The European Physical Journal Special Topics

, Volume 225, Issue 13–14, pp 2533–2547

A tale of two fractals: The Hofstadter butterfly and the integral Apollonian gaskets

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Part of the following topical collections:
  1. Temporal and Spatio-Temporal Dynamic Instabilities: Novel Computational and Experimental Approaches

Abstract

This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden D3 symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions.

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References

  1. 1.
    M.Ya. Azbel’, JETP 19, 634 (1964)Google Scholar
  2. 2.
    D. Langbein, Phys. Rev. 180, 633 (1969)ADSCrossRefGoogle Scholar
  3. 3.
    D. Hofstadter, Phys. Rev. B. 14 2239 (1976)ADSCrossRefGoogle Scholar
  4. 4.
    K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980)ADSCrossRefGoogle Scholar
  5. 5.
    M.V. Berry, Proc. R. Soc. A 392, 45 (1984)ADSCrossRefGoogle Scholar
  6. 6.
    B. Simon, Phys. Rev. Lett. 51, 2167 (1983)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    C.R Dean, et al., Nature 497, 598 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    M. Aidelsburger, Phys. Rev. Lett. 111, 185301 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    H. Miyake, et al., Phys. Rev. Lett. 111, 185302 (2013)ADSCrossRefGoogle Scholar
  10. 10.
    G.H. Wannier, Phys. Stat. Sol. B 88, 757 (1978)ADSCrossRefGoogle Scholar
  11. 11.
    F.H. Claro, G.H. Wannier, Phys. Rev. B 19, 19 (1979)CrossRefGoogle Scholar
  12. 12.
    A. MacDonald, Phys. Rev. B 28, 6713 (1983)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    M. Wilkinson, J. Phys. A: Math. Gen. 20, 4337 (1987)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Wilkinson, J. Phys. A: Math. Gen. 21, 8123 (1994)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    D. Mackenzie, Am. Scientist 98, 10 (2010)Google Scholar
  16. 16.
    L.R. Ford, Am. Math. Monthly 39, 586 (1938)CrossRefGoogle Scholar
  17. 17.
    The theorem is named after Rene Descartes, who stated it in 1643. See R. Descartes. Oeuvres de Descartes, Correspondence IV, edited by C. Adam and P. Tannery (Leopold Cerf, Paris, 1901)Google Scholar
  18. 18.
    I.I. Satija, with contributions by Douglas Hofstadter, Butterfly in the Quantum World: the Story of the Most Fascinating Quantum Fractal (IOP Concise, Morgan and Claypool, San Rafael, 2016)Google Scholar
  19. 19.
    I. Satija, G. Naumis, Phys. Rev. B 88, 054204 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    E. Zhao, N. Bray-Ali, C. Williams, I. Spielman, I.I. Satija, Phys. Rev. A 84, 063629 (2011)ADSCrossRefGoogle Scholar
  21. 21.
    A. Avila, S. Jitomikskaya, C.A. Marx [arXiv:http://arxiv.org/abs/1602.05111] (unpublished)
  22. 22.
    M. Lababidi, I Satija, E. Zhao, Phys. Rev. Lett. 112, 026805 (2014)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2016

Authors and Affiliations

  1. 1.Department of PhysicsGeorge Mason UniversityFairfaxUSA

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