Theory of a single magnetic impurity on a thin metal film in proximity to a superconductor

We argue that the formation of Yu-Shiba-Rusinov excitations in proximitized thin films is largely mediated by a type of Andreev-bound state named after de Gennes and Saint-James. This is shown by studying an experimentally motivated model and computing the overlap of the wave functions of these two subgap states. We find the overlap stays close to unity even as the system moves away from weak coupling across the parity-changing quantum phase transition. Based on this observation, we introduce a single-site model of the bound state coupled to a quantum spin. The adequacy of this description is assessed by reintroducing the coupling to the continuum as a weak perturbation and studying its scaling flow using Anderson's poor man's scaling.