Fermionic superoperators for zero-temperature non-linear transport: real-time perturbation theory and renormalization group for Anderson quantum dots

TL;DR

This paper develops a real-time renormalization group approach using fermionic superoperators to analyze zero-temperature non-linear transport through Anderson quantum dots, revealing new effects like enhanced inelastic cotunneling and non-linear stability diagrams.

Contribution

It introduces a fermionic superoperator formalism in Liouville space for Anderson models, simplifying perturbation theory and enabling comprehensive 2-loop RG analysis at finite voltage and magnetic field.

Findings

Enhanced inelastic cotunneling peaks due to renormalization effects.

Discovery of tunnel-induced non-linearity in stability diagrams.

Systematic analysis excluding the Kondo regime.

Abstract

We study the transport through a strongly interacting Anderson quantum dot at zero-temperature using the real-time renormalization group (RT-RG) in the framework of a kinetic equation for the reduced density operator. We further develop the general finite temperature real-time transport formalism by introducing field superoperators that obey fermionic statistics. This direct second quantization in Liouville-Fock space strongly simplifies the construction of operators and superoperators which transform irreducibly under the Anderson-model symmetry transformations. The fermionic field superoperators naturally arise from the univalence (fermion-parity) superselection rule for the total system. Expressed in these field superoperators, the causal structure of the perturbation theory for the effective time-evolution superoperator-kernel becomes explicit. The causal structure also implies the existence of a fermion-parity protected eigenvector of the exact Liouvillian, explaining a recently reported result on adiabatic driving [Phys. Rev. B 85, 075301 (2012)] and generalizing it to arbitrary order in the tunnel coupling. Furthermore, in the WBL the causal representation exponentially reduces the number of diagrams for the time-evolution kernel. We perform a complete 2-loop RG analysis at finite voltage and magnetic field, while systematically accounting for the dependence on both the quantum dot and reservoir frequencies. Using the second quantization in Liouville-space and symmetry restrictions we obtain analytical RT-RG equations with an efficient numerical solution and we extensively study the model parameter space, excluding the Kondo regime. The incorporated renormalization effects result in an enhancement of the inelastic cotunneling peak. Moreover, we find a tunnel-induced non-linearity of the stability diagrams at finite voltage, both in the SET and ICT regime.