Many-body perturbation theory for the superconducting quantum dot: Fundamental role of the magnetic field

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