Title: Tuning Majorana zero modes with temperature in $π$-phase Josephson junctions,
Abstract: 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.