Effective Hamiltonian of magnetic moments in proximity with an s-wave superconductor

In this paper, we consider two magnetic moments in proximity with an s-wave superconductor, where each magnetic moment has an internal energy level. The effective Hamiltonian of the two magnetic moments is derived in this paper by the method of path integral in the weak coupling limit. Interestingly, there are four new terms by the proximity effect, including the Ruderman-Kittel-Kasuya-Yosida interaction, the hopping term, the cross and on-site superconducting pairing potentials. Additionally, since the parity of electrons is conserved, the Hilbert space is split into two subspaces of even and odd parities, respectively. We also investigate the consequences of the effective Hamiltonian in the Coulomb blockade regime by varying the distance between the two magnetic moments. We found that a quantum phase transition happens whenever there is a level crossing for the ground states. The degeneracy of the ground states is also changed after the quantum phase transition, and the electron occupations for the magnetic moments with both singlet and triplet states are discontinuous at the level crossing, since the electronic states are always entangled with the spin states of the two magnetic moments.