Dynamical spin-spin correlation functions in the Kondo model out of equilibrium
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 h0 are sufficiently large, \V,h0\ TK, where the Kondo temperature TK 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 ≈h, where 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.