Comptes Rendus
Topological insulators/Isolants topologiques
An introduction to topological insulators
[Introduction aux isolants topologiques]
Comptes Rendus. Physique, Volume 14 (2013) no. 9-10, pp. 779-815.

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.

Publié le :
DOI : 10.1016/j.crhy.2013.09.013
Keywords: Topological insulator, Topological band theory, Quantum anomalous Hall effect, Quantum spin Hall effect, Chern insulator, Kane–Mele insulator
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

1 Laboratoire de physique, École normale supérieure de Lyon (UMR CNRS 5672), 46, allée dʼItalie, 69007 Lyon, France
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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/

[1] M. Nakahara Geometry, Topology and Physics, Taylor & Francis, 2003

[2] K.v. Klitzing; G. Dorda; M. Pepper New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett., Volume 45 (1980), pp. 494-497 | DOI

[3] The Quantum Hall Effet, Poincaré Seminar (B. Douçot; B. Duplantier; V. Pasquier; V. Rivasseau, eds.), Birkhäuser, 2004

[4] D.J. Thouless; M. Kohmoto; M.P. Nightingale; M. den Nijs Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., Volume 49 (1982), pp. 405-408 | DOI

[5] F.D.M. Haldane Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett., Volume 61 (1988), pp. 2015-2018 http://faculty.washington.edu/cobden/papers/haldane88.pdf | DOI

[6] C.-Z. Chang; J. Zhang; X. Feng; J. Shen; Z. Zhang; M. Guo; K. Li; Y. Ou; P. Wei; L.-L. Wang; Z.-Q. Ji; Y. Feng; S. Ji; X. Chen; J. Jia; X. Dai; Z. Fang; S.-C. Zhang; K. He; Y. Wang; L. Lu; X.-C. Ma; Q.-K. Xue Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator, Science, Volume 340 (2013) no. 6120, pp. 167-170

[7] C.L. Kane; E.J. Mele Z2 topological order and the quantum spin Hall effect, Phys. Rev. Lett., Volume 95 (2005), p. 146802 | arXiv | DOI

[8] C.L. Kane; E.J. Mele Quantum spin Hall effect in graphene, Phys. Rev. Lett., Volume 95 (2005), p. 226801 | arXiv | DOI

[9] B.A. Bernevig; T.L. Hughes; S.-C. Zhang Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science, Volume 314 (2006), pp. 1757-1761 | arXiv | DOI

[10] M. König; S. Wiedmann; C. Brüne; A. Roth; H. Buhmann; L.W. Molenkamp; X.-L. Qi; S.-C. Zhang Quantum spin Hall insulator state in HgTe quantum wells, Science, Volume 318 (2007) no. 5851, pp. 766-770 | arXiv | DOI

[11] A. Roth; C. Brüne; H. Buhmann; L.W. Molenkamp; J. Maciejko; X.-L. Qi; S.-C. Zhang Nonlocal transport in the quantum spin Hall state, Science, Volume 325 (2009), p. 294

[12] L. Fu; C.L. Kane; E.J. Mele Topological insulators in three dimensions, Phys. Rev. Lett., Volume 98 (2007), p. 106803 | arXiv | DOI

[13] J.E. Moore; L. Balents Topological invariants of time-reversal-invariant band structures, Phys. Rev. B, Volume 75 (2007), p. 121306 | arXiv | DOI

[14] R. Roy Topological phases and the quantum spin Hall effect in three dimensions, Phys. Rev. B, Volume 79 (2009), p. 195322 | arXiv | DOI

[15] M.Z. Hasan; C.L. Kane Colloquium: Topological insulators, Rev. Mod. Phys., Volume 82 (2010), pp. 3045-3067 | arXiv | DOI

[16] X.-L. Qi; S.-C. Zhang Topological insulators and superconductors, Rev. Mod. Phys., Volume 83 (2011), p. 1057

[17] B.A. Bernevig; T.L. Hughes Topological Insulators and Topological Superconductors, Princeton University Press, 2013

[18] G.E. Volovik The Universe in a Helium Droplet, Oxford University Press, 2003

[19] X.G. Wen Quantum Field Theory of Many-Body Systems, Oxford University Press, 2004

[20] J.C. Baez; J.P. Muniain Gauge Fields, Knots, and Gravity, World Scientific, 1994

[21] A. Hatcher Vector bundles and K-theory, 2003 http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html

[22] B. Simon Holonomy, the quantum adiabatic theorem, and Berryʼs phase, Phys. Rev. Lett., Volume 51 (1983), pp. 2167-2170 | DOI

[23] G. Panati Triviality of bloch and Bloch-Dirac bundles, Ann. I. H. Poincaré, Volume 8 (2007), pp. 995-1011 | arXiv | DOI

[24] C. Bena; G. Montambaux Remarks on the tight-binding model of graphene, New J. Phys., Volume 11 (2009), p. 095003 | arXiv | DOI

[25] J.E. Avron; R. Seiler; B. Simon Homotopy and quantization in condensed matter physics, Phys. Rev. Lett., Volume 51 (1983), pp. 51-53 | DOI

[26] P.A.M. Dirac Quantised singularities in the electromagnetic field http://www.jstor.org/stable/95639 (Proc. Roy. Soc. Lond. A)

[27] D. Chruściński; A. Jamiołkowski Geometric Phases in Classical and Quantum Mechanics, Progress in Mathematical Physics, Birkhäuser, 2004

