Quantum phase transitions in proximitized Josephson junctions

We study fermion-parity-changing quantum phase transitions (QPTs) in platform Josephson junctions. These QPTs, associated with zero-energy bound states, are rather widely observed experimentally. They emerge from numerical calculations frequently without detailed microscopic insight. Importantly, they may incorrectly lend support to claims for the observations of Majorana zero modes. In this paper we present a fully self consistent solution of the Bogoliubov-de Gennes equations and show how this consistency provides insights into the origin of the QPTs. This is enabled by an analysis of the spatially-dependent induced magnetization throughout the junction. The junctions we consider can have an arbitrary phase difference and are chosen to mirror conventional, proximitized Josephson experiments; they contain two superconducting leads (separated by a thin insulator). The leads induce superconductivity in a fourth substrate medium which incorporates both Zeeman and spin-orbit coupling as required for a topological phase. We find, quite generally, the wavefunctions of the zero-energy bound states are confined to regions with magnetization inhomogeneities and exist almost exclusively inside the substrate medium where the magnetization is large. We interpret this to suggest that these parity-changing QPTs derive from physics involving spatially extended magnetic "impurities". Our results suggest more generally that QPTs in Josephson junctions generally do not require the existence of spin-orbit coupling and should not be confused with, nor are they indicators of, Majorana physics.