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Singularities in the Andreev spectrum of a multiterminal Josephson junction

机译：多端约瑟夫森结的安德列夫谱中的奇点

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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 In 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.
机译：N端结中形成的Andreev束缚态（ABS）的能量受超导引线之间N-1个独立的宏观相差的影响，由于处于In周期性，因此可以将其视为（N-1）周期固体中的能带在所有阶段。我们结合现象学和分析方法研究所得ABS谱的奇异性，并用数值结果进行说明。我们特别注意自旋轨道（SO）效应。我们考虑在没有SO相互作用的情况下，圆锥形光谱的Weyl奇点位于零能量处。我们证明了SO相互作用像Zeeman场那样将光谱自旋分裂。在远离零能量的同时保留了奇异性。通过SO相互作用，零能点在相（N-1）维空间中形成（N-2）维歧管，而在没有SO相互作用的情况下该维为N-3维。其他类型的奇点位于超导间隙边缘附近。在不存在（存在）SO相互作用的情况下，在间隙边缘处的ABS光谱在数学上类似于在存在（不存在）SO相互作用时零能量的光谱。我们证明，当没有SO相互作用（存在）时，光谱的间隙边缘接触（GET）点原则上形成N-2（N-3）维流形。该阶段的布里渊区中的某些对称线是该规则的例外，因此应单独考虑那里的GET。我们得出并研究了所有考虑中的奇异点的有效哈密顿量。

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