Full Counting Statistics in the Resonant-Level Model

TL;DR

This paper rigorously derives the full counting statistics for charge transfer in the resonant-level model, confirming the Levitov-Lesovik formula in a non-equilibrium steady state for free fermion systems.

Contribution

It provides a precise proof of the Levitov-Lesovik formula for the resonant-level model, including models with point-like and spread impurities, clarifying measurement subtleties.

Findings

Confirmed the Levitov-Lesovik formula in the resonant-level model.

Described the non-equilibrium steady state and its density matrix.

Addressed subtleties of point-like versus spread impurities.

Abstract

We derive the large deviation function, which provides the large-time full counting statistics for the charge transfer, in the non-equilibrium steady state of the resonant-level model. The general form of this function in free fermion models, in terms of transmission coefficients, was proposed by Levitov and Lesovik in 1993 using a particular measurement set-up involving an interacting spin. It was later suggested to hold as well for a proper quantum mechanical measurement of the transferred charge. We give a precise proof of both statements in the resonant-level model. We first give a full description of the model and its steady state. That is, we explain how the decoupled system prepared with a charge differential evolves, with the impurity coupling, towards the Hershfield non-equilibrium density matrix, in the sense of averages of finitely-supported operators. We describe how this holds both for the usual resonant-level model with a point-like impurity, and for a regularised model with an impurity spread on a finite region, shedding light on subtleties associated to the point-like impurity. We then prove Levitov-Lesovik formula by recasting the problem into calculating averages of finitely-supported operators.