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arXiv:cond-mat/0112306 · Physical Review A

Higher-order evaluation of the critical temperature for interacting homogeneous dilute Bose gases

F. Cruz, Marcus Benghi Pinto, Rudnei O. Ramos, Paulo José Sena dos Santos

Published May 9, 2002• 52 citations
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Abstract

We use the nonperturbative linear $\ensuremath{\delta}$ expansion method to evaluate analytically the coefficients ${c}_{1}$ 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 ${T}_{0}$ 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 ${c}_{1}$ to order ${\ensuremath{\delta}}^{2}$ with the result ${c}_{1}=3.06.$ Here, we push the calculation to the next two orders obtaining ${c}_{1}=2.45$ at order ${\ensuremath{\delta}}^{3}$ and ${c}_{1}=1.48$ 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 $1/N$ 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 ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$

DOI
10.1103/physreva.65.053613

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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 ${T}_{\mathrm{c}}\ensuremath{\simeq}{T}_{0}{1+(1.32\ifmmode\pm\else\textpm\fi{}0.02){\mathrm{an}}^{1/3}+[19.7518\mathrm{ln}{(an}^{1/3})+(75.7\ifmmode\pm\else\textpm\fi{}0.4)]{a}^{2}{n}^{2/3}{+O(a}^{3}n)}.$
The leading-order effect of interactions on a homogeneous Bose gas is theoretically predicted to shift the critical temperature by an amount $\ensuremath{\Delta}{T}_{c}\ensuremath{\simeq}{\mathrm{ca}}_{\mathrm{sc}}{n}^{1/3}{T}_{0}$ from the ideal gas result T0{T}_{0}T0​, where asc{a}_{\mathrm{sc}}asc​ is the scattering length, nnn is the density, and ccc is a pure number. There have been several different theoretical estimates for ccc. We claim to settle the issue by measuring the numerical coefficient in a lattice simulation of O(2) ${\ensuremath{\varphi}}^{4}$ field theory in three dimensions---an effective theory which, as observed previously in the literature, can be systematically matched to the dilute Bose gas problem to reproduce nonuniversal quantities such as the critical temperature. We find $c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.32\ifmmode\pm\else\textpm\fi{}0.02$.
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 $\ensuremath{\gamma}=3.059.$
In the framework of the O(N)\mathrm{O}(N)O(N) three-dimensional effective scalar field model for homogeneous dilute weakly interacting Bose gases we use the 1∕N1∕N1∕N expansion, within the large-NNN limit, to evaluate the parameter rc{r}_{c}rc​ which is directly related to the critical chemical potential ${\ensuremath{\mu}}_{c}$. This quantity enters the order-a2n2∕3{a}^{2}{n}^{2∕3}a2n2∕3 coefficient contributing to the critical temperature shift $\ensuremath{\Delta}{T}_{c}$ where aaa represents the sss-wave scattering length and nnn represents the density. Compared to the recent precise numerical lattice simulation results, our calculation suggests that the large-NNN approximation performs rather well even for the physical case N=2N=2N=2. We then calculate the same quantity but using different forms of the optimized perturbative (variational) method, showing that these produce excellent results for both the finite-NNN and large-NNN cases.
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.
This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at T=0.T=0.T=0. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3,\sqrt{{\mathrm{na}}^{3}},na3​, where aaa is the sss-wave scattering length and nnn is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include 1/N1/N1/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.