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arXiv:cond-mat/0206186 · Physical Review A

Convergence of the linear δ expansion for the shift in Tc for Bose-Einstein condensation

Eric Braaten, Eugeniu Radescu

Published Dec 12, 2002• 29 citations
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Abstract

The leading correction from interactions to the transition temperature ${T}_{c}$ for Bose-Einstein condensation can be obtained from a nonperturbative calculation in the critical $\mathrm{O}(N)$ scalar field theory in three dimensions with $N=2.$ We show that the linear $\ensuremath{\delta}$ expansion can be applied to this problem in such a way that in the large-N limit it converges to the exact analytic result. If the principle of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with the order in the $\ensuremath{\delta}$ expansion. For $N=2,$ we calculate the shift in ${T}_{c}$ to fourth order in $\ensuremath{\delta}.$ The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations.

DOI
10.1103/physreva.66.063601

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The leading correction from interactions to the transition temperature Tc{T}_{c}Tc​ for Bose-Einstein condensation can be obtained from a nonperturbative calculation in the critical O(N)\mathrm{O}(N)O(N) scalar field theory in three dimensions with N=2.N=2.N=2. We show that the linear \ensuremathδ\ensuremath{\delta}\ensuremathδ expansion can be applied to this problem in such a way that in the large-N limit it converges to the exact analytic result. If the principle of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with the order in the \ensuremathδ\ensuremath{\delta}\ensuremathδ expansion. For N=2,N=2,N=2, we calculate the shift in Tc{T}_{c}Tc​ to fourth order in \ensuremathδ.\ensuremath{\delta}.\ensuremathδ. The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations.
The leading correction from interactions to the transition temperature Tc for Bose-Einstein condensation can be obtained from a nonperturbative calculation in the critical O( N) scalar field theory in 3 dimensions with N = 2. We show that the linear δ expansion can be applied to this problem in such a way that in the large- N limit it converges to the exact analytic result. If the principal of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with the order in the δ expansion. For N = 2, we calculate the shift in Tc to fifth order in δ. The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations. The convergence can be accelerated by using a resummation method proposed by Kleinert et al. called variational perturbation theory.
We investigate the convergence properties of optimized perturbation theory, or linear $\ensuremath{\delta}$ expansion (LDE), within the context of finite temperature phase transitions. Our results prove the reliability of these methods, recently employed in the determination of the critical temperature Tc{T}_{c}Tc​ for a system of a weakly interacting homogeneous dilute Bose gas. We carry out explicit LDE optimized calculations and also the infrared analysis of the relevant quantities involved in the determination of Tc{T}_{c}Tc​ in the large-NNN limit, when the relevant effective static action describing the system is extended to O(N)\mathrm{O}(N)O(N) symmetry. Then, using an efficient resummation method, we show how the LDE can already exactly reproduce the known large-NNN result for Tc{T}_{c}Tc​ at the first nontrivial order. Next, we consider the finite N=2N=2N=2 case where, using similar resummation techniques, we improve the analytical results for the nonperturbative terms involved in the expression for the critical temperature, allowing comparison with recent Monte Carlo estimates of them. To illustrate the method, we have considered a simple geometric series showing how the procedure as a whole works consistently in a general case.
The transition temperature of the dilute interacting Bose gas for NNN internal degrees of freedom
We calculate explicitly the variation δTc of the Bose-Einstein condensation temperature Tc induced by weak repulsive two-body interactions to leading order in the interaction strength. As shown earlier by general arguments, δTc/Tc is linear in the dimensionless product an1/3 to leading order, where n is the density and a 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 N real fields, and calculating the temperature shift at leading order for large N. The result is explicit and finite. The reliability of the result depends on the relevance of the large N expansion to the situation N = 2, which can in principle be checked by systematic higher order calculations. The large N result agrees remarkably well with recent numerical simulations.
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 ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$
The linear delta expansion (LDE) is applied to the critical O(N) phi4 three-dimensional field theory which has been widely used to study the temperature of condensation of dilute weakly interacting homogeneous Bose gases. We study the higher order convergence of the LDE as it is usually applied to this problem. We show how to improve both the large N and finite N=2 LDE results with an efficient resummation technique which accelerates convergence. In the large N limit, it reproduces the known exact result within numerical integration accuracy. In the finite N=2 case, our improved results support the recent numerical Monte Carlo estimates for the transition temperature.
We apply the nonperturbative optimized linear $\ensuremath{\delta}$ expansion method to the O(N)O(N)O(N) scalar field model in three dimensions to determine the transition temperature of a dilute homogeneous Bose gas. Our results show that the shift of the transition temperature $\ensuremath{\Delta}{T}_{c}{/T}_{c}$ of the interacting model, compared with the ideal-gas transition temperature, really behaves as $\ensuremath{\gamma}{\mathrm{an}}^{1/3}$ where a is the s-wave scattering length and n is the number density. For N=2N=2N=2 our calculations yield the value $\ensuremath{\gamma}=3.059.$