On the Convergence of the Linear Delta Expansion for the Shift in T_c for Bose-Einstein Condensation

TL;DR

This paper demonstrates that the linear delta expansion method converges to the exact shift in the Bose-Einstein condensation temperature in a critical scalar field theory, with optimized convergence showing exponential error reduction.

Contribution

It applies and validates the linear delta expansion for calculating the T_c shift in Bose-Einstein condensation, including for N=2, with convergence analysis and comparison to lattice results.

Findings

Linear delta expansion converges to the exact result in the large-N limit.

Errors decrease exponentially when using the principle of minimal sensitivity.

Results for N=2 show slow convergence consistent with lattice Monte Carlo calculations.

Abstract

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 O(N) scalar field theory in 3 dimensions with N=2. We show that the linear 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 principal of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with the order in the delta expansion. For N=2, we calculate the shift in T_c to fourth order in delta. The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations.