×

zbMATH — the first resource for mathematics

Parseval frames of exponentially localized magnetic Wannier functions. (English) Zbl 07135157
Summary: Motivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension \(d \le 3\), we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states. When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model \(m\) occupied energy bands by a real-analytic and \({{\mathbb{Z}}}^d\)-periodic family \(\left\{ P(\mathbf{k}) \right\} _{\mathbf{k}\in{{\mathbb{R}}}^d}\) of orthogonal projections of rank \(m\). A moving orthonormal basis of \({{\,\text{Ran}\,}}P(\mathbf{k})\) consisting of real-analytic and \({{\mathbb{Z}}}^d\)-periodic Bloch vectors can be constructed if and only if the first Chern number(s) of \(P\) vanish(es). Here we are mainly interested in the topologically obstructed case. First, by dropping the generating condition, we show how to algorithmically construct a collection of \(m-1\) orthonormal, real-analytic, and \({{\mathbb{Z}}}^d\)-periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of \(m+1\) real-analytic and \({{\mathbb{Z}}}^d\)-periodic Bloch vectors which generate \({{\,\text{Ran}\,}}P(\mathbf{k})\). Both algorithms are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. A moving Parseval frame of analytic, \({{\mathbb{Z}}}^d\)-periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of magnetic Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting. Our results are illustrated in crystalline insulators modelled by \(2d\) discrete Hofstadter-like Hamiltonians, but apply to certain continuous models of magnetic Schrödinger operators as well.

