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 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 in the large-
N limit, when the relevant effective static action describing the system is extended to
O(N) symmetry. Then, using an efficient resummation method, we show how the LDE can already exactly reproduce the known large-
N result for
Tc 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.
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 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
Tc to fourth order in $\ensuremath{\delta}.$ 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 use the nonperturbative linear $\ensuremath{\delta}$ expansion method to evaluate analytically the coefficients
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 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 to order ${\ensuremath{\delta}}^{2}$ with the result
c1=3.06. Here, we push the calculation to the next two orders obtaining
c1=2.45 at order ${\ensuremath{\delta}}^{3}$ and
c1=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 ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$
In the framework of the
O(N) three-dimensional effective scalar field model for homogeneous dilute weakly interacting Bose gases we use the
1∕N expansion, within the large-
N limit, to evaluate the parameter
rc which is directly related to the critical chemical potential ${\ensuremath{\mu}}_{c}$. This quantity enters the order-
a2n2∕3 coefficient contributing to the critical temperature shift $\ensuremath{\Delta}{T}_{c}$ where
a represents the
s-wave scattering length and
n represents the density. Compared to the recent precise numerical lattice simulation results, our calculation suggests that the large-
N approximation performs rather well even for the physical case
N=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-
N and large-
N cases.
The phase transition temperature for the Bose-Einstein condensation (BEC) of weakly interacting Bose gases in three dimensions is known to be related to certain nonuniversal properties of the phase transition of three-dimensional O(2) symmetric phi(4) theory. These properties have been measured previously in Monte Carlo lattice simulations. They have also been approximated analytically, with moderate success, by large N approximations to O(N) symmetric phi(4) theory. To begin investigating the region of validity of the large N approximation in this application, the same Monte Carlo technique developed for the O(2) model [P. Arnold and G. Moore, Phys. Rev. E 64, 066113 (2001)] to O(1) and O(4) theories has been applied. The results indicate that there might exist some theoretically unanticipated systematic errors in the extrapolation of the continuum value from lattice Monte Carlo results. The final results show that the difference between simulations and next-to-leading order large N calculations does not improve significantly from N=2 to N=4. This suggests that one would need to simulate yet larger N's to see true large N scaling of the difference. Quite unexpectedly (and presumably accidentally), the Monte Carlo result for N=1 seems to give the best agreement with the large N approximation among the three cases.