Communications in Mathematical Physics

, Volume 371, Issue 3, pp 1179–1230 | Cite as

Parseval Frames of Exponentially Localized Magnetic Wannier Functions

  • Horia D. Cornean
  • Domenico MonacoEmail author
  • Massimo Moscolari


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 \({{\,\mathrm{Ran}\,}}P(\mathbf{k})\) 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 \({{\,\mathrm{Ran}\,}}P(\mathbf{k})\). 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.



The authors would like to thank G. Panati, G. Nenciu, G. De Nittis and P. Kuchment for inspiring discussions. Financial support from Grant 8021-00084B of the Danish Council for Independent Research | Natural Sciences, from the German Science Foundation (DFG) within the GRK 1838 “Spectral theory and dynamics of quantum systems”, and from the ERC Consolidator Grant 2016 “UNICoSM – Universality in Condensed Matter and Statistical Mechanics” is gratefully acknowledged.


  1. 1.
    Auckly, D., Kuchment, P.: On Parseval frames of exponentially decaying composite Wannier functions. In: Bonetto, F., Borthwick, D., Harrell, E., Loss, M. (eds.) Mathematical Problems in Quantum Physics. Volume 717 in Contemporary Mathematics Volume, pp. 227–240. American Mathematical Society, Providence, RI (2018)Google Scholar
  2. 2.
    Avis, S.J., Isham, C.J.: Quantum field theory and fibre bundles in a general space–time. In: Lévy, M., Deser, S. (eds.) Recent Developments in Gravitation—Cargèse 1978, pp. 347–401. Plenum Press, New York (1979)Google Scholar
  3. 3.
    Avron, J.E., Simon, B.: Analytic properties of band functions. Ann. Phys. 110, 85–101 (1978)ADSMathSciNetGoogle Scholar
  4. 4.
    Brynildsen, M., Cornean, H.D.: On the Verdet constant and Faraday rotation for graphene-like materials. Rev. Math. Phys. 25(4), 1350007 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cancès, É., Levitt, A., Panati, G., Stoltz, G.: Robust determination of maximally localized Wannier functions. Phys. Rev. B 95, 075114 (2017)ADSGoogle Scholar
  6. 6.
    Cornean, H.D.: On the Lipschitz continuity of spectral bands of Harper-like and magnetic Schrödinger operators. Ann. Henri Poincaré 11, 973–990 (2010)ADSMathSciNetzbMATHGoogle Scholar
  7. 7.
    Cornean, H.D., Nenciu, G.: On eigenfunction decay of two dimensional magnetic Schrödinger operators. Commun. Math. Phys. 192, 671–685 (1998)ADSzbMATHGoogle Scholar
  8. 8.
    Cornean, H.D., Nenciu, G.: The Faraday effect revisited. Thermodynamic limit. J. Funct. Anal. 257, 2024–2066 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cornean, H.D., Herbst, I., Nenciu, G.: On the construction of composite Wannier functions. Ann. Henri Poincaré 17, 3361–3398 (2016)ADSMathSciNetzbMATHGoogle Scholar
  10. 10.
    Cornean, H.D., Monaco, D., Moscolari, M.: Beyond Diophantine Wannier diagrams: gap labelling for Bloch-Landau Hamiltonians. Preprint arXiv:1810.05623 (2018)
  11. 11.
    Cornean, H.D., Monaco, D., Teufel, S.: Wannier functions and \(\mathbb{Z}_2\) invariants in time-reversal symmetric topological insulators. Rev. Math. Phys. 29, 1730001 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cornean, H.D., Monaco, D.: On the construction of Wannier functions in topological insulators: the 3D case. Ann. Henri Poincaré 18, 3863–3902 (2017)ADSMathSciNetzbMATHGoogle Scholar
  13. 13.
    des Cloizeaux, J.: Energy bands and projection operators in a crystal: Analytic and asymptotic properties. Phys. Rev. 135, A685–A697; Analytical properties of n-dimensional energy bands and Wannier functions. Ibid., A698–A707 (1964)ADSMathSciNetGoogle Scholar
  14. 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)ADSGoogle Scholar
  15. 15.
    Feichtinger, H.G., Strohmer, T. (eds.): Gabor Analysis and Algorithms, Theory and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (1998)zbMATHGoogle Scholar
  16. 16.
    Fiorenza, D., Monaco, D., Panati, G.: Construction of real-valued localized composite Wannier functions for insulators. Ann. Henri Poincaré 17, 63–97 (2016)ADSMathSciNetzbMATHGoogle Scholar
  17. 17.
    Fiorenza, D., Monaco, D., Panati, G.: \(\mathbb{Z}_2\) invariants of topological insulators as geometric obstructions. Commun. Math. Phys. 343, 1115–1157 (2016)ADSzbMATHGoogle Scholar
  18. 18.
