Tuning Majorana zero modes with temperature in $\pi$-phase Josephson junctions

We study a superconductor-normal state-superconductor (SNS) Josephson junction along the edge of a quantum spin Hall insulator (QSHI) with a superconducting $\pi$-phase across the junction. We solve self-consistently for the superconducting order parameter and find both real junctions, where the order parameter is fully real throughout the system, and junctions where the order parameter has a complex phase winding. Real junctions host two Majorana zero modes (MZMs), while phase-winding junctions have no subgap states close to zero energy. At zero temperature we find that the phase-winding solution always has the lowest free energy, which we establish being due to a strong proximity-effect into the N region. With increasing temperature this proximity-effect is dramatically decreased and we find a phase transition into a real junction with two MZMs.