Full counting statistics in a disordered free fermion system

TL;DR

This paper investigates the full counting statistics of one-dimensional free fermion systems, revealing universal logarithmic growth in charge variance and exploring effects of disorder, localization, and bias voltage on charge fluctuations.

Contribution

It provides a detailed analysis of charge variance dynamics in disordered and clean fermion systems, connecting results to conformal invariance and infinite disorder fixed point scaling.

Findings

Charge variance grows as log(t) in clean systems

Disordered systems show log(log t) scaling for charge variance

Bias voltage induces power-law and logarithmic behaviors in charge fluctuations

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.