MSC:
82D20 Statistical mechanical studies of solids
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82D25 Statistical mechanical studies of crystals
82D40 Statistical mechanical studies of magnetic materials
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Auckly, D.; Kuchment, P.; Bonetto, F. (ed.); Borthwick, D. (ed.); Harrell, E. (ed.); Loss, M. (ed.), On Parseval frames of exponentially decaying composite Wannier functions, 227-240 (2018), Providence, RI · Zbl 1428.35417
[2] Avis, SJ; Isham, CJ; Lévy, M. (ed.); Deser, S. (ed.), Quantum field theory and fibre bundles in a general space–time, 347-401 (1979), New York
[3] Avron, JE; Simon, B., Analytic properties of band functions, Ann. Phys., 110, 85-101 (1978)
[4] Brynildsen, M.; Cornean, HD, On the Verdet constant and Faraday rotation for graphene-like materials, Rev. Math. Phys., 25, 1350007 (2013) · Zbl 1272.82023
[5] Cancès, É.; Levitt, A.; Panati, G.; Stoltz, G., Robust determination of maximally localized Wannier functions, Phys. Rev. B, 95, 075114 (2017)
[6] Cornean, HD, On the Lipschitz continuity of spectral bands of Harper-like and magnetic Schrödinger operators, Ann. Henri Poincaré, 11, 973-990 (2010) · Zbl 1208.81089
[7] Cornean, HD; Nenciu, G., On eigenfunction decay of two dimensional magnetic Schrödinger operators, Commun. Math. Phys., 192, 671-685 (1998) · Zbl 0915.35013
[8] Cornean, HD; Nenciu, G., The Faraday effect revisited. Thermodynamic limit, J. Funct. Anal., 257, 2024-2066 (2009) · Zbl 1178.82077
[9] Cornean, HD; Herbst, I.; Nenciu, G., On the construction of composite Wannier functions, Ann. Henri Poincaré, 17, 3361-3398 (2016) · Zbl 1357.82069
[10] Cornean, H.D., Monaco, D., Moscolari, M.: Beyond Diophantine Wannier diagrams: gap labelling for Bloch-Landau Hamiltonians. Preprint arXiv:1810.05623 (2018)
[11] Cornean, HD; Monaco, D.; Teufel, S., Wannier functions and \[\mathbb{Z}_2\] Z2 invariants in time-reversal symmetric topological insulators, Rev. Math. Phys., 29, 1730001 (2017) · Zbl 1370.81081
[12] Cornean, HD; Monaco, D., On the construction of Wannier functions in topological insulators: the 3D case, Ann. Henri Poincaré, 18, 3863-3902 (2017) · Zbl 1386.82068
[13] Cloizeaux, Jacques Des, Energy Bands and Projection Operators in a Crystal: Analytic and Asymptotic Properties, Physical Review, 135, a685-a697 (1964)
[14] Dubail, J.; Read, N., Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension, Phys. Rev. B, 92, 205307 (2015)
[15] Feichtinger, H.G., Strohmer, T. (eds.): Gabor Analysis and Algorithms, Theory and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (1998) · Zbl 0890.42004
[16] Fiorenza, D.; Monaco, D.; Panati, G., Construction of real-valued localized composite Wannier functions for insulators, Ann. Henri Poincaré, 17, 63-97 (2016) · Zbl 1338.82057
[17] Fiorenza, D.; Monaco, D.; Panati, G., \[ \mathbb{Z}_2\] Z2 invariants of topological insulators as geometric obstructions, Commun. Math. Phys., 343, 1115-1157 (2016) · Zbl 1346.81158
[18] Freeman, D.; Poore, D.; Wei, AR; Wyse, M., Moving Parseval frames for vector bundles, Houston J. of Math., 40, 817-832 (2014) · Zbl 1306.42046
[19] Freund, S.; Teufel, S., Peierls substitution for magnetic Bloch bands, Anal. PDE, 9, 773-811 (2016) · Zbl 1343.81088
[20] Galli, G.; Parrinello, M., Large scale electronic structure calculations, Phys. Rev. Lett., 69, 3547 (1992)
[21] Goedecker, S., Linear scaling electronic structure methods, Rev. Mod. Phys., 71, 1085-1123 (1999)
[22] Gröchenig, K.: Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2001) · Zbl 0966.42020
[23] Gontier, D.; Levitt, A.; Siraj-Dine, S., Numerical construction of Wannier functions through homotopy, J. Math. Phys., 60, 031901 (2019) · Zbl 1417.82030
[24] Han, D., Larson, D.R.: Frames, Bases and Group Representations. No. 697 in Memoirs of the American Mathematical Society. American Mathematical Society, Providence (2000)
[25] Hofstadter, DR, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B, 14, 2239-2249 (1976)
[26] Husemoller, D.: Fibre Bundles. No. 20 in Graduate Texts in Mathematics, 3rd edn. Springer, New York (1994)
[27] Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966) · Zbl 0148.12601
[28] Kohmoto, M., Topological invariant and the quantization of the Hall conductance, Ann. Phys., 160, 343-354 (1985)
[29] Kuchment, P.: Floquet Theory for Partial Differential Equations. Birkhäuser, Basel (1993) · Zbl 0789.35002
[30] Kuchment, P., Tight frames of exponentially decaying Wannier functions, J. Phys. A Math. Theor., 42, 025203 (2009) · Zbl 1154.82028
[31] Kuchment, P., An overview of periodic ellipic operators, Bull. AMS, 53, 343-414 (2016) · Zbl 1346.35170
[32] Ludewig, M., Thiang, G.C.: Good Wannier bases in Hilbert modules associated to topological insulators. Preprint arXiv:1904.13051 (2019)
[33] Marzari, N.; Mostofi, A.; Yates, J.; Souza, I.; Vanderbilt, D., Maximally localized Wannier functions: theory and applications, Rev. Mod. Phys., 84, 1419-1475 (2012)
[34] Monaco, D.; Dell’Antonio, G. (ed.); Michelangeli, A. (ed.), Chern and Fu-Kane-Mele invariants as topological obstructions (2017), Cham · Zbl 1374.81101
[35] Monaco, D.; Panati, G., Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry, Acta Appl. Math., 137, 185-203 (2015) · Zbl 1318.82045
[36] Monaco, D.; Panati, G.; Pisante, A.; Teufel, S., Optimal decay of Wannier functions in Chern and Quantum Hall insulators, Commun. Math. Phys., 359, 61-100 (2018) · Zbl 1400.82249
[37] Monaco, D.; Tauber, C., Gauge-theoretic invariants for topological insulators: a bridge between Berry, Wess-Zumino, and Fu-Kane-Mele, Lett. Math. Phys., 107, 1315-1343 (2017) · Zbl 1370.35093
[38] 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
[39] Nenciu, A.; Nenciu, G., Existence of exponentially localized Wannier functions for nonperiodic systems, Phys. Rev. B, 47, 10112-10115 (1993)
[40] Nenciu, A.; Nenciu, G., The existence of generalised Wannier functions for one-dimensional systems, Commun. Math. Phys., 190, 541-548 (1998) · Zbl 0907.34075
[41] Panati, G., Triviality of Bloch and Bloch-Dirac bundles, Ann. Henri Poincaré, 8, 995-1011 (2007) · Zbl 1375.81102
[42] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Volume IV: Analysis of Operators. Academic Press, New York (1978) · Zbl 0401.47001
[43] Resta, R.; Vanderbilt, D.; Rabe, KM (ed.); Ahn, CH (ed.); Triscone, J-M (ed.), Theory of polarization: a modern approach, 31-68 (2007), Berlin
[44] Simon, B.: Harmonic Analysis: A Comprehensive Course in Analysis, Part 3. No. 3 of A Comprehensive Course in Analysis. American Mathematical Society, Providence (2015) · Zbl 1334.00002
[45] Spaldin, NA, A beginner’s guide to the modern theory of polarization, J. Solid State Chem., 195, 2 (2012)
[46] Thouless, DJ, Wannier functions for magnetic sub-bands, J. Phys. C Solid State Phys., 17, l325-l327 (1984)
[47] Thouless, DJ; Kohmoto, M.; Nightingale, MP; Nijs, M., Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., 49, 405-408 (1982)
[48] Yates, J.; Wang, X.; Vanderbilt, D.; Souza, I., Spectral and Fermi surface properties from Wannier interpolation, Phys. Rev. B, 75, 195121 (2007)
[49] Zaidenberg, MG; Krein, SG; Kuchment, P.; Pankov, AA, Banach bundles and linear operators, Russian Math. Surveys, 30, 115-175 (1975) · Zbl 0335.47017
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.