Counting statistics and entanglement in a disordered free fermion system with a voltage bias

The Full Counting Statistics is studied for a disordered one-dimensional system of non-interacting fermions with and without a voltage bias. 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 $\langle \delta N^2\rangle \approx \frac{1}{\pi^2}\log{t}$. 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 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 $z \ne 1$, and also a model for entanglement entropy based upon dynamical scaling at the Infinite Disorder Fixed Point (IDFP). The latter scaling, which predicts $\langle \delta N^2\rangle \propto \log\log{t}$, 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.