Hofstadter’s Cocoon

Abstract

Hofstadter showed that the energy levels of electrons on a lattice plotted as a function of magnetic field form an beautiful structure now referred to as “Hofstadter’s butterfly”. We study a non-Hermitian continuation of Hofstadter’s model; as the non-Hermiticity parameter g increases past a sequence of critical values the eigenvalues successively go complex in a sequence of “double-pitchfork bifurcations” wherein pairs of real eigenvalues degenerate and then become complex conjugate pairs. The associated wavefunctions undergo a spontaneous symmetry breaking transition that we elucidate. Beyond the transition a plot of the real parts of the eigenvalues against magnetic field resembles the Hofstadter butterfly; a plot of the imaginary parts plotted against magnetic fields forms an intricate structure that we call the Hofstadter cocoon. The symmetries of the cocoon are described. Hatano and Nelson have studied a non-Hermitian continuation of the Anderson model of localization that has close parallels to the model studied here. The relationship of our work to that of Hatano and Nelson and to PT transitions studied in PT quantum mechanics is discussed.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    Hofstader, D.R.: Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B14, 2239 (1976)

    ADS  Article  Google Scholar 

  2. 2.

    Kuhl, U., Stöckmann, H.J.: Microwave realization of the Hofstadter butterfly. Phys. Rev. Lett. 80, 3232 (1998)

    ADS  Article  Google Scholar 

  3. 3.

    Dean, C.R., et al.: Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598 (2013)

    ADS  Article  Google Scholar 

  4. 4.

    Ponomarenko, L.A., et al.: Cloning of Dirac fermions in graphene superlattices. Nature 497, 595 (2013)

    ADS  Article  Google Scholar 

  5. 5.

    Hunt, B., et al.: Massive Dirac Fermions and Hofstadter Butterfly in a van der Waals Heterostructure. Science 340, 1427 (2013)

    ADS  Article  Google Scholar 

  6. 6.

    Yuce, C.: PT Symmetric Aubre-Andre Model. http://arxiv.org/pdf/1402.2749v1

  7. 7.

    Dyson, F.J.: General theory of spin-wave interactions. Phys. Rev. 102, 1217 (1956)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Hatano, N., Nelson, D.R.: Localization transitions in non-hermitian quantum mechanics. Phys. Rev. Lett. 77, 570 (1996)

    ADS  Article  Google Scholar 

  9. 9.

    Hatano, N., Nelson, D.R.: Vortex pinning and non-Hermitian quantum mechanics. Phys. Rev. B56, 8651 (1997)

    ADS  Article  Google Scholar 

  10. 10.

    Hatano, N., Nelson, D.R.: Non-Hermitian delocalization and eigenfunctions. Phys. Rev. B58, 8384 (1998)

    ADS  Article  Google Scholar 

  11. 11.

    Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492 (1958)

    ADS  Article  Google Scholar 

  12. 12.

    Bender, C.M., Boettcher, S.: Real spectra in non-Hemitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007)

    ADS  Article  Google Scholar 

  14. 14.

    Guo, A., et al.: Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)

    ADS  Article  Google Scholar 

  15. 15.

    Ruter, C.E., et al.: Observation of parity-time symmetry in optics. Nat. Phys. 6, 1515 (2010)

    Article  Google Scholar 

  16. 16.

    Yoshioka, D., Halperin, B.I., Lee, P.A.: Ground state of two-dimensional electrons in strong magnetic fields and 1/3 quantized hall effect. Phys. Rev. Lett. 50, 1219 (1983)

    ADS  Article  Google Scholar 

  17. 17.

    Haldane, F.D.M., Rezayi, E.H.: Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect. Phys. Rev. B31, 2529 (1985)

    ADS  Article  Google Scholar 

  18. 18.

    Wannier, G.H.: A result not dependent on rationality for Bloch electrons in a magnetic field. Phys. Status Solidi B88, 757 (1978)

    ADS  Article  Google Scholar 

  19. 19.

    MacDonald, A.H.: Landau-level subband structure of electrons on a square lattice. Phys. Rev. B28, 6713 (1983)

    ADS  Article  MathSciNet  Google Scholar 

  20. 20.

    DiVincenzo, D.P., Steinhardt, P.J. (eds.).: Quasicrystals: The State of the Art, vol. 16. World Scientific, Singapore (1999)

  21. 21.

    McKane, A.J., Stone, M.: Localization as an alternative to Goldstone’s theorem. Ann. Phys. 131, 36 (1981)

    ADS  Article  MathSciNet  Google Scholar 

  22. 22.

    Thouless, D.J., Kohomoto, M., Nightingale, M.P., den Nijs, M.: Quantized hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982)

    ADS  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Katherine Jones-Smith.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jones-Smith, K., Wallace, C. Hofstadter’s Cocoon. Int J Theor Phys 54, 219–226 (2015). https://doi.org/10.1007/s10773-014-2216-4

Download citation

Keywords

  • PT quantum mechanics
  • Hofstadter butterfly
  • Harper’s Equation