Effective Equilibrium Description of Nonequilibrium Quantum Transport I: Fundamentals and Methodology

TL;DR

This paper develops an effective equilibrium framework for analyzing nonequilibrium quantum transport in strongly correlated systems, simplifying calculations by reformulating the problem using modified Hamiltonians and scattering operators.

Contribution

It introduces an alternative derivation of Hershfield's effective Hamiltonian and demonstrates its equivalence with the Schwinger-Keldysh approach for quantum impurity models.

Findings

Effective equilibrium approach reproduces nonequilibrium observables.

Derived general expressions for current and charge occupation.

Introduced a finite temperature formalism for Green's functions.

Abstract

The theoretical description of strongly correlated quantum systems out of equilibrium presents several challenges and a number of open questions persist. In this paper we focus on nonlinear electronic transport through a quantum dot maintained at finite bias using a concept introduced by Hershfield [Phys. Rev. Lett. 70, 2134 (1993)] whereby one can express such nonequilibrium quantum impurity models in terms of the system's Lippmann-Schwinger operators. These scattering operators allow one to reformulate the nonequilibrium problem as an effective equilibrium problem associated with a modified Hamiltonian, thus facilitating the implementation of equilibrium many-body techniques. We provide an alternative derivation of the effective Hamiltonian of Hershfield using the concept of an "open system". Furthermore, we demonstrate the equivalence between observables computed using the Schwinger-Keldysh framework and the effective equilibrium approach. For the study of transport, the non-equilibrium spectral function of the dot is identified as the quantity of principal interest and we derive general expressions for the current (the Meir-Wingreen formula) and the charge occupation of the dot. We introduce a finite temperature formalism which is used as a tool for computing real time Green's functions. In a companion paper we elucidate a generic scheme for perturbative calculations of interacting models, with particular reference to the Anderson model.