Advertisement

Proceedings of the Steklov Institute of Mathematics

, Volume 279, Issue 1, pp 181–193 | Cite as

Magnetic Bloch theory and noncommutative geometry

  • A. G. Sergeev
Article
  • 66 Downloads

Abstract

An interpretation of the magnetic Bloch theory in terms of noncommutative geometry is given. As an application we obtain a mathematical interpretation of the quantum Hall effect.

Keywords

STEKLOV Institute Fundamental Domain Noncommutative Geometry Projective Representation Chern Character 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, Philadelphia, 1976).Google Scholar
  2. 2.
    J. Bellissard, A. van Elst, and H. Schulz-Baldes, “The Noncommutative Geometry of the Quantum Hall Effect,” J. Math. Phys. 35(10), 5373–5451 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    F. A. Berezin and M. A. Shubin, The Schrödinger Equation (Kluwer, Dordrecht, 1991).zbMATHCrossRefGoogle Scholar
  4. 4.
    A. L. Carey, K. C. Hannabuss, and V. Mathai, “Quantum Hall Effect and Noncommutative Geometry,” J. Geom. Symmetry Phys. 6, 16–37 (2006); arXiv:math/0008115 [math.OA].MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Connes, Noncommutative Geometry (Academic Press, San Diego, CA, 1994).zbMATHGoogle Scholar
  6. 6.
    M. Gruber, “Noncommutative Bloch Theory,” J. Math. Phys. 42, 2438–2465 (2001).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Y. Kordyukov, V. Mathai, and M. Shubin, “Equivalence of Spectral Projections in Semiclassical Limit and a Vanishing Theorem for Higher Traces in K-Theory,” J. Reine Angew. Math. 581, 193–236 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 5: Statistical Physics (Pergamon, Oxford, 1980), Part 1.Google Scholar
  9. 9.
    R. B. Laughlin, “Quantized Hall Conductivity in Two Dimensions,” Phys. Rev. B 23, 5632–5633 (1981).CrossRefGoogle Scholar
  10. 10.
    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall Conductance in a Two-Dimensional Periodic Potential,” Phys. Rev. Lett. 49, 405–408 (1982).CrossRefGoogle Scholar
  11. 11.
    J. Xia, “Geometric Invariants of the Quantum Hall Effect,” Commun. Math. Phys. 119, 29–50 (1988).zbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations