Topological electronic bands in crystalline solids

TL;DR

This review introduces the principles of topological electronic band structures in crystalline solids, explaining how topology classifies electronic states and influences observable properties like surface states and topological insulators.

Contribution

It provides a comprehensive, accessible overview of topological band theory, including key concepts, invariants, and their implications for crystalline materials, aimed at new researchers.

Findings

Classification of crystalline solids by topological invariants

Explanation of surface states in topological insulators

Identification of Weyl and Dirac semimetals as topological phases

Abstract

Topology is now securely established as a means to explore and classify electronic states in crystalline solids. This review provides a gentle but firm introduction to topological electronic band structure suitable for new researchers in the field. I begin by outlining the relevant concepts from topology, then give a summary of the theory of non-interacting electrons in periodic potentials. Next, I explain the concepts of the Berry phase and Berry curvature, and derive key formulae. The remainder of the article deals with how these ideas are applied to classify crystalline solids according to the topology of the electronic states, and the implications for observable properties. Among the topics covered are the role of symmetry in determining band degeneracies in momentum space, the Chern number and Z2 topological invariants, surface electronic states, two- and three-dimensional topological insulators, and Weyl and Dirac semimetals