Superfluid transition of homogeneous and trapped two-dimensional Bose gases

^†Laboratoire de Physique Théorique de la Matière Condensée, Unité Mixte de Recherche, Centre National de la Recherche Scientifique 7600, Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France;
^‡Department of Physics, University of Illinois at Urbana–Champaign, 1110 West Green Street, Urbana, IL 61801;
^¶European Centre for Theoretical Studies in Nuclear Physics and Related Areas, 38050 Villazzano (Trento), Italy;
^‖Service de Physique Théorique, Commissariat à l'Energie Atomique–Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France; and
^††Laboratoire Kastler Brossel, Ecole Normal Supérieure, 24 rue Lhomond, 75005 Paris, France

Contributed by Gordon Baym, November 15, 2006 (received for review August 29, 2006)

Abstract

Current experiments on atomic gases in highly anisotropic traps present the opportunity to study in detail the low temperature phases of two-dimensional inhomogeneous systems. Although, in an ideal gas, the trapping potential favors Bose–Einstein condensation at finite temperature, interactions tend to destabilize the condensate, leading to a superfluid Kosterlitz–Thouless–Berezinskii phase with a finite superfluid mass density but no long-range order, as in homogeneous fluids. The transition in homogeneous systems is conveniently described in terms of dissociation of topological defects (vortex–antivortex pairs). However, trapped two-dimensional gases are more directly approached by generalizing the microscopic theory of the homogeneous gas. In this paper, we first derive, via a diagrammatic expansion, the scaling structure near the phase transition in a homogeneous system, and then study the effects of a trapping potential in the local density approximation. We find that a weakly interacting trapped gas undergoes a Kosterlitz–Thouless–Berezinskii transition from the normal state at a temperature slightly below the Bose–Einstein transition temperature of the ideal gas. The characteristic finite superfluid mass density of a homogeneous system just below the transition becomes strongly suppressed in a trapped gas.

Footnotes

^§To whom correspondence should be addressed. E-mail: gbaym{at}uiuc.edu
Author contributions: M.H., G.B., J.-P.B., and F.L. performed research and wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS direct submission.
↵ ‡‡ The scattering length in two dimensions is not a well defined concept because the scattering cross-section vanishes in the limit of low energies and low momentum. Nonetheless, for a singular potential such as a hard core, it is sufficient to sum particle–particle scattering processes in the T matrix and to replace the bare potential by the T matrix in the Hamiltonian (36, 37). Because the T matrix depends only logarithmically on energy and momentum, we can still work to leading order with a momentum- and energy-independent coupling constant, g = 2π/(m log 1/na ²), where a corresponds to the hard core diameter. We further neglect the density dependence of the coupling constant in the following, because it does not enter in an essential way, and write g = 2πα/m.
↵ §§ The expected phase transition has been calculated by path integral simulations (35) without introduction of explicit vortex degrees of freedom; the vorticity correlation function in these calculations showed no direct evidence of vortex unbinding at the transition.
↵ ¶¶ Josephson's relation remains valid inside the critical region of a finite size system, with the limit of zero wavevector replaced by k → k ₀.
Abbreviations:

KTB,

Kosterlitz–Thouless–Berezinskii;

BEC,

Bose–Einstein condensate.