Majorana Fermions in Topological Quantum Matter

Abstract

Two-dimensional topological insulators coupled to s-wave superconductors and external magnetic fields have been predicted to support Majorana edge states. The stability of these states against electron-electron interactions and spin-orbit interactions of Rashba and Dresselhaus type are studied using bosonization and renormalization group analysis. To extend an earlier model, a local Umklapp interaction, corresponding to isolated impurities, is first studied. It is shown that the local Umklapp interaction is less relevant than the global Umklapp interaction, and that the Majorana states are stable against this disorder effect. The model is then extended to also include Rashba and Dresselhaus spin-orbit interactions of constant interaction strength. It is shown that the Majorana states are stable also against the Rashba effect, but may be destabilized by the Dresselhaus effect. The Luttinger parameter is found to be a function of the Rashba interaction strength. The thesis also contains an introduction to the theory of topological insulators, with focus on the field theoretical description and interaction processes.