Skip to main content
SpringerLink
Log in

Incompressibility Estimates for the Laughlin Phase, Part II

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

Abstract

We consider fractional quantum Hall states built on Laughlin’s original N-body wave-functions, i.e., they are of the form holomorphic × Gaussian and vanish when two particles come close, with a given polynomial rate. Such states appear naturally when looking for the ground state of 2D particles in strong magnetic fields, interacting via repulsive forces and subject to an external potential due to trapping and/or disorder. We prove that all functions in this class satisfy a universal local density upper bound, in a suitable weak sense. Such bounds are useful to investigate the response of fractional quantum Hall phases to variations of the external potential. Contrary to our previous results for a restricted class of wave-functions, the bound we obtain here is not optimal, but it does not require any additional assumptions on the wave-function, besides analyticity and symmetry of the pre-factor modifying the Laughlin function.

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. Anderson G., Guionnet A., Zeitouni O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  2. Ciftjá O.: Monte Carlo study of Bose Laughlin wave function for filling factors 1/2, 1/4 and 1/6. Europhys.Lett. 74, 486–492 (2006)

    Article  ADS  Google Scholar 

  3. Dyson F.J.: Statistical theory of the energy levels of a complex system. Part I. J. Math. Phys. 3, 140–156 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Forrester, P.: Log-gases and random matrices. In: London Mathematical Society Monographs Series. Princeton University Press, Princeton (2004)

  5. Ginibre J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Girvin S.: Introduction to the fractional quantum Hall effect. Séminaire Poincaré 2, 54–74 (2004)

    Google Scholar 

  7. Laughlin R.B.: Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983)

    Article  ADS  Google Scholar 

  8. Laughlin, R.B.: Elementary theory: the incompressible quantum fluid. In: Prange, R.E., Girvin, S.E. (eds.) The Quantum Hall effect. Springer, Heidelberg (1987)

  9. Lewin M., Seiringer R.: Strongly correlated phases in rapidly rotating Bose gases. J. Stat. Phys. 137, 1040–1062 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  10. Lieb, E.H., Loss, M.: Analysis of Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)

  11. Mehta M.: Random Matrices, 3rd edn. Elsevier/Academic Press, Amsterdam (2004)

    MATH  Google Scholar 

  12. Papenbrock T., Bertsch G.F.: Rotational spectra of weakly interacting Bose–Einstein condensates. Phys. Rev. A 63, 023616 (2001)

    Article  ADS  Google Scholar 

  13. Petrache, M., Serfaty, S.: Next order asymptotics and renormalized energy for Riesz interactions. J. Institut. Math. Jussieu (2014, to appear)

  14. Rota-Nodari, S., Serfaty, S.: Renormalized energy equidistribution and local charge balance in 2d Coulomb systems. Int. Math. Res. Notices 2015, 3035–3093 (2015)

  15. Rougerie, N., Serfaty, S.: Higher dimensional Coulomb gases and renormalized energy functionals. Comm. Pure Appl. Math. (2013, to appear)

  16. Rougerie N., Serfaty S., Yngvason J.: Quantum Hall phases and plasma analogy in rotating trapped bose gases. J. Stat. Phys. 154, 2–50 (2014)

    Article  MathSciNet  Google Scholar 

  17. Rougerie N., Serfaty S., Yngvason J.: Quantum Hall states of bosons in rotating anharmonic traps. Phys. Rev. A 87, 023618 (2013)

    Article  ADS  Google Scholar 

  18. Rougerie N., Yngvason J.: Incompressibility estimates for the Laughlin phase. Commun. Math. Phys. 336(3), 1109–1140 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  19. Sandier, E., Serfaty, S.: 2D Coulomb Gases and the Renormalized Energy. Annals of Proba. (2012, to appear)

  20. Störmer H., Tsui D., Gossard A.: The fractional quantum Hall effect. Rev. Mod. Phys. 71, S298–S305 (1999)

    Article  Google Scholar 

  21. Trugman S., Kivelson S.: Exact results for the fractional quantum Hall effect with general interactions. Phys. Rev. B 31, 5280 (1985)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Grenoble 1 & CNRS, LPMMC (UMR 5493), B.P. 166, 38 042, Grenoble, France

    Nicolas Rougerie

  2. Fakultät für Physik, Universität Wien, Boltzmanngasse 5, 1090, Vienna, Austria

    Jakob Yngvason

  3. Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090, Vienna, Austria

    Jakob Yngvason

Authors
  1. Nicolas Rougerie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jakob Yngvason
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Rougerie.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rougerie, N., Yngvason, J. Incompressibility Estimates for the Laughlin Phase, Part II. Commun. Math. Phys. 339, 263–277 (2015). https://doi.org/10.1007/s00220-015-2400-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-015-2400-2

Keywords

Search

Navigation