Large Block Properties of the Entanglement Entropy of Disordered Fermions

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 $S_{\mathrm{\Lambda }}$ is the entanglement entropy of a lattice cube $\mathrm{\Lambda }$ of side length $L$ of the system, then for any $d\ge 1$ the expectation $\mathbf{ E}{L^{-(d-1)}S_\Lambda}$hasafinitelimitas$L \to \infty$ 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 $L\to \mathrm{\infty }$. 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 $L^{-(d-1)}{S}_{\mathrm{\Lambda }}$ decays polynomially as $L\to \mathrm{\infty }$, i.e., the entanglement entropy is selfaveraging.