Wavefunction of Andreev bound states with topological Weyl singularity in multi-terminal Josephson junction*

We theoretically investigate a four-terminal Josephson junction.
N superconductors can define N-1 independent superconducting phase differences.
The spectrum of Andreev bound states (ABSs) in the junction is $2\pi$ periodic in
all the phase differences $\{ \varphi \}$ and can be regarded as the ``band structure'' in the $\{ \varphi \}$-space
In a previous studies, we have exhibited that the band structure can have
topologically protected singular points (Weyl points) at zero energy in the spectrum.
\footnote{T.\ Yokoyama and Yu.\ V.\ Nazarov, Phys.\ Rev.\ B {\bf 92}, 155437 (2015).}
In this study, we investigate the wavefunction of ABSs with and without the Weyl point.

For two-terminal junction, the ABS wavefunction is delocalized to connect the two terminals
and does not show any structure except an oscillation associated to the coherence length in the normal region.
For multi-terminal case, the wavefunction can be localized between only several terminals.
This localization is relevant to the presence and absence of Josephson current between the terminals.
At the Weyl points, the spectrum has singularity. We discuss the influence of this singularity on
the current and the ABS wavefunction.