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McGill University M. Sc. Thesis: Topological Superconductivity without Proximity Effect

The search for a Majorana Fermion has been an area of intense interest in condensed matter research of late. This elusive particle, predicted to exist in 1937, has been sought after for both fundamental and practical reasons. On the fundamental level, no particle to date has been observed to be a Majorana fermion, meanwhile on the practical level a Majorana fermion, if found, would represent a non-abelian anyon and could thus be used to build a quantum computer. The search for a Majorana Fermion has recently shifted to topological superconductivity. Topological superconductors are categorized by the nontrivial wind- ing of their order parameter phase and for this reason are expected to support Majorana Fermions in their vortex cores. Owing to this, the study of topological superconductors has intensified in recent years. Current proposals for a device that may behave as a topological superconductor are based on semiconductor heterostructures, where the spin-orbit coupled bands of a semiconductor are split by a band gap or Zeeman field and superconductivity is induced by proximity to a conventional superconductor. In this setup, topological superconductivity is obtained in the semiconductor layer and the proposed heterostructures typically include two or three layers of different materials. In this thesis we propose a simplification to these types of devices, suggesting a way in which the superconducting layer can be replaced. Part of our proposal includes a model Hamiltonian for these types of systems. This thesis will also develop several different methods to analyze this model Hamiltonian in various different parameter regimes with the ultimate goal of classifying its topology.