Asymptotically Improved Convergence of Optimized Perturbation Theory in the Bose-Einstein Condensation Problem

TL;DR

This paper demonstrates that optimized perturbation theory can reliably and efficiently compute the critical temperature of Bose-Einstein condensation, matching known results in large-N limits and improving estimates for finite N through resummation techniques.

Contribution

It provides a detailed analysis of the convergence of optimized perturbation theory in Bose-Einstein condensation, including explicit calculations and resummation methods for both large-N and finite N cases.

Findings

LDE reproduces large-N results at first order

Resummation improves finite N critical temperature estimates

Method aligns with Monte Carlo simulations

Abstract

We investigate the convergence properties of optimized perturbation theory, or linear $\delta$ 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 weakly interacting homogeneous dilute Bose gas. We carry out the 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 O(N) symmetry. Then, using an efficient resummation method, we show how the LDE can exactly reproduce the known large-N result for $T_c$ already at the first non-trivial 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.