Dynamical spin-spin correlation functions in the Kondo model out of equilibrium

TL;DR

This paper develops a real-time renormalization group method to compute dynamical spin-spin correlation functions in a non-equilibrium Kondo model, providing analytical results in the weak-coupling regime for large voltage or magnetic field.

Contribution

The authors generalize a real-time RG method to calculate dynamical correlation functions of arbitrary operators in non-equilibrium Kondo systems, enabling analytical solutions in the weak-coupling regime.

Findings

Analytical expressions for spin-spin correlation functions in frequency space.

Identification of peak features in transverse susceptibility at specific magnetic fields.

Method applicability for large voltage or magnetic field regimes.

Abstract

We calculate the dynamical spin-spin correlation functions of a Kondo dot coupled to two noninteracting leads held at different chemical potentials. To this end we generalize a recently developed real-time renormalization group method in frequency space (RTRG-FS) to allow the calculation of dynamical correlation functions of arbitrary dot operators in systems describing spin and/or orbital fluctuations. The resulting two-loop RG equations are analytically solved in the weak-coupling regime. This implies that the method can be applied provided either the voltage $V$ through the dot or the external magnetic field $h_0$ are sufficiently large, $\max\{V,h_0\}\gg T_K$, where the Kondo temperature $T_K$ is the scale where the system enters the strong-coupling regime. Explicitly, we calculate the longitudinal and transverse spin-spin correlation and response functions as well as the resulting fluctuation-dissipation ratios. The correlation functions in real-frequency space can be calculated in Matsubara space without the need of any analytical continuation. We obtain analytic results for the line-shape, the small- and large-frequency limits and several other features like the height and width of the peak in the transverse susceptibility at $\Omega\approx\tilde{h}$, where $\tilde{h}$ denotes the reduced magnetic field. Furthermore, we discuss how the developed method can be generalized to calculate dynamical correlation functions of other operators involving reservoir degrees of freedom as well.