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Topological Insulators from the Perspective of Non-commutative Geometry and Index Theory

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Abstract

Topological insulators are solid state systems of independent electrons for which the Fermi level lies in a mobility gap, but the Fermi projection is nevertheless topologically non-trivial, namely it cannot be deformed into that of a normal insulator. This non-trivial topology is encoded in adequately defined invariants and implies the existence of surface states that are not susceptible to Anderson localization. This non-technical review reports on recent progress in the understanding of the underlying mathematical structures, with a particular focus on index theory.

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References

  1. Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Comment. Phys.-Math. 157, 245–278 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Altland, A., Zirnbauer, M.R.: Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142–1161 (1997)

    Article  Google Scholar 

  3. Andersson, A.: Index pairings for \({\mathbb{R}}^{n}\)-actions and Rieffel deformations. Preprint arXiv:1406.4078 (2014)

  4. Atiyah, M.F., Singer, I.M.: Index theory for skew-adjoint Fredholm operators. Publ. Math. 37, 5–26 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  5. Avila, J.C., Schulz-Baldes, H., Villegas-Blas, C.: Topological invariants of edge states for periodic two-dimensional models. Math. Phys. Anal. Geom. 16, 136–170 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Avron, J., Seiler, R., Simon, B.: The charge deficiency, charge transport and comparison of dimensions. Commun. Math. Phys. 159, 399–422 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bellissard, J.: K-theory of C∗-algebras in solid state physics. In: Dorlas, T., Hugenholtz, M., Winnink, M. (eds.) Lecture Notes in Physics, vol. 257, pp. 99–156. Springer, Berlin (1986)

    Google Scholar 

  8. Bellissard, J.: Ordinary quantum Hall effect and non-commutative cohomology. In: Ziesche, W., Weller, P. (eds.) Proc. of the Bad Schandau Conference on Localization, 1986. Teubner Texte Phys., vol. 16. Teubner, Leipzig (1988)

    Google Scholar 

  9. Bellissard, J., van Elst, A., Schulz-Baldes, H.: The non-commutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5451 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Blackadar, B.: K-Theory for Operator Algebras. Mathematical Sciences Research Institute Publications, vol. 5. Cambridge University Press Cambridge (1998)

    MATH  Google Scholar 

  11. Boersema, J.L., Loring, T.A.: \(K\)-theory for real C∗-algebras via unitary elements with symmetries. arXiv:1504.03284

  12. Bourne, C., Carey, A.L., Rennie, A.: The bulk-edge correspondence for the quantum Hall effect in Kasparov theory. Lett. Math. Phys. 105, 1253–1273 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bourne, C., Carey, A.L., Rennie, A.: A noncommutative framework for topological insulators. Rev. Math. Phys. 28, 1650004 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bourne, C., Kellendonk, J., Rennie, A.: The \(K \)-theoretic bulk-edge correspondence for topological insulators. Preprint arXiv:1604.02337

  15. Carey, A.L., Gayral, V., Rennie, A., Sukochev, F.A.: Index Theory for Locally Compact Noncommutative Geometries. Mem. AMS. (2014)

    MATH  Google Scholar 

  16. Carey, A.L., Phillips, J., Schulz-Baldes, H.: Spectral flow for skew-adjoint Fredholm operators. Preprint arXiv:1604.06994

  17. Connes, A.: Non-commutative differential geometry. Publ. Math. 62, 41–144 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)

    MATH  Google Scholar 

  19. De Nittis, G., Gomi, K.: Classification of “real” Bloch-bundles: topological quantum systems of type AI. J. Geom. Phys. 86, 303–338 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. De Nittis, G., Gomi, K.: Classification of “Quaternionic” Bloch-bundles. Commun. Math. Phys. 339, 1–55 (2015)

