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Incompressibility Estimates for the Laughlin Phase

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Abstract

This paper has its motivation in the study of the Fractional Quantum Hall Effect. We consider 2D quantum particles submitted to a strong perpendicular magnetic field, reducing admissible wave functions to those of the Lowest Landau Level. When repulsive interactions are strong enough in this model, highly correlated states emerge, built on Laughlin’s famous wave function. We investigate a model for the response of such strongly correlated ground states to variations of an external potential. This leads to a family of variational problems of a new type. Our main results are rigorous energy estimates demonstrating a strong rigidity of the response of strongly correlated states to the external potential. In particular, we obtain estimates indicating that there is a universal bound on the maximum local density of these states in the limit of large particle number. We refer to these as incompressibility estimates.

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References

  1. Aftalion A., Blanc X., Dalibard J.: Vortex patterns in a fast rotating Bose–Einstein condensate. Phys. Rev. A 71, 023611 (2005)

    Article  ADS  Google Scholar 

  2. Aftalion A., Blanc X., Nier F.: Vortex distribution in the lowest Landau level. Phys. Rev. A 73, 011601(R) (2006)

    Article  ADS  Google Scholar 

  3. Aftalion A., Blanc X., Nier F.: Lowest Landau level functionals and Bargmann spaces for Bose– Einstein condensates. J. Funct. Anal. 241, 661–702 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bieri S., Fröhlich J.: Physical principles underlying the quantum Hall effect. C. R. Phys. 12, 332–346 (2011)

    Article  ADS  Google Scholar 

  5. Boyarsky A., Cheianov V.V., Ruchayskiy O.: Microscopic construction of the chiral Luttinger liquid theory of the quantum Hall edge. Phys. Rev. B 70, 235309 (2004)

    Article  ADS  Google Scholar 

  6. Caglioti E., Lions P.L., Marchioro C., Pulvirenti M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143, 501–525 (1992)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Carlen E.: Some integral identities and inequalities for entire functions and their application to the coherent state transform. J. Funct. Anal. 97, 231–249 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ciftja 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 

  9. Diaconis P., Freedman D.: Finite exchangeable sequences. Ann. Probab. 8, 754–764 (1980)

    Google Scholar 

  10. Freedman D.: A remark on the difference between sampling with and without replacement. J. Am. Stat. Assoc. 73, 681 (1977)

    Article  Google Scholar 

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

    Google Scholar 

  12. Goerbig, M.O.: Quantum Hall effects. arXiv:0909.1998 (2009)

  13. Golse, F.: On the dynamics of large particle systems in the mean field limit. arXiv:1301.5494 (2013)

  14. Hewitt E., Savage L.J.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jain J.K.: The role of analogy in unraveling the fractional quantum Hall effect mystery. Phys. E 20, 79–88 (2003)

    Article  Google Scholar 

  16. Kiessling M.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure. Appl. Math. 46, 27–56 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kiessling M., Spohn H.: A note on the eigenvalue density of random matrices. Commun. Math. Phys. 199, 683–695 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. 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 

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

  20. Lee P.A., Nagaosa N., Wen X.G.: Doping a Mott insulator: physics of high temperature superconductivity. Rev. Mod. Phys. 78, 17 (2006)

    Article  ADS  Google Scholar 

  21. Levkivskyi I.P., Fröhlich J., Sukhorukov E.V.: Theory of fractional quantum Hall interferometers. Phys. Rev. B 86, 245105 (2012)

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14. AMS, Providence (1997)

  24. Lieb E.H., Seiringer R., Yngvason J.: The Yrast line of a rapidly rotating Bose gas: the Gross– Pitaevskii regime. Phys. Rev. A 79, 063626 (2009)

    Article  ADS  Google Scholar 

  25. Lions, P.-L.: Mean-field games and applications. Lectures at the Collège de France (2007)

  26. Messer J., Spohn H.: Statistical mechanics of the isothermal Lane– Emden equation. J. Stat. Phys. 29, 561–578 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  27. Morris A.G., Feder D.L.: Gaussian potentials facilitate access to quantum Hall states in rotating Bose gases. Phys. Rev. Lett. 99, 240401 (2007)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  29. Roncaglia, M., Rizzi, M., Dalibard, J.: From rotating atomic rings to quantum Hall states. Sci. Rep. 1 (2011). doi:10.1038/srep00043. http://www.nature.com

  30. Rougerie, N., Serfaty, S.: Higher dimensional Coulomb gases and renormalized energy functionals. arXiv:1307.2805 (2013)

  31. 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 

  32. Rougerie, N., Serfaty, S., Yngvason, J.: Quantum Hall phases and plasma analogy in rotating trapped Bose gases. J. Stat. Phys. (2013). doi:10.1007/s10955-013-0766-0

  33. Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der mathematischen Wissenchaften, vol. 316. Springer, Berlin (1997)

  34. Sandier, E., Serfaty, S.: 2D Coulomb gases and the renormalized energy. arxiv:1201.3503 (2012)

  35. Stormer H.L., Tsui D.C., Gossard A.C.: The fractional quantum Hall effect. Rev. Mod. Phys. 71, S298–S305 (1999)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  37. Viefers S.: Quantum Hall physics in rotating Bose– Einstein condensates. J. Phys. C 12, 123202 (2008)

    Google Scholar 

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Authors and Affiliations

  1. Université Grenoble 1 and CNRS, LPMMC, UMR 5493, BP 166, 38042, 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

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  1. Nicolas Rougerie
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  2. Jakob Yngvason
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Correspondence to Nicolas Rougerie.

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Communicated by H. Spohn

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Rougerie, N., Yngvason, J. Incompressibility Estimates for the Laughlin Phase. Commun. Math. Phys. 336, 1109–1140 (2015). https://doi.org/10.1007/s00220-014-2232-5

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  • DOI: https://doi.org/10.1007/s00220-014-2232-5

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