Full counting statistics and entanglement in a disordered free fermion system

The Full Counting Statistics (FCS) is studied for a one-dimensional system of non-interacting fermions with and without disorder. For two $L$ site translationally invariant lattices connected at time $t=0$, the charge variance increases logarithmically in $t$, following the universal expression $\langle \delta N^2\rangle \approx \frac{1}{\pi^2}\log{t}$, for $t$ much shorter than the ballistic time to encounter the boundary, $t_{b} \sim L$. Since the static charge variance for a length $L$ region is given by $\langle \delta N^2\rangle \approx \frac{1}{\pi^2}\log{L}$, this result reflects the underlying relativistic or conformal invariance and dynamical exponent $z=1$. With disorder and strongly localized fermions, the variance is also found to increase logarithmically in time, but saturates at times $t \approx t_d \propto L^2$, a diffusive time scale. Despite the fact that 1-d fermions are fully localized for any disorder strength, the entanglement responsible for charge fluctuations appears to propagate with dynamical exponent $z=2$.