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.

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.

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