Large Block Properties of the Entanglement Entropy of Disordered Fermions

TL;DR

This paper investigates the large-scale behavior of entanglement entropy in disordered fermionic systems, proving the existence of a finite limit in expectation for all dimensions and demonstrating selfaveraging properties in higher dimensions.

Contribution

It establishes the asymptotic behavior and selfaveraging properties of entanglement entropy in disordered free fermion systems across different dimensions.

Findings

Finite limit of expected entanglement entropy per surface area in all dimensions.

Asymptotic form of entanglement entropy for one-dimensional systems with probability 1.

Polynomial decay of variance indicating selfaveraging in dimensions two and higher.

Abstract

We consider a macroscopic disordered system of free $d$-dimensional lattice fermions whose one-body Hamiltonian is a Schr\"{o}dinger operator $H$ with ergodic potential. We assume that the Fermi energy lies in the exponentially localized part of the spectrum of $H$. We prove that if $S_\Lambda$ is the entanglement entropy of a lattice cube $\Lambda$ of side length $L$ of the system, then for any $d \ge 1$ the expectation $\mathbf{ E}\{L^{-(d-1)}S_\Lambda\}$ has a finite limit as $L \to \infty$ and we identify the limit. Next, we prove that for $d=1$ the entanglement entropy admits a well defined asymptotic form for all typical realizations (with probability 1) as $ L \to \infty$. According to numerical results of [33] the limit is not selfaveraging even for an i.i.d. potential. On the other hand, we show that for $d \ge 2$ and an i.i.d. random potential the variance of $L^{-(d-1)}S_\Lambda$ decays polynomially as $L \to \infty$, i.e., the entanglement entropy is selfaveraging.