[Introduction aux isolants topologiques]
Les bandes électroniques dans un cristal sont définies par un ensemble de fonctions dʼonde 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. Lʼensemble des bandes de valence est alors un objet bien défini, qui peut en particulier posséder une topologie non triviale. Lorsque cela se produit, lʼisolant correspondant est appelé isolant topologique. Nous introduisons cette notion dʼordre topologique dʼune bande comme une obstruction à la définition des fonctions dʼondes de Bloch à lʼaide dʼune convention de phase unique. Plusieurs modèles simples dʼisolants topologiques en dimension deux sont considérés. Différentes expressions des indices topologiques correspondants sont finalement discutées.
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 nontrivial 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.
Mot clés : Isolant topologique, Théorie des bandes topologique, Effet Hall quantique anomal, Effet Hall quantique de spin, Isolant de Chern, Isolant de Kane–Mele
Michel Fruchart 1 ; David Carpentier 1
@article{CRPHYS_2013__14_9-10_779_0,
author = {Michel Fruchart and David Carpentier},
title = {An introduction to topological insulators},
journal = {Comptes Rendus. Physique},
pages = {779--815},
publisher = {Elsevier},
volume = {14},
number = {9-10},
year = {2013},
doi = {10.1016/j.crhy.2013.09.013},
language = {en},
}
Michel Fruchart; David Carpentier. An introduction to topological insulators. Comptes Rendus. Physique, Volume 14 (2013) no. 9-10, pp. 779-815. doi : 10.1016/j.crhy.2013.09.013. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2013.09.013/
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