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Physical Review A

Parametric equation of state of a weakly interacting Bose system

Mingzhe Li, Haixiang Fu, Zhipeng Zhang, Jincan Chen

Published Apr 23, 2007• 6 citations
Publisher

Abstract

A uniform dilute Bose system with repulsive interactions is studied in the grand canonical ensemble formalism. We present a parametric equation of state that holds true from high temperatures down to below the transition temperature, thus providing a scheme for exploring the quantum-statistical nature of the Bose-Einstein condensation transition in interacting gases. As an application, the interaction-induced shift of the transition temperature is derived to be $\mathrm{\ensuremath{\Delta}}{T}_{c}∕{T}_{c}^{0}=2.83{n}^{1∕3}a$, where $n$ is the density and $a$ is the $S$-wave scattering length.

DOI
10.1103/physreva.75.045602

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A uniform dilute Bose system with repulsive interactions is studied in the grand canonical ensemble formalism. We present a parametric equation of state that holds true from high temperatures down to below the transition temperature, thus providing a scheme for exploring the quantum-statistical nature of the Bose-Einstein condensation transition in interacting gases. As an application, the interaction-induced shift of the transition temperature is derived to be \ensuremathΔTc∕Tc0=2.83n1∕3a\mathrm{\ensuremath{\Delta}}{T}_{c}∕{T}_{c}^{0}=2.83{n}^{1∕3}a\ensuremathΔTc​∕Tc0​=2.83n1∕3a, where nnn is the density and aaa is the SSS-wave scattering length.
We compute the critical temperature T_c of a weakly interacting uniform Bose gas in the canonical ensemble, extending the criterion of condensation provided by the counting statistics for the uniform ideal gas. Using ordinary perturbation theory, we find in first order (Tc−Tc0)/Tc0=−0.93aρ1/3(T_c-T_c^0)/T_c^0 = -0.93 aρ^{1/3}(Tc​−Tc0​)/Tc0​=−0.93aρ1/3, where T_c^0 is the transition temperature of the corresponding ideal Bose gas, a is the scattering length, and ρρρ is the particle number density.
The Bose-Einstein condensation (BEC) of weakly interacting Bose gases has been realized recently in experiment. The condensation temperature of a uniform interacting Bose gas is an important thermodynamic quantity fot the BEC. In this paper, the pseudopotential method and grand canonical ensemble are used to investigate the condensation temperature of the uniform interacting Bose gas. The results show that, due to the interaction between particles, the shift of the condensation temperature is δT c/T c=-2.9an 1/3 , where a and n are the two-particle scattering length of S- wave and the particle number density, respectively.
We study the Bose-Einstein condensation phase transition in a weakly interacting gas through a perturbative analysis of finite systems. In both the grand canonical and the canonical ensembles, perturbation theory suffers from infrared divergences and cannot directly determine the transition temperature in the thermodynamic limit. However, in conjunction with finite-size scaling, perturbation theory provides a powerful calculation tool. Here we use this combination of perturbation theory and scaling to estimate a shift in the transition temperature in the canonical ensemble consistent with grand canonical calculations.
We calculate the transition temperature of a uniform dilute Bose gas with repulsive interactions, using a known virial expansion of the equation of state. We find that the transition temperature is higher than that of an ideal gas, with a fractional increase K0(na3)1/6{K}_{0}({\mathrm{na}}^{3}{)}^{1/6}K0​(na3)1/6, where nnn is the density, aaa is the SSS-wave scattering length, and K0{K}_{0}K0​ is a constant given in the paper. This disagrees with all existing results, analytical or numerical.
We calculate explicitly the variation δTcδT_cδTc​ of the Bose-Einstein condensation temperature TcT_cTc​ induced by weak repulsive two-body interactions to leading order in the interaction strength. As shown earlier by general arguments, δTc/TcδT_c/T_cδTc​/Tc​ is linear in the dimensionless product an1/3an^{1/3}an1/3 to leading order, where nnn is the density and aaa the scattering length. This result is non-perturbative, and a direct perturbative calculation of the amplitude is impossible due to infrared divergences familiar from the study of the superfluid helium lambda transition. Therefore we introduce here another standard expansion scheme, generalizing the initial model which depends on one complex field to one depending on NNN real fields, and calculating the temperature shift at leading order for large NNN. The result is explicit and finite. The reliability of the result depends on the relevance of the large NNN expansion to the situation N=2, which can in principle be checked by systematic higher order calculations. The large NNN result agrees remarkably well with recent numerical simulations.
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.