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arXiv:cond-mat/0412512 · Philosophical Magazine Letters

Transition from the Bose–Einstein condensate to the Berezinskii–Kosterlitz–Thouless phase

Tapio Simula, M. D. Lee, David A. W. Hutchinson

Published Aug 1, 2005• 29 citations
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Abstract

We obtain a phase diagram for a trapped two-dimensional ultracold Bose gas. We find a critical temperature above which the free energy of a state with a pair of vortices of opposite circulation is lower than that for a vortex-free Bose–Einstein condensed ground state. We identify three distinct phases which are, in order of increasing temperature, a phase coherent Bose–Einstein condensate, a vortex pair plasma with a fluctuating condensate phase, and a thermal Bose gas. The existence of the vortex pair phase could be verified using current experimental set-ups.

DOI
10.1080/09500830500256587

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We obtain a phase diagram for a trapped two-dimensional ultracold Bose gas. We find a critical temperature above which the free energy of a state with a pair of vortices of opposite circulation is lower than that for a vortex-free Bose–Einstein condensed ground state. We identify three distinct phases which are, in order of increasing temperature, a phase coherent Bose–Einstein condensate, a vortex pair plasma with a fluctuating condensate phase, and a thermal Bose gas. The existence of the vortex pair phase could be verified using current experimental set-ups.
We briefly review the theory of Bose-Einstein condensation in the two-dimensional trapped Bose gas and, in particular the relationship to the theory of the homogeneous two-dimensional gas and the Berezinskii-Kosterlitz-Thouless phase. We obtain a phase diagram for the trapped two-dimensional gas, finding a critical temperature above which the free energy of a state with a pair of vortices of opposite circulation is lower than that for a vortex-free Bose-Einstein condensed ground state. We identify three distinct phases which are, in order of increasing temperature, a phase coherent Bose-Einstein condensate, a vortex pair plasma with fluctuating condensate phase and a thermal Bose gas. The thermal activation of vortex-antivortex pair formation is confirmed using finite-temperature classical field simulations.
Three distinct types of behavior have recently been identified in the two-dimensional trapped bosonic gas, namely, a phase-coherent Bose-Einstein condensate (BEC), a Berezinskii-Kosterlitz-Thouless-type (BKT) superfluid, and normal gas phases in order of increasing temperature. In the BKT phase the system favors the formation of vortex-antivortex pairs, since the free energy is lowered by this topological defect. We provide a simple estimate of the free energy of a dilute Bose gas with and without such vortex dipole excitations and show how this varies with particle number and temperature. In this way we can estimate the temperature for crossover from the coherent BEC to the (only) locally ordered BKT-like phase by identifying when vortex-dipole excitations proliferate. Our results are in qualitative agreement with recent, numerically intensive, classical field simulations.
We present an overview of our recent work describing the treatment of ultracold atomic Bose gases confined to two dimensions in a harmonic trap. We show how the gas goes through a number of regimes as the axially confining potential is increased and discuss the scattering properties of the gas in each of these regimes. We then use the formalism developed to investigate the phase-coherent Bose-Einstein condensate that is present in the trapped two-dimensional gas at temperatures that are low compared to the ideal gas critical temperature and to analyze the first-order coherence as a function of temperature. We identify the regime in which the gas loses its global phase coherence, although a macroscopic occupation of the ground state persists at a temperature well below the ideal gas critical temperature. We comment on the possible connection between this fluctuating regime and the Berezinskii-Kosterlitz-Thouless phase.
Recent experiments on ultracold Bose gases in two dimensions have provided evidence for the existence of the Berezinskii-Kosterlitz-Thouless (BKT) phase via analysis of the interference between two independent systems. In this work we study the two-dimensional quantum degenerate Bose gas at finite temperature using the projected Gross-Pitaevskii equation classical field method. Although this describes the highly occupied modes of the gas below a momentum cutoff, we have developed a method to incorporate the higher momentum states in our model. We concentrate on finite-sized homogeneous systems in order to simplify the analysis of the vortex pairing. We determine the dependence of the condensate fraction on temperature and compare this to the calculated superfluid fraction. By measuring the first order correlation function we determine the boundary of the Bose-Einstein condensate and BKT phases, and find it is consistent with the superfluid fraction decreasing to zero. We reveal the characteristic unbinding of vortex pairs above the BKT transition via a coarse-graining procedure. Finally, we model the procedure used in experiments to infer system correlations [Hadzibabic et al., Nature 441, 1118 (2006)], and quantify its level of agreement with directly calculated in situ correlation functions.
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.
We measure the in-plane distribution of thermally activated vortices in a trapped quasi-2D Bose gas, where we enhance the visibility of density-depleted vortex cores by radially compressing the sample before releasing the trap. The pairing of vortices is revealed by the two-vortex spatial correlation function obtained from the vortex distribution. The vortex density decreases gradually as temperature is lowered, and below a certain temperature, a vortex-free region emerges in the center of the sample. This shows the crossover from a Berezinskii-Kosterlitz-Thouless phase containing vortex-pair excitations to a vortex-free Bose-Einstein condensate in a finite-size 2D system.