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arXiv:cond-mat/0004142

Comment on "The Transition Temperature of the Dilute Interacting Bose Gas" and on "Transition Temperature of a Uniform Bose Gas"

Adriaan M. J. Schakel

Published Apr 10, 2000• 0 citations
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Abstract

It is pointed out that first, contrary to the claim by Baym et al.[Phys. Rev. Lett 83, 1703 (1999)], the shift in the transition temperature of a weakly interacting Bose gas can be calculated perturbatively. And second, that the regime considered by Huang [Phys. Rev. Lett 83, 3770 (1999)] while addressing the same issue does not describe the Bose-Einstein condensed phase.

DOI
10.48550/arxiv.cond-mat/0004142

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It is pointed out that first, contrary to the claim by Baym et al.[Phys. Rev. Lett 83, 1703 (1999)], the shift in the transition temperature of a weakly interacting Bose gas can be calculated perturbatively. And second, that the regime considered by Huang [Phys. Rev. Lett 83, 3770 (1999)] while addressing the same issue does not describe the Bose-Einstein condensed phase.
Within a quasiparticle framework, we reconsider the issue of computing the Bose-Einstein condensation temperature (TcT_cTc​) in a weakly non-ideal Bose gas. The main result of this and previous investigations is that TcT_cTc​ increases with the scattering length aaa, with the leading dependence being either linear or log-linear in aaa. The calculation of TcT_cTc​ reduces to that of computing the excitation spectrum near the transition. We report two approaches to regularizing the infrared divergence at the transition point. One leads to a a∣ln⁡a∣a\sqrt{|\ln{a}|}a∣lna∣​-like shift in TcT_cTc​, and the other allows numerical calculations for the shift.
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.
Using a finite-temperature path integral Monte Carlo simulation (PIMC) method and finite-size scaling, we have investigated the interaction-induced shift of the phase-transition temperature for Bose-Einstein condensation of homogeneous weakly interacting Bose gases in three dimensions, which is given by a proposed analytical expression ${T}_{c}={T}_{c}^{0}{1+{c}_{1}a{n}^{1∕3}+[{c}_{2}^{\ensuremath{'}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}(a{n}^{1∕3})+{c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2∕3}+\mathrm{O}({a}^{3}n)}$, where Tc0{T}_{c}^{0}Tc0​ is the critical temperature for an ideal gas, aaa is the sss-wave scattering length, and nnn is the number density. We have used smaller number densities and more time slices than in the previous PIMC simulations [Gruter et al., Phys. Rev. Lett. 79, 3549 (1997)] in order to understand the difference in the value of the coefficient c1{c}_{1}c1​ between their results and the (apparently) other reliable results in the literature. Our results show that ${({T}_{c}\ensuremath{-}{T}_{c}^{0})∕{T}_{c}^{0}}∕(a{n}^{1∕3})$ depends strongly on the interaction strength an1∕3a{n}^{1∕3}an1∕3 while the previous PIMC results are considerably flatter and smaller than our results. We obtain ${c}_{1}=1.32\ifmmode\pm\else\textpm\fi{}0.14$, in agreement with results from recent Monte Carlo methods of three-dimensional O(2) scalar ${\ensuremath{\phi}}^{4}$ field theory and variational perturbation theory.
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.
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.