×

\(\mathbb Z_2\) invariants of topological insulators as geometric obstructions. (English) Zbl 1346.81158

The physical intuition behind the present research is that of a new class of materials, called nowadays time-reversal symmetric topological insulators. Their topological quantum phases are labeled by integers modulo \(2\). The Fu-Kane analysis of the exemplary tight-binding model governing the behavior of electrons on a honeycomb lattice, subject to nearest neighbor interactions, resulted in the identification of the \(\mathbb{Z}_2\) index to label the topological phases of \(2d\) TRS topological insulators. L. Fu and C. L. Kane [“Time reversal polarization and a \(Z_2\) adiabatic spin pump”, Phys. Rev. B 74, No. 19 Article ID 195312, 13 p. (2006; doi:10.1103/PhysRevB.74.195312)] argued that this index measures the obstruction to the existence of a continuous periodic Bloch frame, which is compatible with the time-reversal symmetry. Analogous proposals have been formulated in \(3d\) systems by L. Fu, et al. [“Topological insulators in three dimensions”, Phys. Rev. Lett. 98, No. 10, Article ID 106803, 4 p. (2007; doi:10.1103/PhysRevLett.98.106803)]. In the present paper the above claims are given a mathematically rigorous status. A geometric characterization is given of \(\mathbb{Z}_2\) indices as topological obstructions to the existence of continuous-periodic and time-reversal symmetric Bloch frames. It is proven that the Fu-Kane index actually is the topological invariant. In \(3d\) one arrives at four \(\mathbb{Z}_2\)-valued topological obstructions that give an unambiguous status to the Fu-Kane-Mele indices. When there is no topological obstruction, an explicit algorithm has been proposed to construct global smooth Bloch frames which are periodic and time-reversal symmetric. The main advantage of the method is that it is based on fundamental symmetries of the system which makes the approach model-independent.

MSC:

81V70 Many-body theory; quantum Hall effect
46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces
55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
19L10 Riemann-Roch theorems, Chern characters
30D45 Normal functions of one complex variable, normal families
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Altland A., Zirnbauer M.: Non-standard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142-1161 (1997) · doi:10.1103/PhysRevB.55.1142
[2] Ando Y.: Topological insulator materials. J. Phys. Soc. Jpn. 82, 102001 (2013) · doi:10.7566/JPSJ.82.102001
[3] Avila J.C., Schulz-Baldes H., Villegas-Blas C.: Topological invariants of edge states for periodic two-dimensional models. Math. Phys. Anal. Geometry 16, 136-170 (2013) · Zbl 1271.81210
[4] Carpentier D., Delplace P., Fruchart M., Gawedzki K.: Topological index for periodically driven time-reversal invariant 2d systems. Phys. Rev. Lett. 114, 106806 (2015) · Zbl 1331.82065 · doi:10.1103/PhysRevLett.114.106806
[5] Carpentier, D., Delplace, P., Fruchart, M., Gawedzki, K., Tauber, C.: Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals. Nucl. Phys. B · Zbl 1331.82065
[6] Chang C.-Z. et al.: Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator. Science 340, 167-170 (2013) · doi:10.1126/science.1234414
[7] De Nittis G., Gomi K.: Classification of “Quaternionic” Bloch bundles. Commun. Math. Phys. 339, 1-55 (2015) · Zbl 1326.57047 · doi:10.1007/s00220-015-2390-0
[8] Dubrovin B.A., Novikov S.P., Fomenko A.T.: Modern Geometry—Methods and Applications. Part II: The Geometry and Topology of Manifolds. No. 93 in Graduate Texts in Mathematics. Springer-Verlag, New York (1985) · Zbl 0565.57001
[9] Fiorenza D., Monaco D., Panati G.: Construction of real-valued localized composite Wannier functions for insulators. Ann. Henri Poincaré 17(1), 63-97 (2016) · Zbl 1338.82057 · doi:10.1007/s00023-015-0400-6
[10] Fröhlich J., Werner P.h.: Gauge theory of topological phases of matter. EPL 101, 47007 (2013) · doi:10.1209/0295-5075/101/47007
[11] Fruchart M., Carpentier D.: An introduction to topological insulators. Comput. Rendus Phys. 14, 779-815 (2013) · doi:10.1016/j.crhy.2013.09.013
[12] Fu L., Kane C.L.: Time reversal polarization and a \[{\mathbb{Z}_2}\] Z2 adiabatic spin pump. Phys. Rev. B 74, 195312 (2006) · doi:10.1103/PhysRevB.74.195312
[13] Fu L., Kane C.L., Mele E.J.: Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007) · doi:10.1103/PhysRevLett.98.106803
[14] 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
[15] Graf G.M.: Aspects of the Integer Quantum Hall effect. Proc. Symp. Pure Math. 76, 429-442 (2007) · Zbl 1132.81356 · doi:10.1090/pspum/076.1/2310213
[16] Graf G.M., Porta M.: Bulk-edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324, 851-895 (2013) · Zbl 1291.82120 · doi:10.1007/s00220-013-1819-6
[17] Haldane F.D.M.: Model for a Quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2017 (1988)
[18] Hasan M.Z., Kane C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045-3067 (2010) · doi:10.1103/RevModPhys.82.3045
[19] Hua L.-K.: On the theory of automorphic functions of a matrix variable I-Geometrical basis. Am. J. Math. 66, 470-488 (1944) · Zbl 0063.02919 · doi:10.2307/2371910
[20] Hua L.-K., Reiner I.: Automorphisms of the unimodular group. Trans. Am. Math. Soc. 71, 331-348 (1951) · Zbl 0045.30402 · doi:10.1090/S0002-9947-1951-0043847-X
[21] Husemoller D.: Fibre bundles, 3rd edn. No. 20 in Graduate Texts in Mathematics. Springer-Verlag, New York (1994)
[22] Kane C.L., Mele E.J.: \[{\mathbb{Z}_2}\] Z2 Topological Order and the Quantum Spin Hall Effect. Phys. Rev. Lett. 95, 146802 (2005) · doi:10.1103/PhysRevLett.95.146802
[23] Kane C.L., Mele E.J.: Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005) · doi:10.1103/PhysRevLett.95.226801
[24] Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966) · Zbl 0148.12601 · doi:10.1007/978-3-662-12678-3
[25] Kennedy R., Guggenheim C.: Homotopy theory of strong and weak topological insulators. Phys. Rev. B 91, 245148 (2015) · doi:10.1103/PhysRevB.91.245148
[26] Kennedy, R., Zirnbauer, M.R.: Bott periodicity for \[{\mathbb{Z}_2}\] Z2 Symmetric Ground States of Free-Fermion systems. Commun. Math. Phys. doi:10.1007/s00220-015-2512-8 · Zbl 1346.81159
[27] Kennedy, R., Zirnbauer, M.R.: Bott-Kitaev periodic table and the diagonal map. Phys. Scr. T164, 014010 (2015). doi:10.1088/0031-8949/2015/T164/014010
[28] Kitaev A.: Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22 (2009) · Zbl 1180.82221
[29] Kuiper N.H.: The homotopy type of the unitary group of Hilbert space. Topology 3, 19-30 (1965) · Zbl 0129.38901 · doi:10.1016/0040-9383(65)90067-4
[30] Mackey, D.S., Mackey, N.: On the Determinant of Symplectic Matrices. Numerical Analysis Report 422, Manchester Centre for Computational Mathematics, Manchester, England (2003) · Zbl 1027.15013
[31] Monaco, D., Panati, G.: Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry. Proceedings of the conference “SPT2014—Symmetry and Perturbation Theory”, Cala Gonone, Italy, Acta App. Math. 137, 185-203 (2015) · Zbl 1318.82045
[32] Moore J.E., Balents L.: Topological invariants of time-reversal-invariant band structures. Phys. Rev. B 75, 121306(R) (2007) · doi:10.1103/PhysRevB.75.121306
[33] Panati G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. Henri Poincaré 8, 995-1011 (2007) · Zbl 1375.81102 · doi:10.1007/s00023-007-0326-8
[34] Prodan E.: Robustness of the Spin-Chern number. Phys. Rev. B 80, 125327 (2009) · doi:10.1103/PhysRevB.80.125327
[35] Prodan E.: Disordered topological insulators: a non-commutative geometry perspective. J. Phys. A 44, 113001 (2011) · Zbl 1213.82082 · doi:10.1088/1751-8113/44/11/113001
[36] Prodan E.: Manifestly gauge-independent formulations of the \[{\mathbb{Z}_2}\] Z2 invariants. Phys. Rev. B 83, 235115 (2011) · doi:10.1103/PhysRevB.83.235115
[37] Ryu S., Schnyder A.P., Furusaki A., Ludwig A.W.W.: Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010) · doi:10.1088/1367-2630/12/6/065010
[38] Schulz-Baldes H.: Persistence of spin edge currents in disordered Quantum Spin Hall systems. Commun. Math. Phys. 324, 589-600 (2013) · Zbl 1278.82065 · doi:10.1007/s00220-013-1814-y
[39] Schulz-Baldes \[H.: {\mathbb{Z}_2}\] Z2 Indices and Factorization Properties of Odd Symmetric Fredholm Operators. Doc. Math. 20, 1481-1500 (2015) · Zbl 1341.47014
[40] Steenrod N.: The Topology of Fibre Bundles. No. 14 in Princeton Mathematical Series. Princeton University Press, Princeton (1951) · Zbl 0054.07103
[41] Sticlet D., Péchon F., Fuchs J.-N., Kalugin P., Simon P.: Geometrical engineering of a two-band Chern insulator in two dimensions with arbitrary topological index. Phys. Rev. B 85, 165456 (2012) · doi:10.1103/PhysRevB.85.165456
[42] Soluyanov A.A., Vanderbilt D.: Wannier representation of \[{\mathbb{Z}_2}\] Z2 topological insulators. Phys. Rev. B 83, 035108 (2011) · doi:10.1103/PhysRevB.83.035108
[43] Soluyanov A.A., Vanderbilt D.: Computing topological invariants without inversion symmetry. Phys. Rev. B 83, 235401 (2011) · doi:10.1103/PhysRevB.83.235401
[44] Soluyanov A.A., Vanderbilt D.: Smooth gauge for topological insulators. Phys. Rev. B 85, 115415 (2012) · doi:10.1103/PhysRevB.85.115415
[45] Thouless D.J., Kohmoto M., Nightingale M.P., de Nijs M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405-408 (1982) · doi:10.1103/PhysRevLett.49.405
[46] Wockel Ch.: A generalization of Steenrod’s approximation theorem. Arch. Math. (Brno) 45, 95-104 (2009) · Zbl 1212.58005
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.