Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Comment
  • Published:

Why is the Hall conductance quantized?

Nature Reviews Physics volume 2, pages 392–393 (2020)Cite this article

Subjects

Why the Hall conductance is quantized was an open problem in condensed matter theory for much of the past 40 years. Spyridon Michalakis who worked on the solution — published in 2015 — gives a personal take on how the field evolved.

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

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

References

  1. Avron, J. E., Osadchy, D. & Seiler, R. A Topological look at the quantum Hall effect. Physics Today 56, 38–42 (2003).

    Article  Google Scholar 

  2. Laughlin, R. B. Quantized Hall conductivity in two dimensions. Phys. Rev. B 23, 5632–5633 (1981).

    Article  ADS  Google Scholar 

  3. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    Article  ADS  Google Scholar 

  4. Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A Math. Phys. Sci. 392, 45–57 (1984).

    ADS  MathSciNet  MATH  Google Scholar 

  5. Simon, B. Holonomy, the Quantum Adiabatic Theorem, and Berry’s Phase. Phys. Rev. Lett. 51, 2167–2170 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  6. Chern, S. On the curvatura integra in a Riemannian manifold. Ann. Math. 46, 674–684 (1945).

    Article  MathSciNet  Google Scholar 

  7. Hastings, M. B. & Wen, X.-G. Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance. Phys. Rev. B 72, 045141 (2005).

    Article  ADS  Google Scholar 

  8. Hastings, M. B. & Michalakis, S. Quantization of Hall conductance for interacting electrons on a torus. Commun. Math. Phys. 334, 433–471 (2015).

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA

    Spyridon Michalakis

Authors
  1. Spyridon Michalakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Spyridon Michalakis.

Ethics declarations

Competing interests

The author declares no competing interests.

Additional information

Related link

Aizenman’s list of open problems in mathematical physics: http://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Michalakis, S. Why is the Hall conductance quantized?. Nat Rev Phys 2, 392–393 (2020). https://doi.org/10.1038/s42254-020-0212-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42254-020-0212-6

This article is cited by

Search

Advanced search

Quick links

Nature Briefing AI and Robotics

Sign up for the Nature Briefing: AI and Robotics newsletter — what matters in AI and robotics research, free to your inbox weekly.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing: AI and Robotics