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Hall Conductivity as the Topological Invariant in Phase Space in the Presence of Interactions and Non-uniform Magnetic Field

  • C. X. Zhang
  • M. A. ZubkovEmail author
Article
  • 3 Downloads

Abstract

The quantum Hall conductivity in the presence of constant magnetic field may be represented as the topological TKNN invariant. Recently the generalization of this expression has been proposed for the nonuniform magnetic field. The quantum Hall conductivity is represented as the topological invariant in phase space in terms of the Wigner transformed two-point Green function. This representation has been derived when the inter - electron interactions were neglected. It is natural to suppose, that in the presence of interactions the Hall conductivity is still given by the same expression, in which the non-interacting Green function is substituted by the complete two-point Green function including the interaction contributions. We prove this conjecture within the framework of the 2 + 1 D tight-binding model of rather general type using the ordinary perturbation theory.

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References

  1. 1.
    D.J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).ADSCrossRefGoogle Scholar
  2. 2.
    E. Fradkin, Field Theories of Condensed Matter Physics, Addison Wesley Publishing Company, Redwood City, CA (1991).zbMATHGoogle Scholar
  3. 3.
    D. Tong, arXiv:1606.06687 [hep-ph].Google Scholar
  4. 4.
    Y. Hatsugai, J. Phys.: Condens. Matter 9, 2507 (1997).ADSGoogle Scholar
  5. 5.
    X.-L. Qi, T.L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008).ADSCrossRefGoogle Scholar
  6. 6.
    T. Matsuyama, Prog. Theor. Phys. 77, 711 (1987).ADSCrossRefGoogle Scholar
  7. 7.
    G.E. Volovik, JETP 67, 1804 (1988).Google Scholar
  8. 8.
    G. E. Volovik, The Universe in a Helium Droplet, Clarendon Press, Oxford (2003).zbMATHGoogle Scholar
  9. 9.
    M.A. Zubkov and X. Wu, arXiv:1901.06661 [cond-mat.mes-hall].Google Scholar
  10. 10.
    S. Coleman and B. Hill, Phys. Lett. B 159, 184 (1985).ADSCrossRefGoogle Scholar
  11. 11.
    T. Lee, Phys. Lett. B 171, 247 (1986).ADSCrossRefGoogle Scholar
  12. 12.
    C. X. Zhang and M. A. M. A. Zubkov, arXiv:1902.06545 [cond-mat.mes-hall].Google Scholar
  13. 13.
    R. Kubo, H. Hasegawa, and N. Hashitsume, J. Phys. Soc. Jpn. 14(1), 56 (1959); DOI: 10.1143/JPSJ.14.56.ADSCrossRefGoogle Scholar
  14. 14.
    Q. Niu, D.J. Thouless, and Y. Wu, Phys. Rev. B 31, 3372 (1985).ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    B. L. Altshuler, D. Khmel’nitzkii, A.I. Larkin, and P.A. Lee, Phys. Rev. B 22, 5142 (1980).ADSCrossRefGoogle Scholar
  16. 16.
    B. L. Altshuler and A. G. Aronov, Electron-electron inter-action in disordered systems, ed. by A. L. Efros and M. Pollak, Elsevier, North Holland, Amsterdam (1985).Google Scholar
  17. 17.
    H.J. Groenewold, Physica 12, 405 (1946).ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    J. E. Moyal, Proceedings of the Cambridge Philosophical Society 45, 99 (1949).ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    F. A. Berezin and M. A. Shubin, in Colloquia Mathematica Societatis Janos Bolyai, North-Holland, Amsterdam (1972), p. 21.Google Scholar
  20. 20.
    T. L. Curtright and C. K. Zachos, Asia Pacific Physics Newsletter 01, 37 (2012); arXiv:1104.5269.CrossRefGoogle Scholar
  21. 21.
    M.A. Zubkov, Annals Phys. 373, 298 (2016); arXiv:1603.03665 [cond-mat.mes-hall].ADSCrossRefGoogle Scholar
  22. 22.
    I.V. Fialkovsky and M.A. Zubkov, arXiv:1905.11097.Google Scholar
  23. 23.
    M. Suleymanov and M. A. Zubkov, Nucl. Phys. B 938, 171 (2019); Corrigendum: https://doi.org/10.1016/j.nuclphysb.2019.114674; arXiv:1811.08233 [hep-lat].ADSCrossRefGoogle Scholar
  24. 24.
    M.A. Zubkov and Z.V. Khaidukov, JETP Lett. 106, 172 (2017) [Pisma v ZhETF 106(3), 166 (2017)].ADSCrossRefGoogle Scholar
  25. 25.
    Z.V. Khaidukov and M.A. Zubkov, JETP Lett. 108(10), 670 (2018); doi:10.1134/S0021364018220046; arXiv:1812.00970 [cond-mat.mes-hall].ADSCrossRefGoogle Scholar
  26. 26.
    J. Nissinen and G. E. Volovik, arXiv:1812.03175.Google Scholar

Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.Physics DepartmentAriel UniversityArielIsrael
  2. 2.Institute for Theoretical and Experimental PhysicsMoscowRussia

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