Topological phase transition by a tuning of superconducting phase in multi-terminal Josephson junctions

We theoretically investigate four- and five-terminal Josephson junctions with quantum point contact (QPC) structures.

$N$ superconductors can define $N-1$ independent superconducting phase differences.

The spectrum of Andreev bound states (ABSs) in the junction is $2pi$ periodic in

all the phase differences ${ varphi }$ and can be regarded as the ``band structure.''

In a previous studies, we have demonstrated presence of topologically protected singularities (Weyl points) at zero energy.

footnote{T. Yokoyama and Yu. V. Nazarov, PRB {f 92}, 155437 (2015).}

In this study, we investigate trajectories of Weyl points when the QPC voltages and the fifth superconducting phase are tuned.

Owing to the topological protection, the Weyl points do not disappear solely.

Hence parameter modulations cause the trajectories of Weyl points connecting

the pair annihilation and creation positions in ${ varphi }$-space.

Such pair annihilation and creation coincide with the topological phase transitions between four, two, and no topological charges.

Moreover, the trajectories show two categories:

One is closed type in the first ``Brillouin zone'' and the other is open type for the annihilation and creation with neighboring ``Brillouin zone.''

We consider a phase diagram for the four- and five-terminal junctions and propose a classification of topology by the trajectories.