We develop the general many-body perturbation theory for a superconducting quantum dot represented by a single-impurity Anderson model attached to superconducting leads. We build our approach on a thermodynamically consistent mean-field approximation with a two-particle self-consistency of the parquet type. The two-particle self-consistency leading to a screening of the bare interaction proves substantial for suppressing the spurious transitions of the Hartree-Fock solution. We demonstrate that the magnetic field plays a fundamental role in the extension of the perturbation theory beyond the weakly correlated 0 phase. It controls the critical behavior of the $0\text{−}\pi$ quantum transition and lifts the degeneracy in the $\pi$ phase, where the limits to zero temperature and zero magnetic field do not commute. The response to the magnetic field is quite different in 0 and $\pi$ phases. While the magnetic susceptibility vanishes in the 0 phase it becomes divergent in the $\pi$ phase at zero temperature.

• Received 26 February 2021
• Accepted 7 June 2021

DOI:https://doi.org/10.1103/PhysRevB.103.235163