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Communications in Mathematical Physics

, Volume 343, Issue 2, pp 477–513 | Cite as

Index Pairings in Presence of Symmetries with Applications to Topological Insulators

  • Julian Großmann
  • Hermann Schulz-Baldes
Article

Abstract

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).

Keywords

Dirac Operator Chiral Symmetry Clifford Algebra Fredholm Operator Topological Insulator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. AZ.
    Altland A., Zirnbauer M.: Non-standard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142–1161 (1997)ADSCrossRefGoogle Scholar
  2. ABS.
    Atiyah M.F., Bott R., Shapiro A.: Clifford modules. Topology 3, 3–38 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  3. AS.
    Atiyah M.F., Singer I.M.: Index theory for skew-adjoint Fredholm operators. Publ. IHES 37, 5–26 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  4. ASV.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. ASS.
    Avron J., Seiler R., Simon B.: The index of a pair of projections. J. Funct. Anal. 120, 220–237 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bel.
    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)Google Scholar
  7. BES.
    Bellissard J., van sElst A., Schulz-Baldes H.: The non-commutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5451 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. BL.
    Boersema, J.L., Loring, T.A.: K-theory for real C*-algebras via unitary elements with symmetries. arXiv:1504.03284
  9. BCR.
    Bourne, C., Carey, A.L., Rennie, A.: A noncommutative framework for topological insulators. arXiv:1509.07210
  10. Con1.
    Connes A.: Noncommutative Geometry. Academic Press, New York (1995)zbMATHGoogle Scholar
  11. Con2.
    Connes A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. DG.
    De Nittis G., Gomi K.: Classification of “Real” Bloch-bundles: topological quantum systems of type AI. J. Geom. Phys. 86, 303–338 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. DS.
    De Nittis, G., Schulz-Baldes, H.: Spectral flows associated to flux tubes. Ann. H. Poincaré. arXiv:1405.2054
  14. EG.
    Essin A.M., Gurarie V.: Bulk-boundary correspondence of topological insulators from their Green’s functions. Phys. Rev. B 84, 125132 (2011)ADSCrossRefGoogle Scholar
  15. FM.
    Freed D.S., Moore G.W.: Twisted equivariant matter. Ann. H. Poincaré 14, 1927–2023 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. FKM.
    Fu L., Kane C.L., Mele E.J.: Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007)ADSCrossRefGoogle Scholar
  17. GVF.
    Gracia-Bondía J.M., Várilly J.C., Figueroa H.: Elements of Noncommutative Geometry. Birkhäuser, Boston (2001)CrossRefzbMATHGoogle Scholar
  18. GP.
    Graf G.M., Porta M.: Bulk-edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324, 851–895 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. HK.
    Hasan M.Z., Kane C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)ADSCrossRefGoogle Scholar
  20. HL.
    Hastings M.B., Loring T.A.: Topological insulators and C*-algebras: theory and numerical practice. Ann. Phys. 326, 1699–1759 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. KM.
    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. Kar.
    Karoubi M.: K-Theory: An Introduction. Springer, Berlin (1978)CrossRefzbMATHGoogle Scholar
  23. Kas.
    Kasparov G.G.: The operator K-functor and extensions of C*-algebras. Math. USSR Izv. 16, 513–572 (1981)CrossRefzbMATHGoogle Scholar
  24. Kel.
    Kellendonk, J.: On the C*-algebraic approach to topological phases for insulators. arXiv:1509.06271
  25. KRS.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. KZ.
    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. Kit1.
    Kitaev A.Y.: Unpaired Majorana fermions in quantum wires. Physics-Uspekhi 44, 131–136 (2001)ADSCrossRefGoogle Scholar
  28. Kit2.
    Kitaev A.Y.: Periodic table for topological insulators and superconductors. (Adv. Theor. Phys. Landau Meml. Conf.) AIP Conf. Proc. 1134, 22–30 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. LM.
    Lawson H.B., Michelsohn M.L.: Spin Geometry. Princeton University Press, Princeton (1989)zbMATHGoogle Scholar
  30. Lor.
    Loring T.: K-theory and pseudospectra for topological insulators. Ann. Phys. 356, 383–416 (2015)ADSMathSciNetCrossRefGoogle Scholar
  31. Noe.
    Noether F.: Über eine Klasse singulärer Integralgleichungen. Math. Ann. 82, 42–63 (1920)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Por.
    Porteous I.R.: Clifford Algebras and the Classical Groups. Cambridge Univ. Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  33. PLB.
    Prodan E., Leung B., Bellissard J.: The non-commutative nth Chern number (n \({\geq}\) 0). J. Phys. A Math. Theor. 46, 485202 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. PS.
    Prodan, E., Schulz-Baldes, H.: Non-commutative odd Chern numbers and topological phases of disordered chiral systems. arXiv:1402.5002
  35. QHZ.
    Qi X.L., Hughes T.L., Zhang S.-C.: Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008)ADSCrossRefGoogle Scholar
  36. RLL.
    Rordam M., Larsen F., Laustsen, N.: An Introduction to K-Theory for C*-Algebras. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
  37. RSFL.
    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)ADSCrossRefGoogle Scholar
  38. SRFL.
    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)ADSCrossRefGoogle Scholar
  39. Sch.
    Schröder H.: K-Theory for Real C*-Algebras and Applications. Longman Scientific & Technical, London (1993)zbMATHGoogle Scholar
  40. SB.
    Schulz-Baldes, H.: \({{\mathbb {Z}_{2}}}\)-indices and factorization properties of odd symmetric Fredholm operators. To appear in Doc. Math. arXiv:1311.0379
  41. SCR.
    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)ADSMathSciNetzbMATHGoogle Scholar
  42. Thi.
    Thiang, G.C.: On the K-theoretic classification of topological phases of matter. Ann. H. Poincaré. arXiv:1406.7366
  43. VD.
    Van Daele A.: K-theory for graded Banach algebras I. Q. J. Math. 39, 185–199 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department MathematikFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

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