Abstract
The topologically nontrivial states of matter—charge and spin topological insulators, which exhibit, respectively, properties of the integer quantum Hall effect and the quantum spin Hall effect—are discussed. The topological characteristics (invariant with respect to weak adiabatic changes in the Hamiltonian parameters) which lead to such states are considered. The model of a 2D hexagonal lattice having symmetries broken with respect to time reversal and spatial inversion which was proposed by Haldane and marked the beginning of unprecedented activity in the study of topologically nontrivial states is discussed. This model relates the microscopic nature of the symmetry breaking with respect to the time reversal to the occurrence of spontaneous orbital currents which circulate within a unit cell. Such currents become zero upon summation over the unit cell, but they may form spreading current states at the surface which are similar to the edge current states under the quantum Hall effect. The first model of spontaneous currents (exciton insulator model) is considered, and the possibility of implementing new topologically nontrivial states in this model is discussed.
Similar content being viewed by others
References
F. Bloch, Z. Phys. 52, 555 (1928).
A. H. Wilson, Proc. R. Soc. London, Ser. A 133, 458 (1931); 134, 277 (1931).
K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).
B. I. Halperin, Phys. Rev. B 25, 2185 (1982).
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
L. D. Landau, Zh. Eksp. Teor. Fiz. 7, 19, 627 (1937).
W. Kohn, Phys. Rev. A 133, 171 (1964).
Y. Hatsugai, Phys. Rev. B 48, 11851 (1993).
Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993).
M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984).
R. Resta, Eur. Phys. J. B 79, 121 (2011).
S. I. Vinitskii, V. L. Derbov, V. M. Dubovik, et al., Usp. Fiz. Nauk 160, 1 (1990).
D. N. Klyshko, Usp. Fiz. Nauk 163, 1 (1993).
G. E. Volovik, Sov. Phys. JETP 67, 1804 (1988).
G. E. Volovik, The Universe in a Helium Droplet (Oxford Univ. Press, USA, 2003).
Q. Niu, D. J. Thouless, and Y-S. Wu, Phys. Rev. B 31, 3372 (1985).
S-C. Zhang, T. H. Hanson, and S. A. Kivelson, Phys. Rev. Lett. 62, 82 (1989).
X-G. Wen, Adv. Phys. 44, 405 (1995).
F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
V. M. Yakovenko, Phys. Rev. Lett. 65, 251 (1990).
M. Taillefumier, V. K. Dugaev, B. Canals, et al., Phys. Rev. B 78, 155 330 (2008).
L. V. Keldysh and Yu. V. Kopaev, Sov. Phys. Solid State 6, 2219 (1964).
B. A. Volkov, A. A. Gorbatsevich, Yu. V. Kopaev, and V. V. Tugushev, Sov. Phys. JETP 54, 391 (1981).
V. L. Ginzburg, A. A. Gorbatsevich, Yu. V. Kopaev, and B. A. Volkov, Solid State Commun. 50, 339 (1984).
Yu. V. Kopaev, Usp. Fiz. Nauk 179, 1175 (2009).
A. A. Gorbatsevich, Sov. Phys. JETP 68, 847 (1989).
B. I. Halperin and T. M. Rice, Solid State Phys. 21, 115 (1968).
V. M. Dubovik and V. V. Tugushev, Phys. Rep. 187, 145 (1990).
A. A. Gorbatsevich, V. V. Kapaev, and Yu. V. Kopaev, Ferroelectrics 161, 303 (1994).
S. Raghu, X.-L. Qi, C. Honercamp, and S.-C. Zhang, Phys. Rev. Lett. 100, 156 401 (2008).
C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802, 226801 (2005).
M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
K. S. Novoselov, A. K. Geim, S. V. Morozov, et al., Science 306, 666 (2004).
M. I. D’yakonov and V. I. Perel’, JETP Lett. 13, 467 (1971).
M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971).
Y. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004).
J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047 204 (2005).
B. A. Bernevig, T. L. Hughes, and S-C. Zhang, Science 314, 1757 (2006).
M. Konig, S. Wiedmann, C. Brüne, et al., Science 318, 766 (2007).
L. Fu and C. L. Kane, Phys. Rev. B 76, 045 302 (2007).
D. Hsieh, D. Qian, L. Wray, et al., Nature 452, 970 (2008).
A. A. Gorbatsevich and Yu. V. Kopaev, JETP Lett. 39, 684 (1984).
A. A. Gorbatsevich, V. V. Kapaev, and Yu. V. Kopaev, JETP Lett. 57, 580 (1993).
A. A. Gorbatsevich, O. E. Omel’yanovskii, and V. I. Tsebro, Usp. Fiz. Nauk 179, 887 (2009).
D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M. Haldane, Phys. Rev. Lett. 97, 036 808 (2006).
L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106 803 (2007).
B.-J. Yang and H.-Y. Kee, Phys. Rev. B 82, 195 126 (2010).
K. Sun and E. Fradkin, Phys. Rev. B 78, 245 122 (2008).
S. Kai, Y. Hong, E. Fradkin, and S. A. Kivelson, Phys. Rev. Lett. 103, 046 811 (2009).
B. A. Volkov, A. A. Gorbatsevich, and Yu. V. Kopaev, Usp. Fiz. Nauk 143, 331 (1984).
B. A. Volkov, Yu. V. Kopaev, and M. S. Nunuparov, Sov. Phys. Solid State 21, 1571 (1979).
Yu. V. Kopaev and M. S. Nunuparov, Sov. Phys. Solid State 22, 2108 (1980).
A. A. Gorbatsevich, Yu. V. Kopaev, and V. I. Prokopov, Sov. Phys. Solid State 26, 645 (1986).
B. A. Volkov, V. G. Kantser, and Yu. V. Kopaev, Sov. Phys. JETP 49, 943 (1979).
S. Tewari, S. Das Sarma, C. Nayak, et al., Phys. Rev. Lett. 98, 010506 (2007).
G. E. Volovik, JETP Lett. 66, 555 (1997).
S. Tewari, C. Zhang, V. M. Yakovenko, and S. Das Sarma, Phys. Rev. Lett. 100, 217 004 (2008).
X. L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195 424 (2008).
A. M. Essin, J. E. Moore, and D. Vandrerbilt, Phys. Rev. Lett. 102, 146 805 (2009).
L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096 407 (2008).
A. Kitaev, Ann. Phys. 303, 2 (2003).
F. Wilczek, Nature Phys. 5, 614 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu.V. Kopaev, A.A. Gorbatsevich, V.I. Belyavskii, 2011, published in Kristallografiya, 2011, Vol. 56, No. 5, pp. 906–916.
Rights and permissions
About this article
Cite this article
Kopaev, Y.V., Gorbatsevich, A.A. & Belyavskii, V.I. Charge and spin topological insulators. Crystallogr. Rep. 56, 848–857 (2011). https://doi.org/10.1134/S1063774511050257
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063774511050257