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We apply the unbiased weak-coupling continuous time quantum Monte Carlo(CTQMC) method to review the physics of a single magnetic impurity coupled tos-wave superconducting leads described by the BCS reduced Hamiltonian. As afunction of the superconducting gap $\Delta$, we study the signature of thefirst order transition between the singlet and doublet (local moment) states onvarious quantities. In particular we concentrate on the Josephson current with0 to $\pi$ phase shift, the crossing of the Andreev bound states in the singleparticle spectral function, as well as the local dynamical spin structurefactor. Within DMFT, this impurity problem provides a link to the periodicAnderson model with superconducting conduction electrons (BCS-PAM). The firstorder transition observed in the impurity model is reproduced in the BCS-PAMand is signalized by the crossing of the low energy excitations in the localdensity of states. The momentum resolved single particle spectral function inthe singlet state reveals the coherent, Bloch-like, superposition of Andreevbound states. In the doublet or local moment phase the single particle spectralfunction is characterized by incoherent quasiparticle excitations.

A weak coupling CTQMC study of the single impurity and periodic Anderson models with s-wave superconducting baths