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

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.

References in corpus (6)

• Entanglement in indistinguishable particle systems
• The Page Curve for Fermionic Gaussian States
• On the Area Law for Disordered Free Fermions
• A Proof of Vivo-Pato-Oshanin's Conjecture on the Fluctuation of von Neumann Entropy
• Entanglement Typicality
• Entropy fluctuation formulas of fermionic Gaussian states