Nonuniversal critical quantities from variational perturbation theory and their application to the Bose-Einstein condensation temperature shift

Abstract

For an O͑N͒-symmetric scalar field theory with Euclidean action ͐d 3 x ͓ 1 2 ͉ ٌ ͉ 2 + 1 2 r 2 + 1 4! u 4 ͔ , where = ͑ 1 , . . . , 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 +2͒u 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 ⌬͗ 2 ͘ / Nu from 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 The Appendix contains previously unpublished numerical seven-loop data.

Extrapolation of perturbation-theory expansions by self-similar approximants

“…The corrected iterated root B 2/2 = 1.309 produces the most accurate result, practically coinciding with that found by the Monte Carlo simulations [1,2,30,56,59]. Kastening [32][33][34], using the Kleinert variational perturbation theory involving seven loops, found the value 1.27 ± 0.11, which is close to our results.…”

Representative statistical ensembles for Bose systems with broken gauge symmetry

“…The most accurate of these calculations are those employing Monte Carlo simulations [67][68][69][70][71] and those based on the optimized perturbation theory [72][73][74], as has been done in Refs. [75][76][77][78][79][80]. These investigations show that the first correction to the critical temperature comes from effects beyond the mean-field approximation.…”

Critical temperature in weakly interacting multicomponent field theory

“…One is able to calculate the shift coefficient c 1 involving the loop expansion [15], which yields [16][17][18] an expansion in powers of the variable…”

“…It is worth mentioning that the extrapolation of asymptotic series for a small variable to arbitrary values of the latter can be accomplished in the frame of optimized perturbation theory [19,20]. Different variants of this theory have been employed for finding the shift coefficient c 1 by introducing control functions with a variable change [17,18] or incorporating them into initial approximations [21][22][23][24]. However, the results of such calculations strongly depend on how the control functions are introduced, and also these calculations are rather cumbersome requiring the use of optimization conditions defining control functions at each approximation order.…”

“…We accomplish this procedure for the critical temperature shift (12) using the fifth-order series (15). The coefficients a n , calculated using the data of the seven-loop expansion by Kastening [17,18], for different numbers of the components N, are presented in Table 1.…”