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’s function. This representation has been derived when the interelectron 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’s function is substituted by the complete two-point Green’s 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
E. Fradkin, Field Theories of Condensed Matter Physics (Addison Wesley, Redwood City, CA, 1991).
D. Tong, arXiv:1606.06687 [hep-ph].
Y. Hatsugai, J. Phys.: Condens. Matter 9, 2507 (1997).
X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008).
T. Matsuyama, Prog. Theor. Phys. 77, 711 (1987).
G. E. Volovik, Sov. Phys. JETP 67, 1804 (1988).
G. E. Volovik, The Universe in a Helium Droplet (Clarendon, Oxford, 2003).
M. A. Zubkov and X. Wu, arXiv:1901.06661 [condmat. mes-hall].
S. Coleman and B. Hill, Phys. Lett. B 159, 184 (1985).
T. Lee, Phys. Lett. B 171, 247 (1986).
C. X. Zhang and M. A. Zubkov, arXiv:1902.06545 [cond-mat.mes-hall].
R. Kubo, H. Hasegawa, and N. Hashitsume, J. Phys. Soc. Jpn. 14, 56 (1959). https://doi.org/10.1143/JPSJ.14.56
Q. Niu, D. J. Thouless, and Y. Wu, Phys. Rev. B 31, 3372 (1985).
B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A. Lee, Phys. Rev. B 22, 5142 (1980).
B. L. Altshuler and A. G. Aronov, Electron-Electron Interaction in Disordered Systems, Ed. by A. L. Efros and M. Pollak (Elsevier, North Holland, Amsterdam, 1985).
H. J. Groenewold, Physica (Amsterdam, Neth.) 12, 405 (1946).
J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949).
F. A. Berezin and M. A. Shubin, in Hilbert Space Operators and Operator Algebras, Colloquia Mathematica Societatis Janos Bolyai (North-Holland, Amsterdam, 1972), p. 21.
T. L. Curtright and C. K. Zachos, Asia Pacif. Phys. Newslett. 01, 37 (2012); arXiv:1104.5269.
M. A. Zubkov, Annals Phys. 373, 298 (2016); arXiv:1603.03665 [cond-mat.mes-hall].
I. V. Fialkovsky and M. A. Zubkov, arXiv:1905.11097.
M. Suleymanov and M. A. Zubkov, Nucl. Phys. B 938, 171 (2019); arXiv:1811.08233 [hep-lat]. https://doi.org/10.1016/j.nuclphysb.2019.114674
M. A. Zubkov and Z. V. Khaidukov, JETP Lett. 106, 172 (2017).
Z. V. Khaidukov and M. A. Zubkov, JETP Lett. 108, 670 (2018); arXiv:1812.00970 [cond-mat.mes-hall]. https://doi.org/10.1134/S0021364018220046
J. Nissinen and G. E. Volovik, arXiv:1812.03175.
Acknowledgments
The authors are grateful to I. Fialkovsky, M. Suleymanov, and Xi Wu for useful discussions. M.A. Zubkov kindly acknowledges valuable discussions with G.E. Volovik.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, C.X., Zubkov, M.A. Hall Conductivity as the Topological Invariant in the Phase Space in the Presence of Interactions and a Nonuniform Magnetic Field. Jetp Lett. 110, 487–494 (2019). https://doi.org/10.1134/S0021364019190020
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0021364019190020