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 $<δN^2> \approx \frac{1}{π^2}\log{t}$. Since the static charge variance for a length $l$ region is given by $<δN^2> \approx \frac{1}{π^2}\log{l}$, 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 $z \ne 1$, and also a model for entanglement entropy based upon dynamical scaling at the Infinite Disorder Fixed Point (IDFP). The latter scaling, which predicts $<δN^2> \propto \log\log{t}$, 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 $(\log{t})^{1/ψ}$ and $\log{t}$, respectively. The exponent $ψ$ may be related to the critical exponent characterizing spatial/energy fluctuations at the IDFP.

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