Acta Applicandae Mathematicae

, Volume 162, Issue 1, pp 105–120 | Cite as

Symmetry and Localization for Magnetic Schrödinger Operators: Landau Levels, Gabor Frames and All That

  • Massimo Moscolari
  • Gianluca PanatiEmail author


We investigate the relation between broken time-reversal symmetry and localization of the electronic states, in the explicitly tractable case of the Landau model. We first review, for the reader’s convenience, the symmetries of the Landau Hamiltonian and the relation of the latter with the Segal-Bargmann representation of Quantum Mechanics. We then study the localization properties of the Landau eigenstates by applying an abstract version of the Balian-Low Theorem to the operators corresponding to the coordinates of the centre of the cyclotron orbit in the classical theory. Our proof of the Balian-Low Theorem, although based on Battle’s main argument, has the advantage of being representation-independent.


Magnetic translations Weyl relations Balian-Low Theorem Segal-Bargmann space 



  1. 1.
    Altland, A., Zirnbauer, M.: Non-standard symmetry classes in mesoscopic normalsuperconducting hybrid structures. Phys. Rev. B 55, 1142–1161 (1997) CrossRefGoogle Scholar
  2. 2.
    Avron, J.E., Seiler, R., Simon, B.: Charge deficiency, charge transport and comparison of dimensions. Commun. Math. Phys. 159, 399 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bacry, H., Grossmann, A., Zak, J.: Proof of completeness of lattice states in the \(k \, q\) representation. Phys. Rev. B 12, 1118–1120 (1975) CrossRefGoogle Scholar
  4. 4.
    Balian, R.: Un principe d’incertitude fort en théorie du signal ou en mécanique quantique. C. R. Acad. Sci., Ser. II 292, 1357–1362 (1981) MathSciNetGoogle Scholar
  5. 5.
    Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform part I. Commun. Pure Appl. Math. 14, 187–214 (1961) CrossRefzbMATHGoogle Scholar
  6. 6.
    Bargmann, V., Butera, P., Girardello, L., Klauder, R.: On the completeness of the coherent states. Rep. Math. Phys. 2, 221–228 (1971) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Battle, G.: Heisenberg proof of the Balian-Low theorem. Lett. Math. Phys. 15, 175–177 (1988) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brouder, Ch., Panati, G., Calandra, M., Mourougane, Ch., Marzari, N.: Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98, 046402 (2007) CrossRefGoogle Scholar
  9. 9.
    Boon, M., Zak, J.: Discrete coherent states on the von Neumann lattice. Phys. Rev. B 18, 6744–6751 (1978) CrossRefGoogle Scholar
  10. 10.
    Cancès, É., Levitt, A., Panati, G., Stoltz, G.: Robust determination of maximally-localized Wannier functions. Phys. Rev. B 95, 075114 (2017) CrossRefGoogle Scholar
  11. 11.
    des Cloizeaux, J.: Energy bands and projection operators in a crystal: analytic and asymptotic properties. Phys. Rev. 135, A685–A697 (1964) MathSciNetCrossRefGoogle Scholar
  12. 12.
    des Cloizeaux, J.: Analytical properties of \(n\)-dimensional energy bands and Wannier functions. Phys. Rev. 135, A698–A707 (1964) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cornean, H., Herbst, I., Nenciu, G.: On the construction of composite Wannier functions. Ann. Henri Poincaré 17, 3361–3398 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cornean, H., Monaco, D., Moscolari, M.: Parseval frames of exponentially localized magnetic Wannier functions. Preprint (2018). arXiv:1704.00932
  15. 15.
    Fiorenza, D., Monaco, D., Panati, G.: Construction of real-valued localized composite Wannier functions for insulators. Ann. Henri Poincaré 17, 63–97 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Girvin, S.M., Jach, T.: Formalism for the quantum Hall effect: Hilbert space of analytic functions. Phys. Rev. B 29, 5617–5625 (1984) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hall, B.C.: Quantum Theory for Mathematicians. Graduate Texts in Mathematics. Springer, New York (2013) CrossRefzbMATHGoogle Scholar