    Article  MATH  Google Scholar 

  21. De Nittis, G., Lein, M.: Topological polarization in graphene-like systems. J. Phys. A, Math. Theor. 46, 385001 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. De Nittis, G., Schulz-Baldes, H.: Spectral flows associated to flux tubes. Ann. Henri Poincaré 17, 1–35 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. De Nittis, G., Drabkin, M., Schulz-Baldes, H.: Localization and Chern numbers for weakly disordered BdG operators. Markov Process. Relat. Fields 21, 463–482 (2015)

    MathSciNet  Google Scholar 

  24. Elliott, G.A.: On the K-theory of the C∗-algebra generated by a projective representation of a torsion-free discrete Abelian group. In: Operator Algebras and Group Representations, vol. I, Neptun, 1980. Monographs Stud. Math, vol. 17, pp. 157–184. Pitman, Boston (1984)

    Google Scholar 

  25. Essin, A.M., Gurarie, V.: Bulk-boundary correspondence of topological insulators from their Green’s functions. Phys. Rev. B 84, 125132 (2011)

    Article  Google Scholar 

  26. Fiorenza, D., Monaco, D., Panati, G.: \(Z_{2}\) invariants of topological insulators as geometric obstructions. Commun. Math. Phys. 343, 1115–1157 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Freed, D.S., Moore, G.W.: Twisted equivariant matter. Ann. Henri Poincaré 14, 1927–2023 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Birkhäuser, Boston (2001)

    Book  MATH  Google Scholar 

  30. Graf, G.M., Porta, M.: Bulk-edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324, 851–895 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Grossmann, J., Schulz-Baldes, H.: Index pairings in presence of symmetries with applications to topological insulators. Commun. Math. Phys. 343, 477–513 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hatsugai, Y.: Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697–3700 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kane, C.L., Mele, E.J.: Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005)

    Article  Google Scholar 

  34. Kane, C.L., Mele, E.J.: Z(2) topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005)

    Article  Google Scholar 

  35. Karoubi, M.: \(K\)-Theory: An Introduction. Springer, Berlin (1978)

    Book  MATH  Google Scholar 

  36. Katsura, H., Koma, T.: The \(Z_{2}\) index of disordered topological insulators with time reversal symmetry. J. Math. Phys. 57, 021903 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kellendonk, J.: On the C∗-algebraic approach to topological phases for insulators. Preprint arXiv:1509.06271

  38. Kellendonk, J., Richter, T., Schulz-Baldes, H.: Edge current channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14, 87–119 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kennedy, R., Zirnbauer, M.: Bott periodicity for \({\mathbb {Z}}_{2}\) symmetric ground states of gapped free-fermion systems. Commun. Math. Phys. 342, 909–963 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kitaev, A.: Periodic table for topological insulators and superconductors. In: Advances in Theoretical Physics: Landau Memorial Conference AIP Conference Proceedings, vol. 1134, pp. 22–30 (2009)

    Google Scholar 

  41. Knez, I., Rettner, C.T., Yang, S.H., Parkin, S.S., Du, L., Du, R.R., Sullivan, G.: Observation of edge transport in the disordered regime of topologically insulating InAs/GaSb quantum wells. Phys. Rev. Lett. 112, 026602 (2014)

    Article  Google Scholar 

  42. Kubota, Y.: Controlled topological phases and bulk-edge correspondence. Preprint arXiv:1511.05314

  43. Lawson, H.B., Michelson, M.L.: Spin Geometry. Princeton University Press, Princeton (1989)

    Google Scholar 

  44. Li, D., Kaufmann, R.M., Wehefritz-Kaufmann, B.: Topological insulators and K-theory. Preprint arXiv:1510.08001

  45. Loring, T.A.: K-theory and pseudospectra for topological insulators. Ann. Phys. 356, 383–416 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  46. Loring, T.A., Hastings, M.B.: Topological insulators and C∗-algebras: theory and numerical practice. Ann. Phys. 326, 1699–1759 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ma, E.Y., et al.: Unexpected edge conduction in mercury telluride quantum wells under broken time-reversal symmetry. Nature Commun. 6, 7252 (2015)

