An introduction to topological insulators

Abstract

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. RésuméLes bandes électroniques dans un cristal sont définies par un ensemble de fonctions d'onde de Bloch dépendant du moment défini dans la première zone de Brillouin, ainsi que des énergies associées. Dans un isolant, les bandes de valence sont séparées des bandes de conduction par un gap en énergie. L'ensemble des bandes de valence est alors un objet bien défini, qui peut en particulier posséder une topologie non triviale. Lorsque cela se produit, l'isolant correspondant est appelé isolant topologique. Nous introduisons cette notion d'ordre topologique d'une bande comme une obstruction à la définition des fonctions d'ondes de Bloch à l'aide d'une convention de phase unique. Plusieurs modèles simples d'isolants topologiques en dimension deux sont considérés. Différentes expressions des indices topologiques correspondants sont finalement discutées.Keywords: topological insulator, topological band theory, quantum anomalous Hall effect, quantum spin Hall effect, Chern insulator, Kane-Mele insulator Mots-clés : isolant topologique, théorie des bandes topologique, effet Hall quantique anomal, effet Hall quantique de spin, isolant de Chern, isolant de Kane-Mele

Index of Dirac operators and classification of topological insulators

“…The low energy effective Hamiltonians of topological insulators and superconductors in all dimensions and symmetry classes are characterized by Dirac-like Hamiltonians. They can be written generally in terms of the momentum variables k in the following form [15,21]…”

Laser angle-resolved photoemission as a probe of initial statekzdispersion, final-state band gaps, and spin texture of Dirac states in theBi2Te3topolog

“…1 have continued to draw intense interest since their discovery a few years ago [2][3][4][5][6][7][8][9][10][11][12][13][14] . Bismuth based binary compounds [Bi 2 Te 3 and Bi 2 Se 3 ] were the first materials predicted to host the 3D TI phase, which was subsequently verified experimentally.…”

Adiabatic photo-steering theory in topological insulators

“…For this purpose, among several methods for the Chern number calculation [74–76], we apply a procedure presented by De Nittis and Lein to reproduce the diagram of Chern number for the Haldane model [77]. We first rewrite the model into the following convenient form, which is a linear combination of the Pauli matrix , with where and .…”