An Introduction to Topological Order in Insulators

Michel Fruchart (1) , David Carpentier (1)

1 Phys-ENS - Laboratoire de Physique de l'ENS Lyon

Electronic bands in crystals are described by an ensemble of Bloch wave functions indexed by momenta defined in the first Brillouin Zone, and their associated energies. In an insulator, an energy gap around the chemical potential separates valence bands from conduction bands. The ensemble of valence bands is then a well defined object, which can possess non-trivial or twisted topological properties. In the case of a twisted topology, the insulator is called a topological insulator. We introduce this notion of topological order in insulators as an obstruction to define the Bloch wave functions over the whole Brillouin Zone using a single phase convention. Several simple historical models displaying a topological order in dimension two are considered. Various expressions of the corresponding topological index are finally discussed.

Keywords: topological insulator; band theory; topological band theory; quantum Hall effect; quantum anomalous Hall effect; quantum spin Hall effect; Chern insulator; Kane-Mele insulator

Domains: Physics [physics]; Condensed Matter [cond-mat]; Materials Science [cond-mat.mtrl-sci]; Physics [physics]; Condensed Matter [cond-mat]; Mesoscopic Systems and Quantum Hall Effect [cond-mat.mes-hall]

Michel Fruchart, David Carpentier. An Introduction to Topological Order in Insulators. 2013. ⟨ensl-00868307v1⟩