Full counting statistics 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 unbiased $L$ site lattices connected at time $t=0$, the charge variance increases as the natural logarithm of $t$, following the universal expression $<delta N^2> approx frac{1}{pi^2}log{t}$. Since the static charge variance for a length $l$ region is given by $<delta N^2> approx frac{1}{pi^2}log{l}$, this result reflects the underlying relativistic or conformal invariance and dynamical exponent $z=1$ of the disorder-free lattice. With disorder and strongly localized fermions, we have compared our results to a model with a dynamical exponent $z e 1$, and also a model for entanglement entropy based upon dynamical scaling at the Infinite Disorder Fixed Point (IDFP). The latter scaling, which predicts $<delta N^2> propto loglog{t}$, appears to better describe the charge variance of disordered 1-d fermions. When a bias voltage is introduced, the behavior changes dramatically and the charge and variance become proportional to $(log{t})^{1/psi}$ and $log{t}$, respectively. The exponent $psi$ may be related to the critical exponent characterizing spatial/energy fluctuations at the IDFP.