Full counting statistics of persistent current

TL;DR

This paper introduces a method to calculate the full counting statistics of persistent current in nanostructures, accounting for interactions, and reveals unique periodicity and noise characteristics.

Contribution

It develops a general formula for the cumulant generating function of persistent current, including interaction effects, and applies it to various models to analyze charge transfer statistics.

Findings

Cumulant generating function exhibits doubled periodicity in the counting field.

Second cumulant peaks at switching points and vanishes at zero temperature.

Interactions suppress persistent current and its noise, with notable differences from noninteracting systems.

Abstract

We develop a method for calculation of charge transfer statistics of persistent current in nanostructures in terms of the cumulant generating function (CGF) of transferred charge. We consider a simply connected one-dimensional system (a wire) and develop a procedure for the calculation of the CGF of persistent currents when the wire is closed into a ring via a weak link. For the non-interacting system we derive a general formula in terms of the two-particle Green's functions. We show that, contrary to the conventional tunneling contacts, the resulting cumulant generating function has a doubled periodicity as a function of the counting field. We apply our general formula to short tight-binding chains and show that the resulting CGF perfectly reproduces the known evidence for the persistent current. Its second cumulant turns out to be maximal at the switching points and vanishes identically at zero temperature. Furthermore, we apply our formalism for a computation of the charge transfer statistics of genuinely interacting systems. First we consider a ring with an embedded Anderson impurity and employing a self-energy approximation find an overall suppression of persistent current as well as of its noise. Finally, we compute the charge transfer statistics of a double quantum dot system in the deep Kondo limit using an exact analytical solution of the model at the Toulouse point. We analyze the behaviour of the resulting cumulants and compare them with those of a noninteracting double quantum dot system and find several pronounced differences, which can be traced back to interaction effects.