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On the construction of composite Wannier functions. (English) Zbl 1357.82069
The existence and construction of exponentially localized Wannier functions is one of the fundamental problems in solid-state physics. The authors give a constructive proof for the existence of a Bloch basis of rank \(N\) which is both smooth and periodic with respect to its \(d\)-dimensional quasi-momenta, when \(1\leq d\leq 2\) and \(N\geq 1\). Also it is shown that, by adding a weak, globally bounded but not necessarily constant magnetic field, the existence of a localized basis is preserved.

MSC:
82D20 Statistical mechanical studies of solids
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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[1] Brouder, Ch.; Panati, G.; Calandra, M.; Mourougane, Ch.; Marzari, N., Exponential localization of Wannier functions in insulators, Phys. Rev. Lett., 98, 046402, (2007)
[2] Cornean, H.D., On the Lipschitz continuity of spectral bands of harper-like and magnetic Schrödinger operators, Ann. Henri Poincaré, 11, 973-990, (2010) · Zbl 1208.81089
[3] Cornean, H.D., Monaco, D., Teufel, S.: Wannier functions and \({{\mathbb{Z}}_2}\) invariants in time-reversal symmetric topological insulators. http://arxiv.org/abs/1603.06752 · Zbl 1370.81081
[4] Cornean, H.D.; Nenciu, G., On eigenfunction decay for two dimensional magnetic Schrödinger operators, Commun. Math. Phys., 192, 671-685, (1998) · Zbl 0915.35013
[5] Cornean, H.D.; Nenciu, G., The Faraday effect revisited: thermodynamic limit, J. Funct. Anal., 257, 2024-2066, (2009) · Zbl 1178.82077
[6] Cornean, H.D.; Nenciu, A.; Nenciu, G., Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators, J. Phys. A Math. Theor., 41, 125202, (2008) · Zbl 1137.82338
[7] Nittis, G.de , Gomi, K.: Classification of “Real” Bloch-bundles: topological quantum systems of type AI. J. Geom. Phys. 86, 303-338 (2014) · Zbl 1316.57019
[8] des Cloizeaux, J.: Energy bands and projection operators: analytic and asymptotic properties. Phys. Rev., A685-A697 (1964) · Zbl 0907.34075
[9] des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A698-A707 (1964) · Zbl 1059.81064
[10] Fiorenza, D.; Monaco, D.; Panati, G., Construction of real-valued localized composite Wannier functions for insulators, Ann. H. Poincaré, 17, 63-97, (2016) · Zbl 1338.82057
[11] Fiorenza, D., Monaco, D., Panati, G.: \({{\mathbb{Z}}_2}\) Invariants of topological insulators as geometric obstructions. Commun. Math. Phys. (2016). doi:10.1007/s00220-015-2552-0 · Zbl 1346.81158
[12] Freund, S., Teufel, S.: Peierls substitution for magnetic Bloch bands. Anal. PDE (to appear). http://arxiv.org/abs/1312.5931 · Zbl 1343.81088
[13] Fruchart, M.; Carpentier, D.; Gaw̧edzki, K., Parallel transport and band theory in crystals, EPL, 106, 60002, (2014)
[14] Helffer, B.; Sjöstrand, J., Equation de Schrödinger avec champ magnétique et équation de harper, Springer Lect. Notes Phys., 345, 118-197, (1989) · Zbl 0699.35189
[15] Geller, M.R.; Kohn, W., Theory of generalized Wannier functions for nearly periodic potentials, Phys. Rev., 48, 14085-14088, (1993)
[16] Kivelson, S., Wannier functions in one-dimensional disordered systems, Phys. Rev. B, 26, 4269-4274, (1982) · Zbl 1200.35284
[17] Kohn, W., Analytic properties of Bloch waves and Wannier functions, Phys. Rev., 115, 809-821, (1959) · Zbl 0086.45101
[18] Kohn, W.; Onffroy, J., Wannier functions in a simple nonperiodic system, Phys. Rev. B, 8, 2485-2495, (1973)
[19] Luttinger, J.M., The effect of a magnetic field on electrons in a periodic potential, Phys. Rev., 84, 814-817, (1951) · Zbl 0044.45301
[20] Marzari, N.; Vanderbilt, D., Maximally localized generalized wannietr functions for composite bands, Phys. Rev. B, 56, 12847-12865, (1997)
[21] Marzari, N.; Mostofi, A.; Yates, Y.; Souza, I.; Vanderbilt, D., Maximaly localized Wannier functions: theory and applications, Rev. Modern Phys., 84, 1419-1470, (2012)
[22] Monaco, D.; Panati, G., Topological invariants of eigenvalue intersections and decrease of Wannier functions in graphene, J. Stat. Phys., 155, 1027-1071, (2014) · Zbl 1401.82064
[23] Nenciu, G., Existence of exponentially localized Wannier functions, Commun. Math. Phys., 91, 81-85, (1983) · Zbl 0545.47012
[24] Nenciu, G., Stability of energy gaps under variation of the magnetic field, Lett. Math. Phys., 11, 127-132, (1986) · Zbl 0607.46049
[25] Nenciu, G., Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians, Rev. Modern Phys., 63, 91127, (1991)
[26] Nenciu, G., On asymptotic perturbation theory for quantum mechanics: almost invariant subspaces and gauge invariant magnetic perturbation theory, J. Math. Phys., 43, 1273-1298, (2002) · Zbl 1059.81064
[27] Nenciu, A.; Nenciu, G., The existence of generalized Wannier functions for one dimensional systems, Commun. Math. Phys., 190, 541-548, (1998) · Zbl 0907.34075
[28] Nenciu, A.; Nenciu, G., Existence of exponentially localized Wannier functions for nonperiodic systems, Phys. Rev. B, 47, 10112-10115, (1993)
[29] Panati, G., Triviality of Bloch and Bloch-Dirac bundles, Ann. Henri Poincaré, 8, 995-1011, (2007) · Zbl 1375.81102
[30] Panati, G.; Pisante, A., Bloch bundles, marzari-vanderbilt functional and maximally localized Wannier functions, Commun. Math. Phys., 322, 835-875, (2013) · Zbl 1277.82057
[31] Panati, G.; Spohn, H.; Teufel, S., Effective dynamics for Bloch electrons: Peierls substitution and beyond, Commun. Math. Phys., 242, 547-578, (2003) · Zbl 1058.81020
[32] Peierls, R., On the theory of diamagnetism of conduction electrons, Z. Phys., 80, 763-791, (1933)
[33] Prodan, E., On the generalized Wannier functions, J. Math. Phys., 56, 113511, (2015) · Zbl 1330.35092
[34] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 4. Analysis of Operators. Academic Press, New York (1978) · Zbl 0401.47001
[35] Soluyanov, A.A.; Vanderbilt, D., Wannier representation of \({{\mathbb{Z}}_2}\) topological insulators, Phys. Rev. B, 83, 035108, (2011)
[36] Thouless, D.J., Wannier functions for magnetic sub-bands, J. Phys. C Solid State Phys., 17, l325-l327, (1984)
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