Non-universal Critical Quantities from Variational Perturbation Theory and Their Application to the BEC Temperature Shift

TL;DR

This paper employs high-order variational perturbation theory to compute critical quantities in scalar field theories with applications to Bose gas condensation temperature shifts, achieving results consistent with Monte Carlo simulations.

Contribution

It extends variational perturbation theory to seven loops for multiple N values and applies the results to Bose gas temperature shifts, providing new numerical data and large-N limits.

Findings

Computed critical quantities for N=0,1,2,3,4 using seven-loop variational perturbation theory.

Extended earlier calculations of the interaction-induced shift to additional N values.

Results agree with Monte Carlo simulations for N=1,2,4.

Abstract

For an O(N) symmetric scalar field theory with Euclidean action integral d^3x [1/2 |nabla phi|^2 + 1/2 r phi^2 + 1/4! u phi^4], where phi = (phi_1,...,phi_N) is a vector of N real field components, variational perturbation theory through seven loops is employed for N = 0,1,2,3,4 to compute the renormalized value of r/(N+2)u^2 at the phase transition. Its exact large-N limit is determined as well. We also extend an earlier computation of the interaction-induced shift Delta<phi^2>/Nu for N = 1,2,4 to N = 0,3. For N = 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 = 1,2,4. The appendix contains previously unpublished numerical seven-loop data provided to us by B.Nickel.