Full counting statistics in a disordered free fermion system

Abstract

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 $z\ne 1$, and also a model for entanglement entropy based upon dynamical scaling at the Infinite Disorder Fixed Point (IDFP). The latter scaling, which predicts $<\delta N^2>\propto \log \log t$, 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 $(\log t{)}^{1/\psi }$ and $\log t$, respectively. The exponent $\psi$ may be related to the critical exponent characterizing spatial/energy fluctuations at the IDFP.

10 pages, 14 figures; fixed typos; added references; added IDFP scaling based upon reference [1]; added finite bias section; fixed typos

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