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

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 ͑BR͒, the real-time diagrammatic technique ͑RT͒, and the scattering rate approach based on the T-matrix ͑TM͒. 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 matrix͒ 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-frequency͒ 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.

Subgap features due to quasiparticle tunneling in quantum dots coupled to superconducting leads

“…Since the generalized master equation approach to transport through quantum dots has become rather standard in recent years (see, e.g., the method article by Timm et al 35 or the recent paper by Koller et al 36 ), we only go into details of the derivation of the master equation when the effect of the superconducting leads brings significant differences with respect to the normal conducting theory.…”

Charge fluctuations in nonlinear heat transport

“…We go beyond standard approaches by including the competition of all tunneling rates O( ) and O( 2 ) (Fig. 1) into the stationary master equatioṅ [46] for more details of the calculations [45,50,51] of the transition rate matrix W and the current). We focus on the dominant energy dependence introduced by the interacting quantum dot, assuming a flat spectral density in the wide-band limit for the electrodes.…”

Sources of negative tunneling magnetoresistance in multilevel quantum dots with ferromagnetic contacts

“…3(a) shows the typical conductance peak patterns expected for the degeneracies according to Table I, for both a second-order (sequential tunneling) and a fourth-order (cotunneling, pair tunneling, single charge fluctuations, etc.) 21 truncation in the calculation of the kernel K. Second-order theory predicts a constant TMR value, 14 but this changes upon inclusion of higher-order effects. Within fourth-order perturbation theory, the TMR exhibits an oscillatory gate voltage dependence, albeit the variation is small.…”

“…The time evolution ofρ(t) follows the Liouville equation, and it can for the stationary state (ρ(t) = 0) be cast into the form (see, e.g., Ref. 21) taking matrix elements with respect to the many-body eigenstates of the CNT: ρ aa := a|ρ|a and E a := a|Ĥ |a . Furthermore K aa bb := b|[K|a a |]|b , with the kernel superoperator K arising from the perturbation, i.e., the tunnel coupling between quantum dot and leads.…”