We investigate theoretically the Josephson junction of semiconductor nanowire with strong spin-orbit (SO) interaction in the presence of magnetic field. By using a tight-binding model, the energy levels E n of Andreev bound states are numerically calculated as a function of phase difference ϕ between two superconductors in the case of short junctions. The dc Josephson current is evaluated from the Andreev levels. In the absence of SO interaction, a 0-π transition due to the magnetic field is clearly observed. In the presence of SO interaction, the coexistence of SO interaction and Zeeman effect results in E n (−ϕ) = E n (ϕ), where the anomalous Josephson current flows even at ϕ = 0. In addition, the direction dependence of critical current is observed, in accordance with experimental results.

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The energies of Andreev bound states (ABS) forming in a 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 solid 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 on 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 departed from zero energy. With SO interaction, points of zero-energy form an N − 2 dimensional manifold in N − 1 dimensional space of phases, while this dimension is N − 3 in the absence of SO interaction. The singularities of other type are situated near the superconducting gap edge. In the absence (presence) of SO interaction, the ABS spectrum at the gap edge is mathematically analogues 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 N − 2 (N − 3) dimensional manifold when the SO interaction is absent (present). Certain symmetry lines in the Brillouin zone of the phases are exceptional from this rule, and GET there should be considered separately. We derive and study the effective Hamiltonians for all the singularities under consideration.