Area Law Scaling for the Entropy of Disordered Quasifree Fermions

We study theoretically and numerically the entanglement entropy of the $d$-dimensional free fermions whose one-body Hamiltonian is the Anderson model. Using the basic facts of the exponential Anderson localization, we show first that the disorder averaged entanglement entropy $⟨S_{\mathrm{\Lambda }}⟩$ of the $d$ dimension cube $\mathrm{\Lambda }$ of side length $l$ admits the area law scaling $⟨S_{\mathrm{\Lambda }}⟩\sim l^{(d-1)},l\gg 1$, 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 self-averaging, i.e., has nonvanishing random fluctuations even if $l\gg 1$.

Figure 1

The probability distributions $p_U(x)$ and $p_L(x)$ of the lower (19) and upper (20) bounds obtained from numerical data on 15 000 realizations of disorder. The hopping parameter $a$ of the Anderson model is $1/10$, the potential is uniformly distributed over $[-1,1]$, $\mu =-0.25$, the system size $L=30\hspace{0.17em}000$, and the subsystem size $l=1500$.