zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv
[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
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.