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.
We calculate the quantum phase transition for a homogeneous Bose gas in the plane of s-wave scattering length as and temperature T. This is done by improving a one-loop result near the interaction-free Bose-Einstein critical temperature Tc(0) with the help of recent high-loop results on the shift of the critical temperature due to a weak atomic repulsion based on variational perturbation theory. The quantum phase diagram shows a nose above Tc(0), so that we predict the existence of a reentrant transition above Tc(0), where an increasing repulsion leads to the formation of a condensate.
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.