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arXiv:cond-mat/0401324 · Physical Review A

Improved optimization of perturbation theory: Applications to the oscillator energy levels and Bose-Einstein condensate critical temperature

Jean-Loı̈c Kneur, A. Neveu, Marcus Benghi Pinto

Published May 24, 2004• 52 citations
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Abstract

Improving perturbation theory via a variational optimization has generally produced in higher orders an embarrassingly large set of solutions, most of them unphysical (complex). We introduce an extension of the optimized perturbation method which leads to a drastic reduction of the number of acceptable solutions. The properties of this method are studied and it is then applied to the calculation of relevant quantities in different ${\ensuremath{\phi}}^{4}$ models, such as the anharmonic oscillator energy levels and the critical Bose-Einstein condensation temperature shift $\ensuremath{\Delta}{T}_{c}$ recently investigated by various authors. Our present estimates of $\ensuremath{\Delta}{T}_{c}$, incorporating the most recently available six and seven loop perturbative information, are in excellent agreement with all the available lattice numerical simulations. This represents a very substantial improvement over previous treatments.

DOI
10.1103/physreva.69.053624

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Improving perturbation theory via a variational optimization has generally produced in higher orders an embarrassingly large set of solutions, most of them unphysical (complex). We introduce an extension of the optimized perturbation method which leads to a drastic reduction of the number of acceptable solutions. The properties of this method are studied and it is then applied to the calculation of relevant quantities in different \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 models, such as the anharmonic oscillator energy levels and the critical Bose-Einstein condensation temperature shift \ensuremathΔTc\ensuremath{\Delta}{T}_{c}\ensuremathΔTc​ recently investigated by various authors. Our present estimates of \ensuremathΔTc\ensuremath{\Delta}{T}_{c}\ensuremathΔTc​, incorporating the most recently available six and seven loop perturbative information, are in excellent agreement with all the available lattice numerical simulations. This represents a very substantial improvement over previous treatments.
For an O(N)\mathrm{O}(N)O(N)-symmetric scalar field theory with Euclidean action $\ensuremath{\int}{d}^{3}\phantom{\rule{0.3em}{0ex}}x[\frac{1}{2}\ensuremath{\mid}\ensuremath{\nabla}\ensuremath{\phi}{\ensuremath{\mid}}^{2}+\frac{1}{2}r{\ensuremath{\phi}}^{2}+\frac{1}{4!}u{\ensuremath{\phi}}^{4}]$, where $\ensuremath{\phi}=({\ensuremath{\phi}}_{1},\dots{},{\ensuremath{\phi}}_{N})$ is a vector of NNN real-field components, variational perturbation theory through seven loops is employed for N=0,1,2,3,4N=0,1,2,3,4N=0,1,2,3,4 to compute the renormalized value of r∕(N+2)u2r∕(N+2){u}^{2}r∕(N+2)u2 at the phase transition. Its exact large-NNN limit is determined as well. We also extend an earlier computation of the interaction-induced shift $\ensuremath{\Delta}⟨{\ensuremath{\phi}}^{2}⟩∕Nu$ from N=1,2,4N=1,2,4N=1,2,4 to N=0,3N=0,3N=0,3. For N=2N=2N=2, the results for the two quantities are used to compute the second-order shift of the condensation temperature of a dilute Bose gas, both in the homogenous case and for the wide limit of a harmonic trap. Our results are in agreement with earlier Monte Carlo simulations for N=1,2,4N=1,2,4N=1,2,4. The Appendix contains previously unpublished numerical seven-loop data.
The shift of the Bose-Einstein condensation temperature for a homogenous weakly interacting Bose gas in leading order in the scattering length aaa is computed for given particle density nnn. Variational perturbation theory is used to resum the corresponding perturbative series for $\ensuremath{\Delta}⟨{\ensuremath{\phi}}^{2}⟩∕Nu$ in a classical three-dimensional scalar field theory with coupling uuu and where the physical case of N=2N=2N=2 field components is generalized to arbitrary NNN. Our results for N=1,2,4N=1,2,4N=1,2,4 are in agreement with recent Monte Carlo simulations; for N=2N=2N=2, we obtain $\ensuremath{\Delta}{T}_{c}∕{T}_{0}=(1.27\ifmmode\pm\else\textpm\fi{}0.11)a{n}^{1∕3}$. We use seven-loop perturbative coefficients, extending earlier work by one loop order.
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.
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.