Nonequilibrium Kondo effect by equilibrium numerical renormalization group method: The hybrid Anderson model subject to a finite spin bias.

We investigate Kondo correlations in a quantum dot with normal and superconducting electrodes, where a spin voltage is applied across the device and the local interaction $U$ is either attractive or repulsive. This nonequilibrium strongly-correlated problem maps into an equilibrium model solvable by the numerical renormalization group method. Nonequilibrium Kondo spectra with characteristic splitting due to unbalanced chemical potentials are thus obtained at unprecedented high precision. It is shown that while the bias-induced decoherence of the spin Kondo effect is partially compensated by the superconductivity, the charge Kondo effect is enhanced out of equilibrium and undergoes an additional splitting by the superconducting proximity effect, yielding four Kondo peaks in the spectral density. We find a universal scaling of conductance in this hybrid device under different biases, which is peaked at and hence directly measures the charge Kondo temperature. Our results are relevant in view of recent experiments realizing negative-$U$ charge Kondo effect in hybrid oxide quantum dots [Prawiroatmodjo \textit{et al.}, Nat. Commun. \textbf{8}, 395 (2017)].