Eur. Phys. J. B (2016) 89: 197
https://doi.org/10.1140/epjb/e2016-70299-7

Regular Article

Spin-symmetric solution of an interacting quantum dot attached to superconducting leads: Andreev states and the 0-π transition

¹ Institute of Physics, The Czech Academy of Sciences, Na Slovance 2, 18221 Praha 8, Czech Republic
² Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 5, 12116 Praha 2, Czech Republic

^a e-mail: janis@fzu.cz

Received: 13 May 2016
Received in final form: 7 July 2016
Published online: 12 September 2016

Abstract

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-π transition. We show that the spin-symmetric solution becomes degenerate beyond this interaction, in the π 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.