entanglement entropy behavior.

## 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≪L≲N. A widely accepted mathematical version of this 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 inequality: the regime of asymptotically proportional L and N, i.e., the simultaneous limits L→∞,N→∞,L/N→λ>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 hopping. By using random matrix theory, we show that in this case, the entanglement entropy obeys the volume law known for systems with short-range hopping 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.