Density-operator approaches to transport through interacting quantum dots: Simplifications in fourth-order perturbation theory

TL;DR

This paper compares three quantum transport methods for interacting quantum dots, demonstrating their equivalence, simplifying calculations, and addressing divergences in the T-matrix approach, with implications for higher-order perturbation theory.

Contribution

It establishes the equivalence of the BR and RT methods, simplifies kernel evaluations, and clarifies regularization issues in the T-matrix approach for quantum dot transport.

Findings

BR and RT methods are exactly equivalent.

Simplified rules for zero-frequency kernel calculations.

Regularization issues in T-matrix approach are due to incomplete cancellation.

Abstract

Various theoretical methods address transport effects in quantum dots beyond single-electron tunneling while accounting for the strong interactions in such systems. In this paper we report a detailed comparison between three prominent approaches to quantum transport: the fourth-order Bloch-Redfield quantum master equation (BR), the real-time diagrammatic technique (RT), and the scattering rate approach based on the T-matrix (TM). Central to the BR and RT is the generalized master equation for the reduced density matrix. We demonstrate the exact equivalence of these two techniques. By accounting for coherences (nondiagonal elements of the density matrix) between nonsecular states, we show how contributions to the transport kernels can be grouped in a physically meaningful way. This not only significantly reduces the numerical cost of evaluating the kernels but also yields expressions similar to those obtained in the TM approach, allowing for a detailed comparison. However, in the TM approach an ad hoc regularization procedure is required to cure spurious divergences in the expressions for the transition rates in the stationary (zero-frequency) limit. We show that these problems derive from incomplete cancellation of reducible contributions and do not occur in the BR and RT techniques, resulting in well-behaved expressions in the latter two cases. Additionally, we show that a standard regularization procedure of the TM rates employed in the literature does not correctly reproduce the BR and RT expressions. All the results apply to general quantum dot models and we present explicit rules for the simplified calculation of the zero-frequency kernels. Although we focus on fourth-order perturbation theory only, the results and implications generalize to higher orders. We illustrate our findings for the single impurity Anderson model with finite Coulomb interaction in a magnetic field.