    Freeman, D., Poore, D., Wei, A.R., Wyse, M.: Moving Parseval frames for vector bundles. Houston J. of Math. 40, 817–832 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Freund, S., Teufel, S.: Peierls substitution for magnetic Bloch bands. Anal. PDE 9, 773–811 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Galli, G., Parrinello, M.: Large scale electronic structure calculations. Phys. Rev. Lett. 69, 3547 (1992)ADSGoogle Scholar
  21. 21.
    Goedecker, S.: Linear scaling electronic structure methods. Rev. Mod. Phys. 71, 1085–1123 (1999)ADSGoogle Scholar
  22. 22.
    Gröchenig, K.: Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2001)zbMATHGoogle Scholar
  23. 23.
    Gontier, D., Levitt, A., Siraj-Dine, S.: Numerical construction of Wannier functions through homotopy. J. Math. Phys. 60, 031901 (2019)ADSMathSciNetzbMATHGoogle Scholar
  24. 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)Google Scholar
  25. 25.
    Hofstadter, D.R.: Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976)ADSGoogle Scholar
  26. 26.
    Husemoller, D.: Fibre Bundles. No. 20 in Graduate Texts in Mathematics, 3rd edn. Springer, New York (1994)Google Scholar
  27. 27.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)zbMATHGoogle Scholar
  28. 28.
    Kohmoto, M.: Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160, 343–354 (1985)ADSMathSciNetGoogle Scholar
  29. 29.
    Kuchment, P.: Floquet Theory for Partial Differential Equations. Birkhäuser, Basel (1993)zbMATHGoogle Scholar
  30. 30.
    Kuchment, P.: Tight frames of exponentially decaying Wannier functions. J. Phys. A Math. Theor. 42, 025203 (2009)ADSMathSciNetzbMATHGoogle Scholar
  31. 31.
    Kuchment, P.: An overview of periodic ellipic operators. Bull. AMS 53, 343–414 (2016)zbMATHGoogle Scholar
  32. 32.
    Ludewig, M., Thiang, G.C.: Good Wannier bases in Hilbert modules associated to topological insulators. Preprint arXiv:1904.13051 (2019)
  33. 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)ADSGoogle Scholar
  34. 34.
    Monaco, D.: Chern and Fu–Kane–Mele invariants as topological obstructions. In: Dell’Antonio, G., Michelangeli, A. (eds.) Advances in Quantum Mechanics: Contemporary Trends and Open Problems. Vol. 18 in Springer INdAM Series, Chapter 12. Springer, Cham (2017)Google Scholar
  35. 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)MathSciNetzbMATHGoogle Scholar
  36. 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)MathSciNetzbMATHGoogle Scholar
  37. 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)ADSMathSciNetzbMATHGoogle Scholar
  38. 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)ADSMathSciNetzbMATHGoogle Scholar
  39. 39.
    Nenciu, A., Nenciu, G.: Existence of exponentially localized Wannier functions for nonperiodic systems. Phys. Rev. B 47, 10112–10115 (1993)ADSGoogle Scholar
  40. 40.
    Nenciu, A., Nenciu, G.: The existence of generalised Wannier functions for one-dimensional systems. Commun. Math. Phys. 190, 541–548 (1998)ADSMathSciNetzbMATHGoogle Scholar
  41. 41.
    Panati, G.: Triviality of Bloch and Bloch–Dirac bundles. Ann. Henri Poincaré 8, 995–1011 (2007)ADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Volume IV: Analysis of Operators. Academic Press, New York (1978)zbMATHGoogle Scholar
  43. 43.
    Resta, R., Vanderbilt, D.: Theory of polarization: a modern approach. In: Rabe, K.M., Ahn, C.H., Triscone, J.-M. (eds.) Physics of Ferroelectrics: A modern perspective, pp. 31–68. Springer, Berlin (2007)Google Scholar
  44. 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)zbMATHGoogle Scholar
  45. 45.
    Spaldin, N.A.: A beginner’s guide to the modern theory of polarization. J. Solid State Chem. 195, 2 (2012)ADSGoogle Scholar
  46. 46.
    Thouless, D.J.: Wannier functions for magnetic sub-bands. J. Phys. C Solid State Phys. 17, L325–L327 (1984)ADSMathSciNetGoogle Scholar
  47. 47.
    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)ADSGoogle Scholar
  48. 48.
    Yates, J., Wang, X., Vanderbilt, D., Souza, I.: Spectral and Fermi surface properties from Wannier interpolation. Phys. Rev. B 75, 195121 (2007)ADSGoogle Scholar
  49. 49.
    Zaidenberg, M.G., Krein, S.G., Kuchment, P., Pankov, A.A.: Banach bundles and linear operators. Russian Math. Surveys 30, 115–175 (1975)ADSMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark
  2. 2.Dipartimento di Matematica e FisicaUniversità degli Studi di Roma TreRomeItaly
  3. 3.Dipartimento di Matematica“La Sapienza” Università di RomaRomeItaly

Personalised recommendations