Singularities in the Andreev spectrum of a multiterminal Josephson junction

Abstract

The energies of Andreev bound states (ABS) forming in an N-terminal junction are affected by N−1 independent macroscopic phase differences between superconducting leads and can be regarded as energy bands in (N−1)-periodic solids owing to the 2π periodicity in all phases. We investigate the singularities and peculiarities of the resulting ABS spectrum combining phenomenological and analytical methods and illustrating with the numerical results. We pay special attention to spin-orbit (SO) effects. We consider Weyl singularities with a conical spectrum that are situated at zero energy in the absence of SO interaction. We show that the SO interaction splits the spectrum in spin like a Zeeman field would do. The singularity is preserved while departing from zero energy. With SO interaction, points of zero energy form an (N−2)-dimensional manifold in an (N−1)-dimensional space of phases, while this dimension is N−3 in the absence of SO interaction. Singularities of other types are situated near the superconducting gap edge. In the absence (presence) of SO interaction, the ABS spectrum at the gap edge is mathematically analogous to that at zero energy in the presence (absence) of SO interaction. We demonstrate that the gap edge touching (GET) points of the spectrum in principle form an N−2 (N−3) dimensional manifold when the SO interaction is absent (present). Certain symmetry lines in the Brillouin zone of the phases are exceptions from this rule, and GET there should be considered separately. We derive and study the effective Hamiltonians for all the singularities under consideration.

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