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Quantum Hall Phases and Plasma Analogy in Rotating Trapped Bose Gases

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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.

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Notes

  1. L stands for Landau, referring to the lowest Landau level.

  2. Or rather its fermionic analogue where the exponent of the (z i −z j ) factor is an odd number, ensuring fermionic symmetry.

  3. Here and in the sequel we will drop integration elements from integrals when there is no possible confusion.

  4. We could also consider a more general trapping potential in the 3-direction.

  5. We do not change the notation for the scaled quantities.

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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.

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Authors and Affiliations

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

    N. Rougerie

  2. UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, 75005, France

    S. Serfaty

  3. CNRS, UMR 7598 LJLL, Paris, 75005, France

    S. Serfaty

  4. Courant Institute, New York University, 251 Mercer St, New York, NY, 10012, USA

    S. Serfaty

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

    J. Yngvason

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

    J. Yngvason

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  1. N. Rougerie
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  2. S. Serfaty
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  3. J. Yngvason
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Correspondence to J. Yngvason.

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Dedicated to Herbert Spohn on the occasion of his retirement from the TU München.

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Rougerie, N., Serfaty, S. & Yngvason, J. Quantum Hall Phases and Plasma Analogy in Rotating Trapped Bose Gases. J Stat Phys 154, 2–50 (2014). https://doi.org/10.1007/s10955-013-0766-0

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