# zbMATH — the first resource for mathematics

Classification of “quaternionic” Bloch-bundles: topological quantum systems of type AII. (English) Zbl 1326.57047
The classification of topological quantum systems has been inspired by the study of topological insulators and the quantum Hall effect in particular. According to [A. Altland and M. Zirnbauer, Phys. Rev. B 55, 1142–1161 (1997)], AI and AII classes describe quantum systems that are invariant under time-reversal symmetry, but it is the behavior of the spin that distinguishes between them. Interesting physical effects appear in class AII systems that exhibit the quantum spin Hall effect (QSHE). The principal result of L. Fu, C. L. Kane and E. J. Mele [“Topological insulators in three dimensions”, Phys. Rev. Lett. 98, 106803 (2007)] was an identification of QSHE with the existence of a topological invariant, nowadays known as the Fu-Kane-Mele index. This index characterizes a topology of Bloch energy bands for periodic systems of free fermions, that have the time reversal (TR) symmetry, in the very same way as Chern numbers if the TR symmetry is broken. Major results have been established for two-dimensional lattice systems. The main goal of the present paper is an extension of the topological classification beyond $$d=2$$ (actually up to $$d=4$$). The analysis is based on the construction fo the FKMM invariant (same as the FKM, but this notation gives justice to other independent derivations) which completely classifies so-called quaternionic (actually symplectic) vector bundles in dimension $$d\leq 3$$. In case of $$d=4$$ the classification requires a combined use of the FKKM invariant and the second Chern class.

##### MSC:
 57R22 Topology of vector bundles and fiber bundles 57R56 Topological quantum field theories (aspects of differential topology) 58B05 Homotopy and topological questions for infinite-dimensional manifolds 55P91 Equivariant homotopy theory in algebraic topology 81V70 Many-body theory; quantum Hall effect 82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics 55R50 Stable classes of vector space bundles in algebraic topology and relations to $$K$$-theory
Full Text:
##### References:
 [1] Atiyah, M.F.; Bott, R., On the periodicity theorem for complex vector bundles, Acta Math., 112, 229-247, (1964) · Zbl 0131.38201 [2] Allday C., Puppe V.: Cohomological methods in transformation groups. Cambridge University Press, Cambridge (1993) · Zbl 0799.55001 [3] Avila, J.C.; Schulz-Baldes, H.; Villegas-Blas, C., Topological invariants of edge states for periodic two-dimensional models, Math. Phys. Anal. Geom., 16, 137-170, (2013) · Zbl 1271.81210 [4] Atiyah, M.F., $$K$$-theory and reality, Quart. J. Math. Oxford Ser. (2), 17, 367-386, (1966) · Zbl 0146.19101 [5] Atiyah M.F.: $$K$$-theory. W. A. Benjamin, New York (1967) [6] Altland, A.; Zirnbauer, M., Non-standard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B, 55, 1142-1161, (1997) [7] Biswas, I.; Huisman, J.; Hurtubise, J., The moduli space of stable vector bundles over a real algebraic curve, Math. Ann., 347, 201-233, (2010) · Zbl 1195.14048 [8] Borel, A.: Seminar on transformation groups with contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. Ann. Math. Stud. vol. 46, Princeton University Press, Princeton (1960) [9] De Nittis, G.; Gomi, K., Classification of “real” Bloch-bundles: topological quantum systems of type AI, J. Geom. Phys., 86, 303-338, (2014) · Zbl 1316.57019 [10] De Nittis, G., Gomi, K.: Differential geometric invariants for time-reversal symmetric Bloch-bundles: the “Real” case. arXiv:1502.01232 (2015) · Zbl 1346.57024 [11] De Nittis, G., Gomi, K.: Differential geometric invariants for time-reversal symmetric Bloch-bundles: the “Quaternionic” case (in preparation) · Zbl 1346.57024 [12] Davis, J.F., Kirk, P.: Lecture notes in algebraic topology. AMS, Providence (2001) · Zbl 1018.55001 [13] De Nittis, G.; Lein, M., Topological polarization