On the Area Law for Disordered Free Fermions

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 $l \gg 1$.