Topological Superconductors and Category Theory

Dispersion Chern Insulator

We give a pedagogical introduction to topologically ordered states of matter, with the aim of familiarizing the reader with their axiomatic topological quantum field theory description. We introduce basic noninteracting topological phases of matter protected by symmetries, including the Su-Schrieffer-Heeger model and the one-dimensional p-wave superconductor. The defining properties of topologically ordered states are illustrated explicitly using the toric code and - on a more abstract level - Kitaev's 16-fold classification of two-dimensional topological superconductors. Subsequently, we present a short review of category theory as an axiomatic description of topological order in two-dimensions. Equipped with this structure, we revisit Kitaev's 16-fold way.

B. A. Bernevig, T. Neupert: Lecture notes, for lectures that were in parts held at: Les Houches Summer School 2014, Vietri Training Course in the Physics of Strongly Correlated Systems 2014, Bogota School on Mathematical Physics 2015

PHY 576 Understanding topological phases of matter from toy models

The exploration of topological phases of matter is to a large extend guided by a range of exactly soluble toy models that illustrate the physics at play. In this course, we will study classic models such as the Su-Schrieffer-Heeger model, Kitaev’s Majorana chain, and the toric code. We will use them to understand topological band insulators and the basic concepts of topological order in systems with anyon excitations. The presentation will be as self-contained as possible with an emphasis on explicit derivations of all the relevant properties.

Link to course catalogue