The transition temperature of the dilute interacting Bose gas for N internal states

Abstract

We calculate explicitly the variation δT c of the Bose-Einstein condensation temperature T c induced by weak repulsive two-body interactions to leading order in the interaction strength. As shown earlier by general arguments, δT c /T c is linear in the dimensionless product an 1/3 to leading order, where n is the density and a the scattering length. This result is non-perturbative, and a direct perturbative calculation of the amplitude is impossible due to infrared divergences familiar from the study of the superfluid helium lambda transition. Therefore we introduce here another standard expansion scheme, generalizing the initial model which depends on one complex field to one depending on N real fields, and calculating the temperature shift at leading order for large N . The result is explicit and finite. The reliability of the result depends on the relevance of the large N expansion to the situation N = 2, which can in principle be checked by systematic higher order calculations.The large N result agrees remarkably well with recent numerical simulations.

Bose-Einstein transition in a dilute interacting gas

“…While Refs. [21,16,22] all agree on the functional form, they do not provide definitive quantitative predictions for the prefactor; Ref. [21] provides c ∼ 1, and an estimate in Ref.…”

“…The approach includes, not only the mean field contributions, but also sums of ladder graphs within the many-body T-matrix-approximation. He derives an analytical formula for the modification of the energy spectrum, from which Stoof obtains a relative increase of the critical temperature, 4.66 an 1/3 , exactly twice the value of the large-N calculation [22]. Summing ladders, and neclecting U (k) in the denominator of Eq.…”

“…Similarly, in calculations involving resummations of bubbles or ladder diagrams the cutoff is provided by an effective screening explicitly generated by the infinite resummations. Large N techniques lead to a similar screening, with the advantage of also providing an expansion parameter [22,44].…”

“…4b. Again, the quality of such an approximation can only be gauged by a comparison with an exact calculation, except in large N limit where the bubble summation becomes exact itself [22,44].…”

“…At very low densities the shift of the critical temperature was found to be ∆T c T 0 field theoretical problem in three spatial dimensions. An alternative proof of the linearity in a based on renormalization group arguments is presented in [22]. While Refs.…”