Real-time renormalization group in frequency space: A 2-loop analysis of the nonequilibrium anisotropic Kondo model at finite magnetic field

Abstract

We apply a recently developed nonequilibrium real-time renormalization group method in frequency space to describe nonlinear quantum transport through a small fermionic quantum system coupled weakly to several reservoirs via spin and/or orbital fluctuations. We provide an analytic and well-controlled procedure to solve the RG equations in the weak-coupling regime. Within a 2-loop analysis, we derive analytic formulas for the nonlinear conductance and the kernel determining the time evolution of the reduced density matrix. Most importantly, we present a consistent formalism how the RG flow is cut off by relaxation and dephasing rates, which is necessary to calculate the precise line shape at resonances. We apply the general formalism to the nonequilibrium anisotropic Kondo model at finite magnetic field. We consider the weak-coupling regime, where the maximum of voltage and bare magnetic field is larger than the Kondo-temperature. In this regime, we calculate the nonlinear conductance, the magnetic susceptibility, the renormalized spin relaxation and dephasing rates, and the renormalized g-factor. All quantities are considered up to the first logarithmic contributions at resonance. Up to a redefinition of the Kondo temperature, we confirm previous results for the conductance and the magnetic susceptibility in the isotropic case. In addition, we present a consistent calculation of the line shape at the resonances, including the determination which rate cuts off the logarithmic divergence. Furthermore, we calculate the rates and the renormalized g-factor in nonequilibrium beyond leading order, being quantities characterizing the exponential decay of the time evolution of the magnetization. For all quantities we analyse also the anisotropic case and find interesting nonequilibrium effects at resonance.