Numerical renormalization group studies on single impurity Anderson model in superconductivity: A unified treatment of magnetic, nonmagnetic impurities, and resonance scattering

We investigate a single impurity Anderson model in superconducting state focusing on the localized excited states induced around the impurity site with use of the numerical renormalization group method. Using this model, we present a unified treatment of electronic states around various kinds of impurities embedded in conventional s-wave superconductivity that have been investigated independently so far: (1) magnetic impurity, (2) nonmagnetic impurity with potential scattering, and (3) nonmagnetic impurity with resonance scattering. In the case of the symmetric Anderson model, we clarify how the localized excited state induced around (1) is transformed into the one induced around (3) with a decrease of the strength of the on-site Coulomb repulsion at the impurity site. On the other hand, in the case of the asymmetric Anderson model, we show how the bound state induced by the magnetic impurity disappears when the impurity is gradually transformed from (1) into (2) with an increase of the energy of the impurity level. We also clarify the region where competition between the Kondo effect and superconductivity dominates the properties of the ground and the localized excited states in the parameter space of the Anderson model. In this region, we show that the bound state energy is scaled well by the single parameter T_K/Δ (T_K: Kondo temperature, Δ: superconducting order parameter) as in the case of the antiferromagnetic s-d interaction in superconductivity.