Study of topological and transport properties of spin-orbit coupled Josephson junctions

The experimental pieces of evidence for the existence of Majorana states in topo- logical superconductors have so far been inconclusive despite intense research in the past two decades [Zha+20; Kay+20]. Combined with promising applications in quantum computing [Nay+08; Ali+11] and the resulting technological development of society, the elusiveness of Majorana states keeps motivating theoretical and ex- perimental research to this day. Our analytical findings and numerical explorations in new topological superconducting platforms suggest several tools and solutions for their future realisation in condensed matter systems. The planar Josephson junction (pJJ) introduced in 2017 by F. Pientka et al. [Pie+17] and M. Hell et al. [HLF17] is a versatile platform for topological superconductivity. It harnesses the tunability of the superconducting phase difference across the Josephson junction as an external control parameter that switches the pJJ between the trivial and topological phases of matter. The junction between the (trivial) superconductors is quasi-one-dimensional and hosts one new Majorana zero mode at each of its ends following each topological phase transition. However, the creation of a second Majorana zero mode on one end triggers a return to the trivial regime as both zero modes hybridize into a regular non-topological fermion. It is then crucial to identify the model parameters that lead to topological phases with a single Majorana state per end. Our main result on the pJJ establishes the general constraint on its microscopic parameters—including the phase difference and a magnetic field—to cross the topo- logical phase transitions. The identification of sectors in parameter space leading to a single Majorana mode becomes then straightforward. In some limits the pJJ develops a topological sector at small magnetic field for a phase difference close to the value p while it remains trivial at the same field near zero phase difference. Since the phase is suffic