  18. 18.
    Hasan, M.Z., Kane, C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010) CrossRefGoogle Scholar
  19. 19.
    Helffer, B., Sjostrand, J.: Equation de Schrödinger avec champ magnetique et equation de Harper. In: Holden, H., Jensen, A. (eds.) Schrödinger Operators. Lecture Notes in Physics, vol. 345, pp. 118–197. Springer, Berlin (1989) CrossRefGoogle Scholar
  20. 20.
    Kitaev, A.: Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009) CrossRefzbMATHGoogle Scholar
  21. 21.
    Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, 809 (1959) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Low, F.: Complete sets of wave packets. In: DeTar, C., et al. (eds.) A Passion for Physics: Essays in Honor of Geoffrey Chew, pp. 17–22. World Scientific, Singapore (1985) Google Scholar
  23. 23.
    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) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Marcelli, G., Monaco, D., Moscolari, M., Panati, G.: The Haldane model and its localization dichotomy. Rend. Mat. Appl. 39(2), 307–327 (2018) MathSciNetzbMATHGoogle Scholar
  25. 25.
    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) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Monaco, D., Panati, G., Pisante, A., Teufel, S.: The localization dichotomy for gapped periodic quantum systems (2016). arXiv:1612.09557
  27. 27.
    Moscolari, M., Panati, G.: Ultra generalized Wannier functions for systems without time-reversal symmetry and their relevance to transport. In preparation Google Scholar
  28. 28.
    Moshinsky, M., Quesne, C.: Linear canonical transformations and their unitary representations. J. Math. Phys. 18, 1772–1780 (1998) MathSciNetzbMATHGoogle Scholar
  29. 29.
    Nenciu, G.: Existence of the exponentially localised Wannier functions. Commun. Math. Phys. 91, 81–85 (1983) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nenciu, A., Nenciu, G.: Dynamics of Bloch electrons in external electric fields. II. The existence of Stark-Wannier ladder resonances. J. Phys. A 15, 3313–3328 (1982) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Panati, G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. Henri Poincaré 8, 995–1011 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Perelomov, A.M.: On the completeness of a system of coherent states. Theor. Math. Phys. 6, 213–224 (1971) Google Scholar
  33. 33.
    Rashba, E.I., Zhukov, L.E., Efros, A.L.: Orthogonal localized wave functions of an electron in a magnetic field. Phys. Rev. B 55, 5306–5312 (1997) CrossRefGoogle Scholar
  34. 34.
    Segal, I.: Mathematical problems of relativistic physics, Chap. VI. In: Kac, M. (ed.): Proceedings of the Summer Seminar, Vol. II. Boulder, Colorado 1960. Lectures in Applied Mathematics. American Math. Soc., Providence (1963). Google Scholar
  35. 35.
    Simon, B.: Harmonic Analysis: A Comprehensive Course in Analysis, Part 3. American Mathematical Society, Providence (2015) CrossRefzbMATHGoogle Scholar
  36. 36.
    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) CrossRefGoogle Scholar
  37. 37.
    Thouless, D.J.: Wannier functions for magnetic sub-bands. J. Phys. C 17, L325–L327 (1984) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Zak, J.: Finite translations in solid-state physics. Phys. Rev. Lett. 19, 1385–1387 (1967) CrossRefGoogle Scholar
  39. 39.
    Zak, J.: Dynamics of electrons in solids in external fields. Phys. Rev. 168, 686–695 (1968) CrossRefGoogle Scholar
  40. 40.
    Zak, J.: Balian-Low theorem for Landau levels. Phys. Rev. Lett. 79, 533–536 (1997) CrossRefGoogle Scholar
  41. 41.
    Zak, J.: Orthonormal sets of localized functions for a Landau level. J. Math. Phys. 39, 4195–4200 (1998) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica“La Sapienza” Università di RomaRomeItaly

Personalised recommendations