The transition temperature of the dilute interacting Bose gas for $N$ internal degrees of freedom

TL;DR

This paper calculates the leading-order change in the Bose-Einstein condensation temperature due to weak repulsive interactions, using a large N expansion to overcome infrared divergence issues, and finds good agreement with numerical simulations.

Contribution

It introduces a large N expansion method to explicitly compute the temperature shift in dilute interacting Bose gases, addressing non-perturbative infrared divergences.

Findings

The temperature shift is linear in the interaction parameter $an^{1/3}$.

The large N expansion yields a finite, explicit result.

Results agree well with recent numerical simulations.

Abstract

We calculate explicitly the variation $\delta T_c$ 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, $\delta T_c/T_c$ 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.