Nature of Andreev bound states in Josephson junctions of triple-point semimetals

We study superconductor-barrier-superconductor (S-B-S) Josephson junctions constructed out of two-dimensional and three-dimensional triple-point semimetals, which feature a threefold degeneracy at a single nodal point. We assume a weak and homogeneous s-wave pairing in each superconducting region, and a potential difference is applied across a piece of normal-state semimetal to create the barrier region. We compute the wavefunctions of the Andreev bound states (ABSs), considering the thin-barrier limit. The appropriate boundary conditions at the S-B and B-S junctions allow us to compute the discrete energy eigenvalues $\pm |\varepsilon| $ of the ABSs. We get two distinct solutions for $|\varepsilon| $. This result differs from that in graphene and Weyl semimetals, where one obtains only one solution for $|\varepsilon| $. The multifold nature of the triple-point fermions is responsible for this difference. We also illustrate the behaviour of the Josephson current flowing across the S-B-S junction.