We study the interplay and competition between Kondo overscreening and superconductivity. The model we consider consists of a single $\mathrm{spin}\text{−}\frac{1}{2}$ quantum impurity at the edge of a quantum wire coupled to $\mathrm{spin}\text{−}1$ bulk fermions that interact attractively, generating a (superconducting) mass gap. The competition leads to a rich phase structure. We find that for strong Kondo coupling, there is a regime of phase space where the Kondo phase is stable with the impurity overscreened by a multiparticle Kondo effect, with a Kondo scale dynamically generated. When the bulk and boundary interaction strengths are comparable, we find that a midgap state appears in the spectrum and screens the impurity, while in the ground state, the impurity is unscreened. This midgap state is akin to the Yu-Shiba-Rusinov (YSR) states that exist in the entire phase space in the BCS superconductor. When the bulk superconducting interaction strength is stronger than the boundary Kondo interaction strength, the impurity can no longer be screened. Further, between the Kondo and YSR phases, we find a unique phase where, while the Kondo cloud overscreens the impurity, a boundary excitation exists that has vanishing energy in the thermodynamic limit.

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• Kondo effect
• Quantum field theory
• Quantum wires
• Bethe ansatz