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International Journal of Modern Physics B

CALCULATION OF THE CRITICAL TEMPERATURE OF AN INTERACTING BOSE GAS BY MEANFIELD APPROXIMATION

HIDENORI SUZUKI, Masuo Suzuki

Published Jul 10, 2001• 3 citations
Publisher

Abstract

The critical temperature of a uniform interacting Bose gas is shown to be increased by the repulsive interaction in the frame work of meanfield approximation, which has been regarded by many authors to be useless for studying the shift of the critical point. Furthermore, the critical temperature of a Bose gas with the general dispersion εp=cpk is also increased by the repulsive interaction between quasiparticles. The critical temperature of a trapped gas is also estimated in the same meanfield approximation. The present results agree qualitatively with those obtained perturbationally by other authors.

DOI
10.1142/s0217979201007075

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The critical temperature of a uniform interacting Bose gas is shown to be increased by the repulsive interaction in the frame work of meanfield approximation, which has been regarded by many authors to be useless for studying the shift of the critical point. Furthermore, the critical temperature of a Bose gas with the general dispersion εp=cpk is also increased by the repulsive interaction between quasiparticles. The critical temperature of a trapped gas is also estimated in the same meanfield approximation. The present results agree qualitatively with those obtained perturbationally by other authors.
We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory – in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k . These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a ; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.
We show that the critical temperature of a uniform dilute Bose gas increases linearly with the sss-wave scattering length describing the repulsion between the particles. Because of infrared divergences, the magnitude of the shift cannot be obtained from perturbation theory, even in the weak coupling regime; rather, it is proportional to the size of the critical region in momentum space. By means of a self-consistent calculation of the quasiparticle spectrum at low momenta at the transition, we find an estimate of the effect in reasonable agreement with numerical simulations.
The critical temperature T c of a weakly interacting Bose gas in an isotropic power-low potential is investigated in the mean-field approximation by taking into account the fact that the particle density distribution function appearing in the mean-field depends on the chemical potential. We derive the general formula of the shift of T c from that of the ideal gas to the lowest order of an interaction. In three-dimensional space, we show that the shift of T c changes its sign from a negative value for n < 3 to a positive one for n > 3, where n is the exponent of the power-low potential.
The thermodynamical properties of interacting Bose atoms in a harmonic potential are studied within the mean-field approximation. For weak interactions, the quantum statistics is equivalent to an ideal gas in an effective mean-field potential. The eigenvalue of the Gross–Pitaevskii equation is identified as the chemical potential of the ideal gas. The condensation temperature and density profile of atoms are calculated. It is found that the critical temperature Tc decreases as the interactions increase. Below the critical point, the condensation fraction exhibits a universal relation of N0/N = 1-(T/Tc)γ, with the index γ ≈ 2.3 independent of the interaction strength, the chemical potential, as well as the frequency of the confining potential.
By using a mean-field approach, based on the Popov approximation, we calculate the temperature dependence of the condensate fraction of an interacting Bose gas of N atoms confined in an anisotropic harmonic trap. For systems interacting with repulsive forces we find a significant decrease of the condensate fraction and of the critical temperature with respect to the predictions of the noninteracting model with the same value of N. An analytic result for the shift of the critical temperature holding to first order in the scattering length is also derived. \textcopyright{} 1996 The American Physical Society.
We investigate the critical temperature of an interacting Bose gas confined in a trap described by a generic isotropic power-law potential. We compare the results with respect to the non-interacting case. In particular, we derive an analytical formula for the shift of the critical temperature holding to first order in the scattering length. We show that this shift scales as Nn/3(n+2), where N is the number of Bosons and n is the exponent of the power-law potential. Moreover, the sign of the shift critically depends on the power-law exponent n. Finally, we find that the shift of the critical temperature due to finite-size effects vanishes as N-2n/3(n+2).