Band Representations and Topological Quantum Chemistry

TL;DR

This paper reviews the theory of topological quantum chemistry and crystalline insulators, explaining how symmetry and band representations help classify topological phases and presenting new phase discoveries.

Contribution

It introduces a framework combining band representations and symmetry constraints to classify and discover new topological phases.

Findings

New topological phases identified using band representations

Framework for classifying topological crystalline insulators

Pedagogical overview of symmetry in topological materials

Abstract

In this article, we provide a pedagogical review of the theory of topological quantum chemistry and topological crystalline insulators. We begin with an overview of the properties of crystal symmetry groups in position and momentum space. Next, we introduce the concept of a band representation, which quantifies the symmetry of topologically trivial band structures. By combining band representations with symmetry constraints on the connectivity of bands in momentum space, we show how topologically nontrivial bands can be catalogued and classified. We present several examples of new topological phases discovered using this paradigm, and conclude with an outlook towards future developments.