Journal of Statistical Physics

, Volume 154, Issue 1–2, pp 2–50 | Cite as

Quantum Hall Phases and Plasma Analogy in Rotating Trapped Bose Gases

Article

Abstract

A bosonic analogue of the fractional quantum Hall effect occurs in rapidly rotating trapped Bose gases: There is a transition from uncorrelated Hartree states to strongly correlated states such as the Laughlin wave function. This physics may be described by effective Hamiltonians with delta interactions acting on a bosonic N-body Bargmann space of analytic functions. In a previous paper (Rougerie et al. in Phys. Rev. A 87:023618, 2013) we studied the case of a quadratic plus quartic trapping potential and derived conditions on the parameters of the model for its ground state to be asymptotically strongly correlated. This relied essentially on energy upper bounds using quantum Hall trial states, incorporating the correlations of the Bose-Laughlin state in addition to a multiply quantized vortex pinned at the origin. In this paper we investigate in more details the density of these trial states, thereby substantiating further the physical picture described in (Rougerie et al. in Phys. Rev. A 87:023618, 2013), improving our energy estimates and allowing to consider more general trapping potentials. Our analysis is based on the interpretation of the densities of quantum Hall trial states as Gibbs measures of classical 2D Coulomb gases (plasma analogy). New estimates on the mean-field limit of such systems are presented.

Keywords

Rotating Bose gases Bosonic Quantum Hall Effect 2D Coulomb gases 

Notes

Acknowledgements

NR thanks Xavier Blanc and Mathieu Lewin for helpful discussions during the early stages of this project. SS is supported by an EURYI award. Funding from the CNRS in the form of a PEPS-PTI project is also acknowledged. JY thanks the Institute Mittag Leffler for hospitality during his stay in the fall of 2012.

References

  1. 1.
    Aftalion, A.: Vortices in Bose-Einstein Condensates. Progress in Nonlinear Differential Equations and Their Applications, vol. 67. Birkhäuser, Basel (2006) MATHGoogle Scholar
  2. 2.
    Aftalion, A., Blanc, X.: Reduced energy functionals for a three dimensional fast rotating Bose-Einstein condensate. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25(2), 339–355 (2008) ADSCrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Aftalion, A., Blanc, X., Nier, F.: Vortex distribution in the lowest Landau level. Phys. Rev. A 73, 011601(R) (2006) ADSCrossRefGoogle Scholar
  4. 4.
    Aftalion, A., Blanc, X., Nier, F.: Lowest Landau level functionals and Bargmann spaces for Bose-Einstein condensates. J. Funct. Anal. 241, 661–702 (2006) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Blanc, X., Rougerie, N.: Lowest-Landau-level vortex structure of a Bose-Einstein condensate rotating in a harmonic plus quartic trap. Phys. Rev. A 77, 053615 (2008) ADSCrossRefGoogle Scholar
  6. 6.
    Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Commun. Pure Appl. Math. 14, 187–214 (1961) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Ben Arous, G., Zeitouni, O.: Large deviations from the circular law. ESAIM, Probab. Stat. 2, 123–134 (1998) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bolley, F., Villani, C.: Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. Ann. Fac. Sci. Toulouse 6, 331–352 (2005) CrossRefMathSciNetGoogle Scholar
  9. 9.
    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) ADSCrossRefGoogle Scholar
  10. 10.
    Bretin, V., Stock, S., Seurin, Y., Dalibard, J.: Fast rotation of a Bose-Einstein condensate. Phys. Rev. Lett. 92, 050403 (2004) ADSCrossRefGoogle Scholar
  11. 11.
    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) ADSCrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Cappelli, A., Trugenberger, C.A., Zemba, G.R.: Large N limit in the quantum Hall effect. Phys. Lett. B 306, 100 (1993) ADSCrossRefGoogle Scholar
  13. 13.
    Cooper, N.R.: Rapidly rotating atomic gases. Adv. Phys. 57, 539–616 (2008) ADSCrossRefGoogle Scholar
  14. 14.
    Correggi, M., Yngvason, J.: Energy and vorticity in fast rotating Bose-Einstein condensates. J. Phys. A, Math. Theor. 41, 445002 (2008) ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    Correggi, M., Pinsker, F., Rougerie, N., Yngvason, J.: Critical rotational speeds for superfluids in homogeneous traps. J. Math. Phys. 53, 095203 (2012) ADSCrossRefMathSciNetGoogle Scholar
  16. 16.
    Correggi, M., Pinsker, F., Rougerie, N., Yngvason, J.: Rotating superfluids in anharmonic traps: from vortex lattices to giant vortices. Phys. Rev. A 84, 053614 (2011) ADSCrossRefGoogle Scholar
  17. 17.
    Dalibard, J., Gerbier, F., Juzeliūnas, G., Öhberg, P.: Artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523 (2011) ADSCrossRefGoogle Scholar
  18. 18.
    Di Francesco, P., Gaudin, M., Itzykson, C., Lesage, F.: Laughlin’s wave functions, Coulomb gases and expansions of the discriminant. Int. J. Mod. Phys. A 9, 4257–4351 (1994) ADSCrossRefMATHGoogle Scholar
  19. 19.
    Draxler, D.: Bosons in the lowest Landau level in an anharmonic trap: derivation of the mean-field energy functional. Diploma Thesis, University of Vienna (2010) Google Scholar
  20. 20.
    Fetter, A.L.: Rotating trapped Bose-Einstein condensates. Rev. Mod. Phys. 81, 647–691 (2009) ADSCrossRefGoogle Scholar
  21. 21.
    Forrester, P.J.: Log-Gases and Random Matrices. London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton (2010) MATHGoogle Scholar
  22. 22.
    Girvin, S.: Introduction to the fractional quantum Hall effect. Sémin. Poincaré 2, 54–74 (2004) Google Scholar
  23. 23.
    Girvin, S., Jach, T.: Formalism for the quantum Hall effect: Hilbert space of analytic functions. Phys. Rev. B 29, 5617–5625 (1984) ADSCrossRefMathSciNetGoogle Scholar
  24. 24.
    de Gail, R., Regnault, N., Goerbig, M.O.: Plasma picture of the fractional quantum Hall effect with internal SU(K) symmetries. Phys. Rev. B 77, 165310 (2008) ADSCrossRefGoogle Scholar
  25. 25.
    Jansen, S.: Fermionic and bosonic Laughlin state on thick cylinders. J. Math. Phys. 53, 123306 (2012) ADSCrossRefMathSciNetGoogle Scholar
  26. 26.
