Anomalous Josephson current, incipient time-reversal symmetry breaking, and Majorana bound states in interacting multilevel dots

We study the combined effects of spin-orbit interaction, magnetic field, and Coulomb charging on the Josephson current-phase relation, I(phi), for a multilevel quantum dot tunnel contacted by two conventional s-wave superconductors with phase difference phi. A general model is formulated and analyzed in the cotunneling regime (weak tunnel coupling) and in the deep subgap limit, fully taking into account interaction effects. We determine the conditions for observing a finite anomalous supercurrent I-a = I(phi = 0). For a two-level dot with spin-orbit coupling and arbitrarily weak Zeeman field B, we find the onset behavior I-a proportional to sgn(B) in the presence of interactions, suggesting the incipient spontaneous breakdown of time-reversal symmetry. We also provide conditions for realizing spatially separated (but topologically unprotected) Majorana bound states in a double dot variant of this system. Here, Majoranas are predicted to leave a clear signature in the 2 pi-periodic current-phase relation.