Singularities of Andreev spectrum in multi-terminal Josephson junction

TL;DR

This paper analyzes the singularities in the Andreev bound state spectrum of multi-terminal Josephson junctions, focusing on how spin-orbit interaction influences the topological features and dimensionality of these singularities.

Contribution

It provides a detailed analytical and numerical study of ABS spectrum singularities, especially Weyl points, under spin-orbit effects, revealing how these affect the topology and dimensionality of singular manifolds.

Findings

Spin-orbit interaction splits the spectrum similar to a Zeeman field.

Zero-energy singularities form an (N-2)-dimensional manifold without SO, and (N-3) with SO.

Gap edge touching points form (N-2) or (N-3) dimensional manifolds depending on SO presence.

Abstract

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 $2\pi$ 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.