Full counting statistics in a disordered free fermion system

Abstract

The full counting statistics is studied for a one-dimensional system of noninteracting 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 $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$. Since the static charge variance for a length $l$ region is given by $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}l$, this result reflects the underlying relativistic or conformal invariance and dynamical exponent $z=1$ of the disordered free lattice. With disorder and strongly localized fermions, we have compared our results to a model with a dynamical exponent $z\ensuremath{\ne}1$ and a model for entanglement entropy based upon dynamical scaling at the infinite disorder fixed point (IDFP). The latter scaling, which predicts $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\propto}\mathrm{ln}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$, appears to better describe the charge variance of disordered one-dimensional fermions. When a bias voltage is introduced, the behavior changes dramatically, and the charge and variance become proportional to ${(\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t)}^{1/\ensuremath{\psi}}$ and $\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$, respectively. The exponent $\ensuremath{\psi}$ may be related to the critical exponent, characterizing spatial/energy fluctuations at the IDFP.