On the Area Law for Disordered Free Fermions

TL;DR

This paper demonstrates that disordered free fermions in any dimension follow an area law for entanglement entropy on average, and reveals that in one dimension, the entropy exhibits persistent fluctuations even at large scales.

Contribution

It provides a theoretical and numerical analysis showing the area law for entanglement entropy in disordered free fermions and uncovers non self-averaging behavior in one dimension.

Findings

Average entanglement entropy follows the area law in all dimensions.

In 1D, entanglement entropy has non-vanishing fluctuations at large scales.

The area law holds even in the gapless case due to Anderson localization.

Abstract

We study theoretically and numerically the entanglement entropy of the $d$-dimensional free fermions whose one body Hamiltonian is the Anderson model. Using basic facts of the exponential Anderson localization, we show first that the disorder averaged entanglement entropy $\langle S_\Lambda \rangle$ of the $d$ dimension cube $\Lambda$ of side length $l$ admits the area law scaling $\langle S_\Lambda \rangle \sim l^{(d-1)}, \ l \gg 1$ even in the gapless case, thereby manifesting the area law in the mean for our model. For $d=1$ and $l\gg 1$ we obtain then asymptotic bounds for the entanglement entropy of typical realizations of disorder and use them to show that the entanglement entropy is not selfaveraging, i.e., has non vanishing random fluctuations even if $l \gg 1$.