We derive and apply a general scheme for mapping a setup consisting of a half-filled single-level quantum dot coupled to one normal metallic and two superconducting phase-biased leads onto an ordinary half-filled single impurity Anderson model with single modified tunneling density of states. The theory allows for the otherwise unfeasible application of the standard numerical renormalization group and enables us to obtain phase-dependent local spectral properties as well as phase-dependent induced pairing and Josephson current. The resulting transport properties match well with the numerically exact continuous-time hybridization-expansion quantum Monte Carlo. For weakly coupled normal electrode, the spectral properties can be interpreted in terms of normal-electrode-broadened Andreev bound states with phase-dependent position analogous to the superconducting Anderson model, which coexist in the $\pi$-like phase with a Kondo peak whose phase-dependent Kondo temperature is extracted.

• Received 11 June 2020
• Revised 5 November 2020
• Accepted 22 December 2020

DOI:https://doi.org/10.1103/PhysRevB.103.035419