    Article  Google Scholar 

  48. Macris, N.: On the equality of edge and bulk conductance in the integer quantum Hall effect: microscopic analysis. Unpublished manuscript (2003)

  49. Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence. Commun. Math. Phys. (2016)

  50. Phillips, J.: Self-adjoint Fredholm operators and spectral flow. Can. Math. Bull. 39, 460–467 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  51. Pimsner, M., Voiculescu, D.: Exact sequences for K-groups of certain cross-products of C∗ algebras. J. Oper. Theory 4, 93–118 (1980)

    MathSciNet  MATH  Google Scholar 

  52. Prodan, E.: Robustness of the spin-Chern number. Phys. Rev. B 80, 125327 (2009)

    Article  Google Scholar 

  53. Prodan, E., Schulz-Baldes, H.: Bulk and Boundary Invariants for Complex Topological Insulators: From \(K\)-Theory to Physics. Springer, Berlin (2016)

    Book  MATH  Google Scholar 

  54. Prodan, E., Leung, B., Bellissard, J.: The non-commutative \(n\)-th Chern number \((n\geq 0)\). J. Phys. A, Math. Theor. 46, 485202 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  55. Qi, X.L., Zhang, S.-C.: Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1111 (2011)

    Article  Google Scholar 

  56. Qi, X.L., Hughes, T.L., Zhang, S.-C.: Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008)

    Article  Google Scholar 

  57. Rammal, R., Bellissard, J.: An algebraic semi-classical approach to Bloch electrons in a magnetic field. J. Phys. (Paris) 51, 1803–1830 (1990)

    Article  Google Scholar 

  58. Rordam, M., Larsen, F., Laustsen, N.: An Introduction to K-Theory for C∗-Algebras. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  59. Ryu, S., Schnyder, A.P., Furusaki, A., Ludwig, A.W.W.: Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010)

    Article  Google Scholar 

  60. Schnyder, A.P., Ryu, S., Furusaki, A., Ludwig, A.W.W.: Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008)

    Article  Google Scholar 

  61. Schulz-Baldes, H.: Persistence of spin edge currents in disordered quantum spin Hall systems. Commun. Math. Phys. 324, 589–600 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  62. Schulz-Baldes, H.: \(\mathbb{Z} _{2}\)-Indices and factorization properties of odd symmetric Fredholm operators. Doc. Math. 20, 1481–1500 (2015)

    MathSciNet  MATH  Google Scholar 

  63. Schulz-Baldes, H., Teufel, S.: Orbital polarization and magnetization for independent particles in disordered media. Commun. Math. Phys. 319, 649–681 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  64. Stone, C.-K., Chiu, M., Roy, A.: Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock. J. Phys. A, Math. Theor. 44, 045001 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  65. Streda, P.: Theory of quantized Hall conductivity in two dimensions. J. Phys. C 15, L717–721 (1982)

    Article  Google Scholar 

  66. Su, W.P., Schrieffer, J.R., Heeger, A.J.: Soliton excitations in polyacetylene. Phys. Rev. B 22, 2099–2111 (1980)

    Article  Google Scholar 

  67. Thiang, G.C.: On the K-theoretic classification of topological phases of matter. Ann. Henri Poincaré 17, 757–794 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  68. Wegge-Olsen, N.E.: K-Theory and C∗-Algebras. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

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Acknowledgements

The author thanks his main collaborators Jean Bellissard, Johannes Kellendonk and Emil Prodan as well as further coauthors Giuseppe De Nittis, Stefan Teufel, Julian Grossmann, Carlos Villegas, Julio Cesar Avila, Alan Carey and John Phillips for inspiring intellectual input and persistence. This work is in part supported by the DFG.

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  1. Department Mathematik, FAU Erlangen-Nürnberg, Erlangen, Germany

    Hermann Schulz-Baldes

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Schulz-Baldes, H. Topological Insulators from the Perspective of Non-commutative Geometry and Index Theory. Jahresber. Dtsch. Math. Ver. 118, 247–273 (2016). https://doi.org/10.1365/s13291-016-0142-5

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