Skip to main content
Log in

Index Pairings in Presence of Symmetries with Applications to Topological Insulators

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript


In a basic framework of a complex Hilbert space equipped with a complex conjugation and an involution, linear operators can be real, quaternionic, symmetric or anti-symmetric, and orthogonal projections can furthermore be Lagrangian. This paper investigates index pairings of projections and unitaries submitted to such symmetries. Various scenarios emerge: Noether indices can take either arbitrary integer values or only even integer values or they can vanish and then possibly have secondary \({{\mathbb {Z}_{2}}}\)-invariants. These general results are applied to prove index theorems for the strong invariants of disordered topological insulators. The symmetries come from the Fermi projection (K-theoretic part of the pairing) and the Dirac operator (K-homological part of the pairing depending on the dimension of physical space).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


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

    Article  ADS  Google Scholar 

  2. Atiyah M.F., Bott R., Shapiro A.: Clifford modules. Topology 3, 3–38 (1964)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. Avron J., Seiler R., Simon B.: The index of a pair of projections. J. Funct. Anal. 120, 220–237 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bellissard, J.: K-theory of C*-algebras in solid state physics. In: Dorlas, T.C., Hugenholtz, M.N., Winnink, M. (eds.) Statistical Mechanics and Field Theory, Mathematical Aspects. Lecture Notes in Physics, vol. 257, pp. 99–156 (1986)

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Boersema, J.L., Loring, T.A.: K-theory for real C*-algebras via unitary elements with symmetries. arXiv:1504.03284

  9. Bourne, C., Carey, A.L., Rennie, A.: A noncommutative framework for topological insulators. arXiv:1509.07210

  10. Connes A.: Noncommutative Geometry. Academic Press, New York (1995)

    MATH  Google Scholar 

  11. Connes A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. De Nittis, G., Schulz-Baldes, H.: Spectral flows associated to flux tubes. Ann. H. Poincaré. arXiv:1405.2054

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Fu L., Kane C.L., Mele E.J.: Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007)

    Article  ADS  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Hasan M.Z., Kane C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Kane C.L., Mele E.J.: \({{\mathbb {Z}_{2}}}\) topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 145805–146802 (2005)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  23. Kasparov G.G.: The operator K-functor and extensions of C*-algebras. Math. USSR Izv. 16, 513–572 (1981)

    Article  MATH  Google Scholar 

  24. Kellendonk, J.: On the C*-algebraic approach to topological phases for insulators. arXiv:1509.06271

  25. 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 

  26. Kennedy, R., Zirnbauer, M.: Bott periodicity for \({{\mathbb {Z}_{2}}}\) symmetric ground states of gapped free-fermion systems.Commun. Math. Phys. (2015).doi:10.1007/s00220-015-2512-8

  27. Kitaev A.Y.: Unpaired Majorana fermions in quantum wires. Physics-Uspekhi 44, 131–136 (2001)

    Article  ADS  Google Scholar 

  28. Kitaev A.Y.: Periodic table for topological insulators and superconductors. (Adv. Theor. Phys. Landau Meml. Conf.) AIP Conf. Proc. 1134, 22–30 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  31. Noether F.: Über eine Klasse singulärer Integralgleichungen. Math. Ann. 82, 42–63 (1920)

    Article  MathSciNet  MATH  Google Scholar 

  32. Porteous I.R.: Clifford Algebras and the Classical Groups. Cambridge Univ. Press, Cambridge (1995)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Prodan, E., Schulz-Baldes, H.: Non-commutative odd Chern numbers and topological phases of disordered chiral systems. arXiv:1402.5002

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

    Article  ADS  Google Scholar 

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

    Book  MATH  Google Scholar 

  37. 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  ADS  Google Scholar 

  38. 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, 195125–295144 (2008)

    Article  ADS  Google Scholar 

  39. Schröder H.: K-Theory for Real C*-Algebras and Applications. Longman Scientific & Technical, London (1993)

    MATH  Google Scholar 

  40. Schulz-Baldes, H.: \({{\mathbb {Z}_{2}}}\)-indices and factorization properties of odd symmetric Fredholm operators. To appear in Doc. Math. arXiv:1311.0379

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

    ADS  MathSciNet  MATH  Google Scholar 

  42. Thiang, G.C.: On the K-theoretic classification of topological phases of matter. Ann. H. Poincaré. arXiv:1406.7366

  43. Van Daele A.: K-theory for graded Banach algebras I. Q. J. Math. 39, 185–199 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Hermann Schulz-Baldes.

Additional information

Communicated by Y. Kawahigashi

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Großmann, J., Schulz-Baldes, H. Index Pairings in Presence of Symmetries with Applications to Topological Insulators. Commun. Math. Phys. 343, 477–513 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: