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Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry. (English) Zbl 1318.82045
Summary: We describe some applications of group- and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e.g. insulators and semiconductors.
We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, called the Bloch bundle. The rôle of additional \(\mathbb{Z}_{2}\)-symmetries, as time-reversal and space-reflection symmetry, is discussed, showing how time-reversal symmetry implies the triviality of the Bloch bundle, both in the bosonic and in the fermionic case. Moreover, the same \(\mathbb{Z}_{2}\)-symmetry allows to define a finer notion of isomorphism and, consequently, to define new topological invariants, which agree with the indices introduced by Fu, Kane and Mele in the context of topological insulators.

MSC:
82D20 Statistical mechanical studies of solids
82D37 Statistical mechanical studies of semiconductors
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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[1] Bellissard, J.; Herrmann, D.J.L.; Zarrouati, M.; Baake, M. (ed.); Moody, R.V. (ed.), Hulls of aperiodic solids and gap labelling theorems, (2000), Providence · Zbl 0972.52014
[2] Bellissard, J.; Elst, A.; Schulz-Baldes, H., The noncommutative geometry of the quantum Hall effect, J. Math. Phys., 35, 5373, (1994) · Zbl 0824.46086
[3] Blount, E.I.; Seitz, F. (ed.); Turnbull, D. (ed.), Formalism of band theory, No. 13, 305-373, (1962), San Diego
[4] Brouder, Ch.; Panati, G.; Calandra, M.; Mourougane, Ch.; Marzari, N., Exponential localization of Wannier functions in insulators, Phys. Rev. Lett., 98, (2007)
[5] des Cloizeaux, J., Analytical properties of \(n\)-dimensional energy bands and Wannier functions, Phys. Rev., 135, a698-a707, (1964)
[6] Fiorenza, D., Monaco, D., Panati, G.: Construction of real-valued localized composite Wannier functions for insulators. Preprint available at arXiv:1408.0527 · Zbl 1338.82057
[7] Fiorenza, D., Monaco, D., Panati, G.: \(\mathbb{Z}_{2}\)-invariants of topological insulators as geometric obstructions. Preprint available at arXiv:1408.1030 · Zbl 1346.81158
[8] Fritzsche, K., Grauert, H.: From Holomorphic Functions to Complex Manifolds. Springer, Berlin (2002) · Zbl 1005.32002
[9] Fu, L.; Kane, C.L., Time reversal polarization and a \(\mathbb{Z}_{2}\) adiabatic spin pump, Phys. Rev. B, 74, (2006)
[10] Fu, L.; Kane, C.L.; Mele, E.J., Topological insulators in three dimensions, Phys. Rev. Lett., 98, (2007)
[11] Graf, G.M.; Porta, M., Bulk-edge correspondence for two-dimensional topological insulators, Commun. Math. Phys., 324, 851-895, (2013) · Zbl 1291.82120
[12] Hasan, M.Z.; Kane, C.L., Colloquium: topological insulators, Rev. Mod. Phys., 82, 3045-3067, (2010)
[13] Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966) · Zbl 0148.12601
[14] Kitaev, A., Periodic table for topological insulators and superconductors, AIP Conf. Proc., 1134, 22, (2009) · Zbl 1180.82221
[15] Kittel, C.: Introduction to Solid State Physics. Wiley, New York (1996) · Zbl 0052.45506
[16] Kohn, W., Analytic properties of Bloch waves and Wannier functions, Phys. Rev., 115, 809, (1959) · Zbl 0086.45101
[17] Marzari, N.; Vanderbilt, D., Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B, 56, 12847-12865, (1997)
[18] Marzari, N.; Mostofi, A.A.; Yates, J.R.; Souza, I.; Vanderbilt, D., Maximally localized Wannier functions: theory and applications, Rev. Mod. Phys., 84, 1419, (2012)
[19] Nenciu, G., Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians, Rev. Mod. Phys., 63, 91-127, (1991)
[20] Panati, G., Triviality of Bloch and Bloch-Dirac bundles, Ann. Henri Poincaré, 8, 995-1011, (2007) · Zbl 1375.81102
[21] Panati, G.; Pisante, A., Bloch bundles, marzari-vanderbilt functional and maximally localized Wannier functions, Commun. Math. Phys., 322, 835-875, (2013) · Zbl 1277.82057
[22] Panati, G.; Sparber, C.; Teufel, S., Geometric currents in piezoelectricity, Arch. Ration. Mech. Anal., 91, 387-422, (2009) · Zbl 1161.74024
[23] Panati, G.; Sparber, C.; Teufel, S., Effective dynamics for Bloch electrons: Peierls substitution and beyond, Commun. Math. Phys., 242, 547-578, (2003) · Zbl 1058.81020
[24] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Volume IV: Analysis of Operators. Academic Press, San Diego (1978) · Zbl 0401.47001
[25] Ryu, S.; Schnyder, A.P.; Furusaki, A.; Ludwig, A.W.W., Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys., 12, (2010)
[26] Zak, J., Magnetic translation group, Phys. Rev., 134, (1964) · Zbl 0131.45404
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