A perturbative nonequilibrium renormalization group method for dissipative quantum mechanics: Real-time RG in frequency space (RTRG-FS)

TL;DR

This paper introduces a perturbative nonequilibrium renormalization group method in frequency space for dissipative quantum systems, enabling calculation of stationary states, observables, and relaxation rates in complex reservoir couplings.

Contribution

It develops a formally exact RG approach in Liouville space for dissipative quantum mechanics, extending real-time RG to arbitrary reservoir-system couplings and nonequilibrium conditions.

Findings

Derived RG equations for relaxation and dephasing rates.

Applied method to the nonequilibrium Kondo model at 1-loop.

Provided analytical solutions in the weak-coupling regime.

Abstract

We study a generic problem of dissipative quantum mechanics, a small local quantum system with discrete states coupled in an arbitrary way (i.e. not necessarily linear) to several infinitely large particle or heat reservoirs. For both bosonic or fermionic reservoirs we develop a quantum field-theoretical diagrammatic formulation in Liouville space by expanding systematically in the reservoir-system coupling and integrating out the reservoir degrees of freedom. As a result we obtain a kinetic equation for the reduced density matrix of the quantum system. Based on this formalism, we present a formally exact perturbative renormalization group (RG) method from which the kernel of this kinetic equation can be calculated. It is demonstrated how the nonequilibrium stationary state (induced by several reservoirs kept at different chemical potentials or temperatures), arbitrary observables such as the transport current, and the time evolution into the stationary state can be calculated. Most importantly, we show how RG equations for the relaxation and dephasing rates can be derived and how they cut off generically the RG flow of the vertices. The method is based on a previously derived real-time RG technique but formulated here in Laplace space and generalized to arbitrary reservoir-system couplings. Furthermore, for fermionic reservoirs with flat density of states, we make use of a recently introduced cutoff scheme on the imaginary frequency axis which has several technical advantages. We demonstrate the method by applying it to the nonequilibrium isotropic Kondo model in 1-loop. We present a systematic way to solve the RG equations analytically in the weak-coupling limit and provide an outlook of the applicability to the strong-coupling case.