Lecture Notes on Topological Crystalline Insulators

TL;DR

This paper introduces topological crystalline insulators, exploring their properties using non-Abelian Wilson loops, and discusses higher-order topological insulators with hinge and corner states, including interacting bosonic models.

Contribution

It provides a comprehensive introduction to topological crystalline insulators and details new models for higher-order topological phases with interactions.

Findings

Topological crystalline insulators are characterized by symmetries like mirror or rotation.

Non-Abelian Wilson loops are used to deduce topological properties.

Interacting bosonic models for higher-order topological insulators are presented.

Abstract

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.