Quantum Phase Transitions in Proximity Coupled Three-Dimensional Josephson Junctions

Josephson junctions are becoming of greater interest in the search for and confirmation of topological superconductors. Here we address a realistic configuration of a three dimensional SIS junction where two conventional s-wave superconductors (separated by an insulator) induce superconductivity by proximity coupling onto a substrate. This substrate effectively contains spin-orbit and Zeeman coupling and can host a topological phase [Phys. Rev. B 95, 014519]. While the substrate here is taken to be a conical magnet, our findings should apply more generally. We solve the Bogoliubov-de Gennes equation to determine the energy dispersion of the system and find multiple zero energy crossings which are the signatures of parity-changing quantum phase transitions. Important to our work is to clarify the extent to which these zero-energy crossings are associated with the topological phases in the system. Additional insight is provided through simple one-dimensional models of SIS junctions which host a variety of Shiba-like bound states.