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

We study the combined effects of spin-orbit interaction, magnetic field, and Coulomb charging on the Josephson current-phase relation, I(φ), for a multi-level quantum dot tunnel-contacted by two conventional s-wave superconductors with phase difference φ. 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(φ=0). For a two-level dot with spin-orbit coupling and arbitrarily weak Zeeman field B, we find the onset behavior I_a\propto {\rm 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 this system, which have a clear signature in the 2π-periodic current-phase relation.