in graphene-like systems, J. Phys. A, 46, 385001, (2013) · Zbl 1277.82082 [14] Dos Santos, P.F.; Lima-Filho, P., Quaternionic algebraic cycles and reality, Trans. Am. Math. Soc., 356, 4701-4736, (2004) · Zbl 1052.55013 [15] Dupont, J.L., Symplectic bundles and KR-theory, Math. Scand., 24, 27-30, (1969) · Zbl 0184.48401 [16] Edelson, A.L., Real vector bundles and spaces with free involutions, Trans. Am. Math. Soc., 157, 179-188, (1971) · Zbl 0217.49001 [17] Essin, A.M.; Moore, J.E.; Vanderbilt, D., Magnetoelectric polarizability and axion electrodynamics in crystalline insulators, Phys. Rev. Lett., 102, 146805, (2009) [18] Fu, L.; Kane, C.L., Time reversal polarization and a $${\mathbb{Z}_2}$$ adiabatic spin pump, Phys. Rev. B, 74, 195312, (2006) [19] Fu, L.; Kane, C.L.; Mele, E.J., Topological insulators in three dimensions, Phys. Rev. Lett., 98, 106803, (2007) [20] Furuta, M., Kametani, Y., Matsue, H., Minami, N.: Stable-homotopy Seiberg-Witten invariants and Pin bordisms. UTMS Preprint Series 2000, UTMS 2000-46 (2000) · Zbl 1117.57030 [21] Griffiths P., Harris J.: Principles of algebraic geometry. Wiley, New York (1978) · Zbl 0408.14001 [22] Gomi, K., A variant of K-theory and topological T-duality for real circle bundles, Commun. Math. Phys., 334, 923-975, (2015) · Zbl 1320.19001 [23] Graf G.M., Porta M.: Bulk-edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324, 851-895 (2013) · Zbl 1291.82120 [24] Hatcher A.: Algebraic topology. Cambridge University Press, Cambridge (2002) · Zbl 1044.55001 [25] Hughes, T.L.; Prodan, E.; Bernevig, B.A., Inversion-symmetric topological insulators, Phys. Rev. B, 83, 245132, (2011) [26] Hsiang W.Y.: Cohomology theory of topological transformation groups. Springer, Berlin (1975) · Zbl 0429.57011 [27] Husemoller D.: Fibre bundles. Springer, New York (1994) · Zbl 0794.55001 [28] Kahn, B., Construction de classes de Chern équivariantes pour un fibré vectoriel Réel, Commun. Algebra., 15, 695-711, (1987) · Zbl 0615.57012 [29] Karoubi M.: $$K$$-theory. An introduction. Springer, New York (1978) · Zbl 0382.55002 [30] Kane, C.L.; Mele, E.J., Quantum spin Hall effect in graphene, Phys. Rev. Lett., 95, 226801, (2005) [31] Kane, C.L.; Mele, E.J., $${\mathbb{Z}_2}$$topological order and the quantum spin Hall effect, Phys. Rev. Lett., 95, 146802, (2005) [32] Kitaev, A., Periodic table for topological insulators and superconductors, AIP Conf. Proc., 1134, 22-30, (2009) · Zbl 1180.82221 [33] Luke G., Mishchenko A.S: Vector bundles and their applications. Kluwer Academic Publishers, Dordrecht (1998) · Zbl 0907.55002 [34] Lawson, H.B.; Lima-Filho, P.; Michelsohn, M.-L., Algebraic cycles and the classical groups. part II: quaternionic cycles, Geom. Topol., 9, 1187-1220, (2005) · Zbl 1081.14013 [35] Lin, H.; Yau, S.-T., On exotic sphere fibrations, topological phases, and edge states in physical systems, Int. J. Mod. Phys. B, 27, 1350107, (2013) · Zbl 1277.81092 [36] Matumoto, T., On $$G$$-CW complexes and a theorem of J, H. C. Whitehead. J. Fac. Sci. Univ. Tokyo, 18, 363-374, (1971) · Zbl 0232.57031 [37] Moore, J.E.; Balents, L., Topological invariants of time-reversal-invariant band structures, Phys. Rev. B, 75, 121306, (2007) [38] Maciejko, J.; Hughes, T.L.; Zhang, S.-C., The quantum spin Hall effect, Annu. Rev. Condens. Matter Phys., 2, 31-53, (2011) [39] Milnor J., Stasheff J.D.: Characteristic classes. Princeton University Press, Princeton (1974) · Zbl 0298.57008 [40] Roy, R., Topological phases and the quantum spin Hall effect in three dimensions, Phys. Rev. B, 79, 195322, (2009) [41] Seymour, R.M., The real K-theory of Lie groups and homogeneous spaces, Quart. J. Math. Oxford, 24, 7-30, (1973) · Zbl 0258.57021 [42] 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, (2008) [43] Vaisman, I., Exotic characteristic classes of quaternionic bundles, Israel J. Math., 69, 46-58, (1990) · Zbl 0713.57015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.