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Recently, superconductor junction attracts great interests in theoretical and experimental studies. Topologically protected state, e.g. Majorana fermion, is one of the main topics for such junctions. Spin-orbit (SO) interaction plays an essential role to induce topological physics. SO interaction is strong in narrow gap semiconductors such as InAs and InSb, and many spin physics are investigated in the quantum well, nanowire, etc. In this study, we investigate Josephson junctions using the semiconductor nanostructures.
We examine multi-terminal Josephson junction. Such junction can be fabricated using crossed nanowires so-called nanocross [1]. For N superconductors, number of the independent phase differences is N-1. The ABS energies are 2π periodic for all phases. By regarding the phases as “quasi momenta” of the energies, we can consider an “energy band” of the multi-terminal junction. The band shows topological singularities at zero energy, properties of which indicates indicate the Weyl singularity [2,3]. We investigate properties of the Andreev spectrum, e.g. peculiar points and lines at the superconducting gap edge in the presence of strong SO interaction, which have mathematical analogy with the Weyl singular point at zero energy [3]. In addition, we consider protection of gaps, crossing of bunching levels, etc [4].

[1] S. R. Plissard et al., Nature Nanotech. 8, 859 (2013).
[2] R.-P. Riwar et al., arXiv: 1503.06862.
[3] T. Yokoyama and Yu. V. Nazarov, PRB 92, 155437 (2015).
[4] T. Yokoyama et al., in preparation.

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