Topological Invariants

  • Shun-Qing ShenEmail author
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 187)


There are two classes of topological invariants for topological phases of matter. The first is characterized by the elements of the group Z , which consists of all integers. For example, the integer quantum Hall effect is characterized by the integer n, i.e., the filling factor of electrons. The second class is characterized by the elements of the group \(\mathrm {Z}_{2}\), which consists of 0 and 1, or 1 and \(-1\) depending on convention. In a topological insulator with time reversal symmetry, 0 and 1 represent the existence of odd and even numbers of the surface states in three dimensions or even and odd numbered pairs of helical edge states in two dimensions, respectively.


Topological Insulators Time-reversal Invariant Momenta Berry Curvature Spin Pumping Charge Pump 
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.


  1. 1.
    C. Kittel, Introduction to Solid State Physics, 7th edn. (Willey, New York, 1996)zbMATHGoogle Scholar
  2. 2.
    D. Xiao, M.C. Chang, Q. Niu, Rev. Mod. Phys. 82, 1959 (2010)ADSCrossRefGoogle Scholar
  3. 3.
    M. Kohmoto, Ann. Phys. 160, 343 (1985)ADSCrossRefGoogle Scholar
  4. 4.
    A. Messiah, Quantum Mechanics (Interscience, New York, 1961)Google Scholar
  5. 5.
    S.Q. Shen, Phys. Rev. B 70, 081311 (2004)Google Scholar
  6. 6.
    M.C. Chang, Q. Niu, Phys. Rev. Lett. 75, 1348 (1995)ADSCrossRefGoogle Scholar
  7. 7.
    G. Sundaram, Q. Niu, Phys. Rev. B 59, 14915 (1999)ADSCrossRefGoogle Scholar
  8. 8.
    R. Resta, D. Vanderbilt, A modern perspective, in Physics of Ferroelectrics, ed. by K. Rabe, C.H. Ahn, J.M. Triscone (Springer, Berlin, 2007), p. 31CrossRefGoogle Scholar
  9. 9.
    R.D. King-Smith, D. Vanderbilt, Phys. Rev. B 47, 1651 (1993)ADSCrossRefGoogle Scholar
  10. 10.
    M.J. Rice, E.J. Mele, Phys. Rev. Lett. 49, 1455 (1982)ADSCrossRefGoogle Scholar
  11. 11.
    L. Fu, C.L. Kane, Phys. Rev. B 74, 195312 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    R.B. Laughlin, Phys. Rev. B 23, 5632 (1981)ADSCrossRefGoogle Scholar
  13. 13.
    L. Fu, C.L. Kane, Phys. Rev. B 76, 045302 (2007)ADSCrossRefGoogle Scholar
  14. 14.
    J.E. Moore, L. Balents, Phys. Rev. B 75, 121306(R) (2007)ADSCrossRefGoogle Scholar
  15. 15.
    T. Fukui, Y. Hatsugai, J. Phys. Soc. Jpn. 76, 053702 (2007)ADSCrossRefGoogle Scholar
  16. 16.
    L. Fu, C.L. Kane, E.J. Mele, Phys. Rev. Lett. 98, 106803 (2007)ADSCrossRefGoogle Scholar
  17. 17.
    A. Cayley, Cambridge Dublin Math. J. 7, 40–51 (1852)Google Scholar
  18. 18.
    C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 146802 (2005)ADSCrossRefGoogle Scholar
  19. 19.
    J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, Inc., New York, 1964)zbMATHGoogle Scholar
  20. 20.
    G.E. Volovik, JETP Lett. 91, 55 (2010)ADSCrossRefGoogle Scholar
  21. 21.
    G.E. Volovik, The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003)zbMATHGoogle Scholar
  22. 22.
    S.Q. Shen, W.Y. Shan, H.Z. Lu, SPIN 01, 33 (2011)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of PhysicsThe University of Hong KongHong KongChina

Personalised recommendations