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

Abstract

We apply a recently developed nonequilibrium real-time renormalization group (RG) 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. Within a weak-coupling two-loop analysis, we derive analytic formulas for the nonlinear conductance and the kernel determining the time evolution of the reduced density matrix. A consistent formalism is presented how the RG flow is cut off by relaxation and dephasing rates. 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 correction beyond leading order 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 resonant line shapes, including the determination whether the spin relaxation or dephasing rate cuts off the logarithmic divergence. Furthermore, we calculate quantities characterizing the exponential decay of the time evolution of the magnetization. In contrast to the conductance, we find that the derivative of the spin relaxation (dephasing) rate with respect to the magnetic field is logarithmically enhanced (suppressed) for voltages smaller (larger) than the renormalized magnetic field, and that the logarithmic divergence is cut off by the opposite rate. The renormalized $g$ factor is predicted to show a symmetric logarithmic suppression at resonance, which is cut off by the spin relaxation rate. We propose a three-terminal setup to measure the suppression at resonance. For all quantities, we analyze also the anisotropic case and find additional nonequilibrium effects at resonance.