Abstract
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 sufficient to turn on and off the topology, we call this feature “switchable topology”. Looking for switchable topology is experimentally relevant as it makes the topology easily tunable while keeping intact the proximitized su- perconductivity otherwise jeopardized by the applied field. Concretely, we found switchable topology in three configurations: in wide junctions with a transparent interface with the superconducting regions, in fine-tuned narrow junctions weakly coupled to the superconducting regions, and in junctions with a strong Zeeman energy when they are ultranarrow and transparent. Thanks to our exact analytical results, setups interpolating between these limits can adjust the desired properties at will. The other important finding about the pJJ concerns the stability of its topological phases, by which we mean the presence of a sizable spectral gap in the topological sector. We observed that the Rashba spin-orbit coupling is responsible for strongly decreasing the gap in the relevant topological sector at low Zeeman field, but sym- metry arguments justify that wide, transparent junctions are generically immune to this effect for large enough Rashba coupling. After 2017, other platforms started to use the Josephson superconducting phase difference as a knob to trigger topological superconductivity [Liu+19; JY21]. We introduce here the stacked Josephson junction (sJJ) as a new platform for topological superconductivity, which is made of two non-centrosymmetric superconductors sandwiching a two-dimensional magnet around which chiral Majorana edge modes propagate. Unlike the Majorana zero modes in the pJJ, chiral Majorana modes can add to each other if they propagate in the same direction, as indicated by the integer Chern number of their topological phase. The bulk-edge correspondence, however, only constrains the net number of topological edge states and allows room for other non-topological states to coexist with the chiral Majorana states without interacting with them. We found that the presence of trivial chiral edge modes in the sJJ restricts access to the Majorana states themselves. The symmetry protection of the trivial modes, fortunately, disappears with an in-plane magnetic field applied through the magnet or with superconducting leads different on the top and at the bottom of the stacked junction. The theoretical investigations of topological platforms have currently outnum- bered the experiments with convincing signatures of Majorana edge states. This imbalance calls for new ways to probe the agreement between topological models and laboratory setups. The critical current of a Josephson junction acts as a link between the microscopic description and macroscopic observables. Thermoelectric measurements, which distinguish between supercurrent and quasiparticle current, modify this model-dependent connection, and would provide an electrical probe to estimate the validity of a model like that of the pJJ. We computed the contribution to the thermoelectric coefficient of the bulk states of a uniform superconductor, that has a similar environment to that of the pJJ (i.e., Rashba coupling and in-plane Zeeman field). The results were not conclusive and motivated us to suggest new analytical and numerical approaches to obtain the thermoelectric response of the pJJ, in particular by including the contribution of the Andreev bound states and non-linear effects.
