Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0207295 · Physical Review A

Asymptotically improved convergence of optimized perturbation theory in the Bose-Einstein condensation problem

Jean-Loı̈c Kneur, Marcus Benghi Pinto, Rudnei O. Ramos

Published Oct 13, 2003• 41 citations
arXivHTMLPDF

Abstract

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 ${T}_{c}$ 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 ${T}_{c}$ in the large-$N$ limit, when the relevant effective static action describing the system is extended to $\mathrm{O}(N)$ symmetry. Then, using an efficient resummation method, we show how the LDE can already exactly reproduce the known large-$N$ result for ${T}_{c}$ at the first nontrivial order. Next, we consider the finite $N=2$ 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.

DOI
10.1103/physreva.68.043615

Related papers

Convergent Resummed Linear δ Expansion in the Critical O(N) (ϕi2)3d2 Model

Jean-Loı̈c Kneur, Marcus Benghi Pinto, Rudnei O. Ramos

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.

arXiv:cond-mat/0207089•Physical Review Letters•Nov 1, 2002•56 citations

Semi-analytic calculation of the shift in the critical temperature for bose-einstein condensation

Eugeniu Radescu

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.

•Jan 1, 2004•0 citations

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

Eric Braaten, Eugeniu Radescu

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.

arXiv:cond-mat/0206186•Physical Review A•Dec 12, 2002•29 citations

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

F. Cruz, Marcus Benghi Pinto, Rudnei O. Ramos +1 more

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 ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$

arXiv:cond-mat/0112306•Physical Review A•May 9, 2002•52 citations

Finite-size scaling and the role of the thermodynamic ensemble in the transition temperature of a dilute Bose gas

Erich J. Mueller, Gordon Baym, Markus Holzmann

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.

arXiv:cond-mat/0105359•Journal of Physics B Atomic Molecular and Optical Physics•Nov 23, 2001•14 citations

The transition temperature of the dilute interacting Bose gas for N internal states

Gordon Baym, Jean-Paul Blaizot, Jean Zinn‐Justin

We calculate explicitly the variation $δT_c$ of the Bose-Einstein condensation temperature $T_c$ induced by weak repulsive two-body interactions to leading order in the interaction strength. As shown earlier by general arguments, $δT_c/T_c$ is linear in the dimensionless product $an^{1/3}$ 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.

arXiv:cond-mat/9907241•Europhysics Letters (EPL)•Jan 15, 2000•107 citations
We investigate the convergence properties of optimized perturbation theory, or linear \ensuremathδ\ensuremath{\delta}\ensuremathδ 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 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.
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.
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 $\ensuremath{\delta}$ 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,N=2,N=2, we calculate the shift in Tc{T}_{c}Tc​ to fourth order in $\ensuremath{\delta}.$ The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations.
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 ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$
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 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.