[28] M.V. Berry Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. Lond. A, Volume 392 (1984), pp. 45-57 http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry120.pdf | DOI

[29] D. Sticlet; F. Piéchon; J.-N. Fuchs; P. Kalugin; P. Simon Geometrical engineering of a two-band Chen insulator in two dimensions with arbitrary topological index, Phys. Rev. B, Volume 85 (2012), p. 165456 | arXiv | DOI

[30] E. Fradkin Field Theories of Condensed Matter Physics, Cambridge University Press, 2013

[31] H.B. Nielsen; M. Ninomiya No go theorem for regularizing chiral fermions, Phys. Lett., Volume 105 (1981), p. 219

[32] M. Le Bellac Quantum Physics, Cambridge University Press, 2012

[33] J.J. Sakurai Modern Quantum Mechanics, Addison Wesley, 1993

[34] O. Madelung Introduction to Solid-State Theory, Springer Series in Solid-State Sciences, Springer, 1996

[35] L. Fu; C.L. Kane Topological insulators with inversion symmetry, Phys. Rev. B, Volume 76 (2007), p. 045302 | arXiv | DOI

[36] G.M. Graf; M. Porta Bulk-edge correspondence for two-dimensional topological insulators, July 2012 | arXiv

[37] A. Alexandradinata; X. Dai; B.A. Bernevig Wilson-loop characterization of inversion-symmetric topological insulators, August 2012 | arXiv

[38] A. Kitaev Periodic table for topological insulators and superconductors (V. Lebedev; M. FeigelʼMan, eds.), American Institute of Physics Conference Series, American Institute of Physics Conference Series, vol. 1134, 2009, pp. 22-30 | arXiv | DOI

[39] S. Ryu; A.P. Schnyder; A. Furusaki; A.W.W. Ludwig Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys., Volume 12 (2010), p. 065010 | arXiv | DOI

[40] D.S. Freed; G.W. Moore Twisted equivariant matter, Ann. I. H. Poincaré (2013) | arXiv | DOI

[41] R. Jackiw; C. Rebbi Solitons with fermion number 1/2, Phys. Rev. D, Volume 13 (1976) no. 13, pp. 3398-3409

[42] L. Fu; C.L. Kane Time reversal polarization and a Z2 adiabatic spin pump, Phys. Rev. B, Volume 74 (2006), p. 195312 | arXiv | DOI

[43] A.A. Soluyanov; D. Vanderbilt Wannier representation of Z2 topological insulators, Phys. Rev. B, Volume 83 (2011), p. 035108 | arXiv | DOI

[44] S.-S. Lee; S. Ryu Many-body generalization of the Z2 topological invariant for the quantum spin Hall effect, Phys. Rev. Lett., Volume 100 (2008), p. 186807 | arXiv | DOI

[45] R. Roy Z2 classification of quantum spin Hall systems: An approach using time-reversal invariance, Phys. Rev. B, Volume 79 (2009), p. 195321 | arXiv | DOI

[46] X.-L. Qi; T.L. Hughes; S.-C. Zhang Topological field theory of time-reversal invariant insulators, Phys. Rev. B, Volume 78 (2008), p. 195424 | DOI

[47] Z. Wang; X.-L. Qi; S.-C. Zhang Equivalent topological invariants of topological insulators, New J. Phys., Volume 12 (2010) no. 6, p. 065007 http://stacks.iop.org/1367-2630/12/i=6/a=065007 | DOI

[48] J.E. Avron; L. Sadun; J. Segert; B. Simon Chern numbers, quaternions, and Berryʼs phases in Fermi systems, Commun. Math. Phys., Volume 124 (1989), pp. 595-627 http://projecteuclid.org/euclid.cmp/1104179297 | DOI

[49] L. Fu Topological crystalline insulators, Phys. Rev. Lett., Volume 106 (2011) no. 10, p. 106802

[50] A.M. Turner; A. Vishwanath Beyond band insulators: Topology of semi-metals and interacting phases, 2013 | arXiv

[51] N. Bourbaki Algèbre: Chapitre 9, Springer, 2006

[52] J.H.B. Lawson The Theory of Gauge Fields in Four Dimensions, American Mathematical Society, 1985

[53] R.W.R. Darling Differential Forms and Connections, Cambridge University Press, 1994

[54] V. Guillemin; A. Pollack Differential Topology, American Mathematical Society, 1974 reprint (2010) edition

[55] B. Dubrovin; A. Fomenko; S. Novikov Modern Geometry – Methods and Applications: Part II: The Geometry and Topology of Manifolds, Springer, 1985

[56] H. Flanders Differential Forms with Applications to the Physical Sciences, Mathematics in Science and Engineering, vol. 11, Academic Press, New York, 1989

[57] G. Dinca; J. Mawhin Brouwer degree and applications, 2009 http://www.ljll.math.upmc.fr/~smets/ULM/Brouwer_Degree_and_applications.pdf

[58] D. Bleecker; B. Booss-Bavnbek Index theory with applications to mathematics and physics, 2004 http://milne.ruc.dk/~Booss/A-S-Index-Book/BlckBss_2012_04_25.pdf (preprint)

[59] W. Fulton Algebraic Topology: A First Course, Springer, 2008

[60] D. Ullrich Complex Made Simple, Graduate Studies in Mathematics, American Mathematical Society, 2008

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