For ETH Zurich students, please consult with the ETHZ Vorlesungsverzeichnis.
Topological Crystalline Insulators
We give an introduction to topological crystalline insulators, that is, gapped ground states of quantum matter that are not adiabatically connected to an atomic limit without breaking symmetries that include spatial transformations, like mirror or rotational symmetries. To deduce the topological properties, we use non- Abelian Wilson loops. We also discuss in detail higher-order topological insulators with hinge and corner states, and in particular present interacting bosonic models for the latter class of systems.
T. Neupert and F. Schindler, Lecture notes, for lectures given at the San Sebastián Topological Matter School 2017, published in "Topological Matter. Springer Series in Solid-State Sciences, vol 190. Springer, Cham"
Topological Superconductors and Category Theory
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