Abstract
There are two classes of topological invariants for topological insulators. One is characterized by the elements of the group Z, which consists of all integers. For example, the integer quantum Hall effect is characterized by an integer n, that is, the filling factor of electrons. The other is by the elements of the group Z2, which consists of 0 and 1 or 1 and −1. 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 the even and odd number pairs of helical edge states in two dimensions, respectively.
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References
C. Kittel, Introduction to Solid State Physics, 7th edn. (Willey, New York, 1996)
D. Xiao, M.C. Chang, Q. Niu, Rev. Mod. Phys. 82, 1959 (2010)
M. Kohmoto, Ann. Phys. 160, 343 (1985)
A. Messiah, Quantum Mechanics, Chap. XVII (Interscience, New York, 1961)
S.Q. Shen, Phys. Rev. B 70, 081311(R) (2004)
M.C. Chang, Q. Niu, Phys. Rev. Lett. 75, 1348 (1995)
G. Sundaram, Q. Niu, Phys. Rev. B 59, 14915 (1999)
R. Resta, D. Vanderbilt, In: Physics of Ferroelectrics: A modern Perspective, ed. by K.M. Rabe, C.H. Ahn, J.M. Triscone (Springer, Berlin, 2007) p. 31.
R.D. King-Smith, D. Vanderbilt, Phys. Rev. B 47, 1651 (1993)
M.J. Rice, E.J. Mele, Phys. Rev. Lett. 49, 1455 (1982)
L. Fu, C.L. Kane, Phys. Rev. B 74, 195312 (2006)
R.B. Laughlin, Phys. Rev. B 23, 5632 (1981)
L. Fu, C.L. Kane, Phys. Rev. B 76, 045302 (2007)
J.E. Moore, L. Balents, Phys. Rev. B 75, 121306(R) (2007)
A. Cayley, Cambridge and Dublin Mathematical Journal VII, 40 (1852). Also see the item “Pfaffian” in Wikipedia
T. Fukui, Y. Hatsugai, J. Phys. Soc. Jpn. 76, 053702 (2007)
L. Fu, C.L. Kane, E.J. Mele, Phys. Rev. Lett. 98, 106803 (2007)
C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 146802 (2005)
J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), p. 72
G.E. Volovik, JETP Lett. 91, 55 (2010)
G.E. Volovik, The Universe in a Helium Droplet (Clarendon, Oxford, 2003)
S.Q. Shen, W.Y. Shan, H.Z. Lu, SPIN 1, 33 (2011)
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Shen, SQ. (2012). Topological Invariants. In: Topological Insulators. Springer Series in Solid-State Sciences, vol 174. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32858-9_4
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DOI: https://doi.org/10.1007/978-3-642-32858-9_4
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