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 apply the BdG formalism to compute the wavefunctions of the subgap Andreev bound states (ABSs), considering the thin-barrier limit, in a piecewise-continuous manner. The appropriate boundary conditions at the S-B and B-S junctions allow us to compute the discrete energy eigenvalues ± |ε| of the ABSs. We get two distinct solutions for |ε|. This result is different from the S-B-S set-ups in graphene and Weyl semimetals (each featuring twofold-degenerate nodal points), where one obtains only one solution for |ε|. Although the three bands of the triple-point fermions comprise a nondispersive flat band, the multifold nature of the bandstructure is responsible for this difference. We also illustrate the behaviour of the Josephson current flowing across the S-B-S junction, contributed by the two pairs of ABSs.