Scattering theory of topological phase transitions

Abstract

This thesis deals with characterizing topological phases as well as the transitions between them, focusing on transport properties and the effects of disorder.

In Chapters 2 and 3 we derived scattering matrix expressions for the topological invariants of systems. This approach is oftentimes numerically easier to evaluate than Hamiltonian expressions.

In Chapter 4 we predict novel transport features of the quantum Hall plateau transition, and efficiently estimate the associated critical exponent.

In Chapter 5 we examine the universal properties of phase transitions in two-dimensional helical topological superconductors. We compute the critical exponents characterizing the divergence of the localization length, as well as the critical conductance. In Chapter 6, we model a one-dimensional topological superconductor in a bottom-up fashion, as an array of coupled quantum dots. We show how to tune this system deep within the non-trivial phase, with well localized Majorana bound states at its ends. In Chapter 7, we find a new class of disordered topological insulators protected not by an exact symmetry, but by an average symmetry of the disordered ensemble. This greatly increases the range of non-trivial phases, as every topological phase transition gives rise to infinitely many higher-dimensional topological phases