Skip to main content
SpringerLink
Log in

The quantum hall effect for electrons in a random potential

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We assume the existence of sufficiently localised states, near the edges of each Landau band. We then prove that the Hall conductivity is quantised and the parallel conductivity vanishes, when the filling factor stays close to an integer. The Hall integer is a topological invariant, given by the Landau band index.

We also prove that at weak disorder, the localisation length diverges in each Landau band.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Von Klitzing, K., Dorda, G., Pepper, M.: New method for high-accuracy determination of the fine-structure of constant based on quantized Hall resistance. Phys. Rev. Lett.45, 494 (1980)

    Google Scholar 

  2. Laughlin, R.B.: Quantized Hall conductivity in two dimensions. Phys. Rev. B23, 5632 (1981)

    Google Scholar 

  3. Halperin, B.I.: Quantized Hall conductance current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B25, 2185 (1982)

    Google Scholar 

  4. Thouless, D.J., Kohmoto, M., Nightingale, P.P., Deh-Nijs, M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett.49, 1405 (1982)

    Google Scholar 

  5. Avron, J.E., Seiler, R., Simon, B.: Homotopy and quantization in condensed matter physics. Phys. Rev. Lett.51, 51 (1983)

    Google Scholar 

  6. Dubrovin, B.A., Novikov, S.P.: Ground states of a two-dimensional electron in a periodic magnetic field. Sov. Phys. JETP52, 511 (1980)

    Google Scholar 

  7. Novikov, S.P.: Magnetic Bloch functions and vector bundles. Sov. Math. Dokl.23, 298 (1981)

    Google Scholar 

  8. Niu, Q., Thouless, D.J.: Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction. J. Phys. A17, 2453 (1984)

    Google Scholar 

  9. Niu, Q., Thouless, D.J., Wu, Y.S.: Quantized Hall conductance as a topological invariant. Phys. Rev. B31, 3372 (1985)

    Google Scholar 

  10. Avron, J.E., Seiler, R.: Quantization of the Hall conductance for general, multiparticle Schrödinger Hamiltonians. Phys. Rev. Lett.54, 259 (1985)

    Google Scholar 

  11. Avron, J.E., Seiler, R., Shapiro, B.: Generic properties of quantum Hall Hamiltonians for finite systems. Nucl. Phys. B265 [FS15], 364 (1986)

    Google Scholar 

  12. Tao, R., Haldane, F.D.M.: Impurity effect, degeneracy, and topological invariant in the quantum Hall effect. Phys. Rev. B33, 3844 (1986)

    Google Scholar 

  13. Grümm, H.R., Narnhofer, H., Thirring, W.: On the Hall current in quantum theory. Acta Physica Austriaca57, 175 (1985)

    Google Scholar 

  14. Kunz, H. Zumbach, G.: The static Hall conductivity in the presence of a strong magnetic field and random impurities, Nucl. Phys. B270 [FS16], 347 (1986)

    Google Scholar 

  15. Landau, L., Lifchitz, E.: Mécanique quantique — MIR (1966)

  16. Delyon, F., Lévy, Y., Souillard, B.: Anderson localization for multi-dimensional systems at large disorder or large energy. Commun. Math. Phys.100, 463 (1985)

    Google Scholar 

  17. Fröhlich, J., Martinelli, F., Scoppola, F., Spencer, T.: Constructive proof of localization in the Anderson tight binding model. Commun. Math. Phys.101, 21 (1985)

    Google Scholar 

  18. Simon, B., Taylor, M., Wolff, T.: Some rigorous results for the Anderson model. Phys. Rev. Lett.54, 1589 (1985)

    Google Scholar 

  19. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys.88, 151 (1983)

    Google Scholar 

  20. Martinelli, F., Holden, H.: On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator onL 2(R v). Commun. Math. Phys.93, 197 (1984)

    Google Scholar 

  21. Aoki, H., Ando, T.: Critical localization in two-dimensional Landau quantization. Phys. Rev. Lett.54, 831 (1985)

    Google Scholar 

  22. Weidmann, J.: Linear operators in Holbert spaces. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  23. Thouless, D.J.: Localisation and the two-dimensional Hall effect. J. Phys. C14, 3475 (1981)

    Google Scholar 

  24. Kato, T.: Perturbation theory for linear operators, Sect. 4. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Physique Théorique, Ecole Polytechnique Fédérale de Lausanne, PHB-Ecublens, CH-1015, Lausanne, Switzerland

    H. Kunz

Authors
  1. H. Kunz
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by E. Lieb

Dedicated to Walter Thirring on his 60th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kunz, H. The quantum hall effect for electrons in a random potential. Commun.Math. Phys. 112, 121–145 (1987). https://doi.org/10.1007/BF01217683

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01217683

Keywords

Search

Navigation