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

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

2010-08-02

arxiv.org/abs/1008.0347v2

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

29 pages, 12 figures; replacement due to compilation error in pdf (redundant pages at the end)

Scientific paper

Various theoretical methods address transport effects in quantum dots beyond single-electron tunneling, while accounting for the strong interactions in such systems. 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 between non-secular 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 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 a single Zeeman-split level with finite Coulomb interaction.

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