Topology of Bands in Solids : From Insulators to Dirac Matter

TL;DR

This paper reviews how the topological properties of electronic band vector bundles in crystals reveal new characteristics of insulators and semi-metals, advancing understanding of their electronic phases.

Contribution

It provides a comprehensive review of the topological aspects of band theory in solids, connecting mathematical topology with physical electronic phases.

Findings

Topological invariants characterize insulators and semi-metals.

Vector bundle topology reveals new electronic phase properties.

Topological classification aids in understanding material behaviors.

Abstract

Bloch theory describes the electronic states in crystals whose energies are distributed as bands over the Brillouin zone. The electronic states corresponding to a (few) isolated energy band(s) thus constitute a vector bundle. The topological properties of these vector bundles provide new characteristics of the corresponding electronic phases. We review some of these properties in the case of (topological) insulators and semi-metals.