Spectrum of a particle on a polyhedron enclosing a synthetic magnetic monopole

  • M. Ö. OktelEmail author
Regular Article


We consider a single particle hopping on a tight binding lattice formed by the vertices of a regular polyhedron and discuss the effect of a magnetic monopole enclosed in the polyhedron. The presence of the monopole induces phases on the hopping terms, given by Peierls substitution. By requiring the flux through each face of a regular polyhedron to be the same, Dirac’s quantization condition is obtained in this discrete setting. For each regular polyhedron, we calculate the energy spectrum for an arbitrary value of the flux through a Dirac string coming in from one of the faces. We find that the energy levels are degenerate only when the flux through the Dirac string corresponds to a quantized monopole. We show that the degeneracies in the presence of the monopole can be classified using the double group of the symmetry of the polyhedron and label all energy levels with corresponding irreducible representations.


Cold Matter and Quantum Gas 


  1. 1.
    P.A.M. Dirac, Proc. R. Soc. Lond. 133, 60 (1931)ADSCrossRefGoogle Scholar
  2. 2.
    F.D.M. Haldane, Phys. Rev. Lett. 51, 605 (1983)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    D.R. Hofstadter, Phys. Rev. B 14, 2239 (1976)ADSCrossRefGoogle Scholar
  4. 4.
    D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)ADSCrossRefGoogle Scholar
  5. 5.
    A.S. Sørensen, E. Demler, M.D. Lukin, Phys. Rev. Lett. 94, 086803 (2005)ADSCrossRefGoogle Scholar
  6. 6.
    M. Hafezi, A.S. Sørensen, E. Demler, M.D. Lukink, Phys. Rev. A 76, 023613 (2007)ADSCrossRefGoogle Scholar
  7. 7.
    D. Jaksch, P. Zoller, New J. Phys. 5, 56 (2003)ADSCrossRefGoogle Scholar
  8. 8.
    R.N. Palmer, D. Jaksch, Phys. Rev. Lett. 96, 180407 (2006)ADSCrossRefGoogle Scholar
  9. 9.
    R.O. Umucalılar, M.Ö. Oktel, Phys. Rev. A 76, 055601 (2007)ADSCrossRefGoogle Scholar
  10. 10.
    S. Tung, V. Schweikhard, E.A. Cornell, Phys. Rev. Lett. 97, 240402 (2006)ADSCrossRefGoogle Scholar
  11. 11.
    R.A. Williams, S. Al-Assam, C.J. Foot, Phys. Rev. Lett. 104, 050404 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Y.-J. Lin, R.L. Compton, K. Jiménez-García, J.V. Porto, I.B. Spielman, Nature 462, 628 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    R.O. Umucalılar, H. Zhai, M.Ö. Oktel, Phys. Rev. Lett. 100, 070402 (2008)ADSCrossRefGoogle Scholar
  14. 14.
    R. Peierls, Z. Phys. 80, 763 (1933)ADSzbMATHCrossRefGoogle Scholar
  15. 15.
    G.M. Obermair, H.-J. Schellnhuber, Phys. Rev. B 23, 5185 (1981)ADSCrossRefGoogle Scholar
  16. 16.
    H.-J. Schellnhuber, G.M. Obermair, A. Rauh, Phys. Rev. B 23, 5191 (1981)ADSCrossRefGoogle Scholar
  17. 17.
    R.O. Umucalilar, M.Ö. Oktel, Phys. Rev. A 78, 033602 (2008)ADSCrossRefGoogle Scholar
  18. 18.
    J. Zak, Phys. Rev. 134, A1602 (1964)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    E. Brown, Phys. Rev. 133, A1038 (1964)ADSCrossRefGoogle Scholar
  20. 20.
    T. Janssen, Crystallographic groups (North Holland, Amsterdam, 1974)Google Scholar
  21. 21.
    L.L. Boyle, K.F. Green, Phil. Trans. R. Soc. Lond. 288, 237 (1978)MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    H. Weyl, The theory of groups and quantum mechanics (Methuen, London, 1931)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of PhysicsBilkent UniversityAnkaraTurkey

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