Spin-symmetric solution of an interacting quantum dot attached to   superconducting leads: Andreev states and the $0-\pi$ transition
  Behavior of Andreev gap states in a quantum dot with Coulomb repulsion symmetrically attached to superconducting leads is studied via the perturbation expansion in the interaction strength. We find the exact asymptotic form of the spin-symmetric solution for the Andreev states continuously approaching the Fermi level. We thereby derive a critical interaction at which the Andreev states at zero temperature merge at the Fermi energy, being the upper bound for the $0-\pi$ transition. We show that the spin-symmetric solution becomes degenerate beyond this interaction, in the $\pi$ phase, and the Andreev states do not split unless the degeneracy is lifted. We further demonstrate that the degeneracy of the spin-symmetric state extends also into the $0$ phase in which the solutions with zero and non-zero frequencies of the Andreev states may coexist. 
V. Jani\v{s}, V. Pokorn\'y, and M. \v{Z}onda
cond-mat.mes-hall cond-mat.str-el cond-mat.supr-con