Visualizing Topological Phase Transitions in Planar Josephson Junctions

We investigate the signatures of topological superconductivity in the current-phase relation (CPR) of planar Josephson junctions (JJs). We show how the magnetic-field dependence of the CPR in phase-biased JJs can be used to obtain a good description and visualization of topological phase transitions. In particular, we propose a new method for obtaining the topological phase diagram in the phase and magnetic field space from CPR measurements. This allows to identify the superconducting phase difference and magnetic field at which the junction transits into the topological superconducting state. Furthermore, the magnetic-field dependence of the CPR in phase-biased JJs can also be used to predict the self-tuning of phase-unbiased JJs into the topological state. We discuss the reliability of the proposed method by comparing to direct numerical calculations of the topological phase diagram and to signatures provided by other physical quantities, like conductance and spin susceptibility.