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arXiv:cond-mat/0005197 · Physical Review A

T_{c} for dilute Bose gases: Beyond leading order in 1/N

Peter B. Arnold, Boris Tomášik

Published Nov 1, 2000• 80 citations
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Abstract

Baym, Blaizot, and Zinn-Justin have recently used the large-$N$ approximation to calculate the effect of interactions on the transition temperature of dilute Bose gases. We extend their calculation to next-to-leading order in $1/N$ and find a relatively small correction of $\ensuremath{-}26%$ to the leading order result. This suggests that the large-$N$ approximation works surprisingly well in this application.

DOI
10.1103/physreva.62.063604

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Baym, Blaizot, and Zinn-Justin have recently used the large-NNN approximation to calculate the effect of interactions on the transition temperature of dilute Bose gases. We extend their calculation to next-to-leading order in 1/N1/N1/N and find a relatively small correction of \ensuremath−26\ensuremath{-}26%\ensuremath−26 to the leading order result. This suggests that the large-NNN approximation works surprisingly well in this application.
We consider the nonrelativistic interacting Bose gas at zero and finite temperature. Using the 1/N-expansion, we derive expressions for the free energy density and the number density to next-to-leading order in 1/N. Outside the critical region and at weak coupling, our calculations reduce to the well-known mean-field results for the dilute Bose gas. We also rederive the nonperturbative critical density for Bose condensation which was first calculated by Baym, Blaizot, Holzmann, Lalöe, and Vautherin.
The transition temperature for a dilute, homogeneous, three-dimensional Bose gas has the expansion ${T}_{\mathrm{c}}{=T}_{0}{{1+c}_{1}{\mathrm{an}}^{1/3}+[{c}_{2}^{\ensuremath{'}}\mathrm{ln}{(an}^{1/3}{)+c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2/3}{+O(a}^{3}n)},$ where a is the scattering length, n the number density, and T0{T}_{0}T0​ the ideal gas result. The first-order coefficient c1{c}_{1}c1​ depends on nonperturbative physics. In this paper, we show that the coefficient ${c}_{2}^{\ensuremath{'}}$ may be computed perturbatively. We also show that the remaining second-order coefficient ${c}_{2}^{\ensuremath{''}}$ depends on non-perturbative physics but may be related, by a perturbative calculation, to quantities that have previously been measured using lattice simulations of three-dimensional O(2)O(2)O(2) scalar field theory. Making use of those simulation results, we find ${T}_{\mathrm{c}}\ensuremath{\simeq}{T}_{0}{1+(1.32\ifmmode\pm\else\textpm\fi{}0.02){\mathrm{an}}^{1/3}+[19.7518\mathrm{ln}{(an}^{1/3})+(75.7\ifmmode\pm\else\textpm\fi{}0.4)]{a}^{2}{n}^{2/3}{+O(a}^{3}n)}.$
We use the nonperturbative linear $\ensuremath{\delta}$ expansion method to evaluate analytically the coefficients c1{c}_{1}c1​ and ${c}_{2}^{\ensuremath{''}}$ that appear in the expansion for the transition temperature for a dilute, homogeneous, three-dimensional Bose gas given by ${T}_{c}{=T}_{0}({1+c}_{1}{\mathrm{an}}^{1/3}+[{c}_{2}^{\ensuremath{'}}\mathrm{ln}{(an}^{1/3}{)+c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2/3}{+O(a}^{3}n)),$ where T0{T}_{0}T0​ is the result for an ideal gas, a is the s-wave scattering length, and n is the number density. In a previous work the same method has been used to evaluate c1{c}_{1}c1​ to order ${\ensuremath{\delta}}^{2}$ with the result c1=3.06.{c}_{1}=3.06.c1​=3.06. Here, we push the calculation to the next two orders obtaining c1=2.45{c}_{1}=2.45c1​=2.45 at order ${\ensuremath{\delta}}^{3}$ and c1=1.48{c}_{1}=1.48c1​=1.48 at order ${\ensuremath{\delta}}^{4}.$ Analyzing the topology of the graphs involved we discuss how our results relate to other nonperturbative analytical methods such as the self-consistent resummation and the 1/N1/N1/N approximations. At the same orders we obtain ${c}_{2}^{\ensuremath{''}}=101.4,$ ${c}_{2}^{\ensuremath{''}}=98.2,$ and ${c}_{2}^{\ensuremath{''}}=82.9.$ Our analytical results seem to support the recent Monte Carlo estimates ${c}_{1}=1.32\ifmmode\pm\else\textpm\fi{}0.02$ and ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$
For some time, the theoretical result for the transition temperature of a dilute three-dimensional Bose gas in an arbitrarily wide harmonic trap has been known to first order in the interaction strength. We extend that result to second order. The first-order result for a gas trapped in a harmonic potential can be computed in mean-field theory (in contrast to the first-order result for a uniform gas, which cannot). We show that, at second order, perturbation theory suffices for relating the transition temperature to the chemical potential at the transition, but the chemical potential is nonperturbative at the desired order. The necessary information about the chemical potential can be extracted, however, from recent lattice simulations of uniform Bose gases.
Beyond standard approaches in existing literature, we explore the relative shift in the transition temperature of a homogeneous dilute Bose gas using the Cornwall-Jackiw-Tomboulis effective action formalism within the improved Hartree-Fock approximation. The first correction to the enhancement of the transition temperature, as compared to that of the ideal Bose gas, is expressed in a universal form: it is linear with respect to the scattering length. The slope of this linear relationship shows excellent agreement with the exact numerical calculations presented in previous studies. Additionally, we identify non-universal terms contributing to the shift.