On topological phases in disordered p-wave superconducting wires

Topological phases of matter have been the subject of intense experimental and theoretical research during the last years. Prominent examples are the Quantum Hall Effect, Topological Insulators or Topological Superconductors. The latter host special excitations, the Majorana states, at their boundaries, which can be thought of as the halves of an electron that can exist separately in this special case. These Majorana states have attracted great interest as they exhibit so-called non-Abelian braiding statistics, which could make them useful tools in the search for fault-tolerant quantum computation. In this context topologically superconducting wires are particularly useful as the Majorana states are located unambiguously at the wire's end, where they form localized end states. Topologically superconducting wires are not known to exist in nature but they can be engineered from commonly available ingredients: semiconductor or ferromagnet nano- wires and conventional superconductors. The nano-wires can inherit superconductivity by the proximity effect and can then exhibit a topologically nontrivial phase. By now, several experiments have been performed on such hybrid structures, reporting measurements that are consistent with the existence of a topologically superconducting phase in the nanowire. Most theoretical investigations on these systems, so far, have been restricted to a one-dimensional effective model: The one-dimensional p-wave superconductor, which is the prototype of a topologically superconducting wire. A nanowire, however, is in general in a quasi-one dimensional regime, with a continuous longitudinal but a quantized transverse degree of freedom. In this Thesis we study the multichannel generalization of a topologically superconducting wire by means of a two-dimensional p + ip-superconductor that is restricted to a narrow-strip geometry. Such systems can be in a topological phase, characterized by the existence of a zero-energy excitation at the wires end—the Majorana bound state. We study the effect of various geometrical terminations on the low-energy spectrum of such a wire and find that subgap states tend to accumulate around zero energy. In a density-of- states measurement, these states potentially obscure the Majorana state thereby hindering the detection of the topological phase. We further investigate the effect of disorder on a multichannel wire and find that it induces a series of phase transitions with a reentrant topological phase. Due to disorder-localized states accumulating in the superconducting gap, the low-energy spectrum for a disordered wire contains a signature of the topological phase transitions as well: a singularity in the density of states, which is the well-known Dyson-singularity.