    Jansen, S., Lieb, E.H., Seiler, R.: Symmetry breaking in Laughlin’s state on a cylinder. Commun. Math. Phys. 285, 503–535 (2009) ADSCrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Kiessling, M.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Math. 46, 27–56 (1993) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Kiessling, M., Spohn, H.: A note on the eigenvalue density of random matrices. Commun. Math. Phys. 199, 683–695 (1999) ADSCrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Laughlin, R.B.: Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983) ADSCrossRefGoogle Scholar
  30. 30.
    Laughlin, R.B.: Elementary theory: the incompressible quantum fluid. In: Prange, R.E., Girvin, S.M. (eds.) The Quantum Hall Effect. Springer, Heidelberg (1987) Google Scholar
  31. 31.
    Levkivskyi, I.P., Fröhlich, J., Sukhorukov, E.V.: Theory of fractional quantum Hall interferometers. Phys. Rev. B 86, 245105 (2012) ADSCrossRefGoogle Scholar
  32. 32.
    Lewin, M., Seiringer, R.: Strongly correlated phases in rapidly rotating Bose gases. J. Stat. Phys. 137, 1040–1062 (2009) ADSCrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Lieb, E.H.: A lower bound for Coulomb energies. Phys. Lett. A 70, 444–446 (1979) ADSCrossRefMathSciNetGoogle Scholar
  34. 34.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14. AMS, Providence (1997) Google Scholar
  35. 35.
    Lieb, E.H., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge (2010) Google Scholar
  36. 36.
    Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and Its Condensation. Oberwolfach Seminar Series, vol. 34. Birkhäuser, Basel (2005) MATHGoogle Scholar
  37. 37.
    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) ADSCrossRefGoogle Scholar
  38. 38.
    Messer, J., Spohn, H.: Statistical mechanics of the isothermal Lane-Emden equation. J. Stat. Phys. 29, 561–578 (1982) ADSCrossRefMathSciNetGoogle Scholar
  39. 39.
    Morris, A.G., Feder, D.L.: Gaussian potentials facilitate access to quantum hall states in rotating Bose gases. Phys. Rev. Lett. 99, 240401 (2007) ADSCrossRefGoogle Scholar
  40. 40.
    Neri, C.: Statistical mechanics of the N-point vortex system with random intensities on a bounded domain. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21, 382–399 (2004) ADSMathSciNetGoogle Scholar
  41. 41.
    Papenbrock, T., Bertsch, G.F.: Rotational spectra of weakly interacting Bose-Einstein condensates. Phys. Rev. A 63, 023616 (2001) ADSCrossRefGoogle Scholar
  42. 42.
    Regnault, N., Jolicoeur, T.: Quantum hall fractions in rotating Bose-Einstein condensates. Phys. Rev. Lett. 91, 030402 (2004) CrossRefGoogle Scholar
  43. 43.
    Regnault, N., Jolicoeur, T.: Quantum Hall fractions for spinless bosons. Phys. Rev. B 69, 235309 (2004) ADSCrossRefGoogle Scholar
  44. 44.
    Regnault, N., Chang, C.C., Jolicoeur, T., Jain, J.K.: Composite fermion theory of rapidly rotating two-dimensional bosons. J. Phys. B 39, S89–S99 (2006) ADSCrossRefGoogle Scholar
  45. 45.
    Roncaglia, M., Rizzi, M., Dalibard, J.: From rotating atomic rings to quantum hall states. Sci. Rep. 1, 43 (2011). doi:10.1038/srep00043. www.nature.com ADSCrossRefGoogle Scholar
  46. 46.
    Rougerie, N.: Annular Bose-Einstein condensates in the lowest Landau level. Appl. Math. Res. Express 2011, 95–121 (2011) MATHMathSciNetGoogle Scholar
  47. 47.
    Rougerie, N., Serfaty, S., Yngvason, J.: Quantum Hall states of bosons in rotating anharmonic traps. Phys. Rev. A 87, 023618 (2013) ADSCrossRefGoogle Scholar
  48. 48.
    Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften, vol. 316. Springer, Berlin (1997) CrossRefMATHGoogle Scholar
  49. 49.
    Sandier, E., Serfaty, S.: 2D Coulomb gases and the renormalized energy. arxiv:1201.3503 (2012)
  50. 50.
    Smith, R.A., Wilkin, N.K.: Exact eigenstates for repulsive bosons in two dimensions. Phys. Rev. A 62, 061602(R) (2000) ADSCrossRefGoogle Scholar
  51. 51.
    Stormer, H.L., Tsui, D.C., Gossard, A.C.: The fractional quantum Hall effect. Rev. Mod. Phys. 71, S298–S305 (1999) CrossRefGoogle Scholar
  52. 52.
    Viefers, S.: Quantum Hall physics in rotating Bose-Einstein condensates. J. Phys. C 12, 123202 (2008) Google Scholar
  53. 53.
    Viefers, S., Hansson, T.H., Reimann, S.M.: Bose condensates at high angular momenta. Phys. Rev. A 62, 053604 (2000) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.LPMMC, UMR 5493Université Grenoble 1 and CNRSGrenobleFrance
  2. 2.UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis LionsParisFrance
  3. 3.CNRS, UMR 7598 LJLLParisFrance
  4. 4.Courant InstituteNew York UniversityNew YorkUSA
  5. 5.Fakultät für PhysikUniversität WienViennaAustria
  6. 6.Erwin Schrödinger Institute for Mathematical PhysicsViennaAustria

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