Advertisement

Communications in Mathematical Physics

, Volume 324, Issue 3, pp 851–895 | Cite as

Bulk-Edge Correspondence for Two-Dimensional Topological Insulators

  • Gian Michele Graf
  • Marcello Porta
Article

Abstract

Topological insulators can be characterized alternatively in terms of bulk or edge properties. We prove the equivalence between the two descriptions for two-dimensional solids in the single-particle picture. We give a new formulation of the \({\mathbb{Z}_{2}}\)-invariant, which allows for a bulk index not relying on a (two-dimensional) Brillouin zone. When available though, that index is shown to agree with known formulations. The method also applies to integer quantum Hall systems. We discuss a further variant of the correspondence, based on scattering theory.

Keywords

Vector Bundle Edge State Topological Insulator Quantum Hall Effect Fermi Line 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avila, J.C., Schulz-Baldes, H., Villegas-Blas, C.: Topological invariants of edge states for periodic two-dimensional models. http://arXiv.org/abs/1202.0537v1 [math ph], 2012, to appear in Math. Phys., Anal. Geom
  2. 2.
    Bernevig B.A., Hughes T.L., Zhang S.-C.: Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006)ADSCrossRefGoogle Scholar
  3. 3.
    Bräunlich G., Graf G.M., Ortelli G.: Equivalence of topological and scattering approaches to quantum pumping. Commun. Math. Phys. 295, 243–259 (2010)ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Essin A.M., Gurarie V.: Bulk-boundary correspondence of topological insulators from their Green’s functions. Phys. Rev. B 84, 125132 (2011)ADSCrossRefGoogle Scholar
  5. 5.
    Fröhlich J., Kerler T.: Universality in quantum Hall systems. Nucl. Phys. B 354, 369–417 (1991)ADSCrossRefGoogle Scholar
  6. 6.
    Fröhlich J., Studer U.M.: Gauge invariance and current algebra in nonrelativistic many-body theory. Rev. Mod. Phys 65, 733 (1993)ADSCrossRefGoogle Scholar
  7. 7.
    Fröhlich, J., Studer, U.M., Thiran, E.: Quantum theory of large systems of non-relativistic matter. Les Houches Lectures 1994, London, New York: Elsevier (1995) available at http://arXiv.org/abs/cond-mat/9508062v1, 1995
  8. 8.
    Fröhlich J., Zee A.: Large scale physics of the quantum Hall fluid. Nucl. Phys. B 364, 517–540 (1991)ADSCrossRefGoogle Scholar
  9. 9.
    Fu L., Kane C.L.: Time reversal polarization and a Z 2 adiabatic spin pump. Phys. Rev. B 74, 195312 (2006)ADSCrossRefGoogle Scholar
  10. 10.
    Fujita M., Wakabayashi K., Nakada K., Kusakabe K.: Peculiar localized state at zigzag graphite edge. J. Phys. Soc. Jpn. 65, 1920–1923 (1996)ADSCrossRefGoogle Scholar
  11. 11.
    Haldane F.D.M: Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Hasan M.Z., Kane C.L.: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    Hatsugai Y.: Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697 (1993)MathSciNetADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Hatsugai Y., Ryu S.: Topological origin of zero-energy edge states in particle-hole symmetric systems. Phys. Rev. Lett. 89, 077002 (2002)ADSCrossRefGoogle Scholar
  15. 15.
    Hsieh D., Qian D., Wray L., Xia Y., Hor Y.S., Cava R.J., Hasan M.Z.: A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970 (2008)ADSCrossRefGoogle Scholar
  16. 16.
    Kane C.L., Mele E.J.: Z 2 Topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005)ADSCrossRefGoogle Scholar
  17. 17.
    Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer-Verlag, 1980Google Scholar
  18. 18.
    Kohn W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, 809–821 (1959)MathSciNetADSCrossRefzbMATHGoogle Scholar
  19. 19.
    König M., Wiedmann S., Brüne C., Roth A., Buhmann H., Molenkamp L.W., Qi X.-L., Zhang S.-C.: Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766 (2007)ADSCrossRefGoogle Scholar
  20. 20.
    Moore J.E., Balents L.: Topological invariants of time-reversal-invariant band structures. Phys. Rev. B 75, 121306(R) (2007)ADSCrossRefGoogle Scholar
  21. 21.
    Nakada K., Fujita M., Dresselhaus G., Dresselhaus M.S.: Edge state in graphene ribbons: Nanometer size effect and edge shape dependence. Phys. Rev. B. 54, 17954 (1996)ADSCrossRefGoogle Scholar
  22. 22.
    Nakahara, M.: Geometry, Topology and Physics. Graduate Student Series in Physics, London: Institute of Physics Publishing, 1990Google Scholar
  23. 23.
    Pfeffer W.F.: More on involutions of a circle. Amer. Math. Monthly 81, 613 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Prodan E.: Robustness of the spin-Chern number. Phys. Rev. B 80, 125327 (2009)ADSCrossRefGoogle Scholar
  25. 25.
    Qi X.-L., Wu Y.-S., Zhang S.-C.: Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors. Phys. Rev. B 74, 085308 (2006)ADSCrossRefGoogle Scholar
  26. 26.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III. Scattering Theory. New York: Academic Press, 1979Google Scholar
  27. 27.
    Roy R.: Z 2 classification of quantum spin Hall systems: An approach using time-reversal invariance. Phys. Rev. B 79, 195321 (2009)ADSCrossRefGoogle Scholar
  28. 28.
    Schulz-Baldes H., Kellendonk J., Richter T.: Simultaneous quantization of edge and bulk Hall conductivity. J. Phys. A: Math. Gen. 33, L27 (2000)MathSciNetADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Sheng D.N., Weng Z.Y., Sheng L., Haldane F.D.M.: Quantum spin-Hall effect and topologically invariant Chern numbers. Phys. Rev. Lett. 97, 036808 (2006)ADSCrossRefGoogle Scholar
  30. 30.
    Thouless D.J.: Quantisation of particle transport. Phys. Rev. B 27, 6083–6087 (1983)MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    Wen X.G.: Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states. Phys. Rev. B 41, 12838–12844 (1990)ADSCrossRefGoogle Scholar
  32. 32.
    Zhang S.-C.: The Chern-Simons-Landau-Ginzburg theory of the fractional quantum Hall effect. Int. J. Mod. Phys. B 6, 25–58 (1992)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Theoretische PhysikETH ZurichZurichSwitzerland

Personalised recommendations