Bose-Einstein transition temperature in a dilute repulsive gas

TL;DR

This paper analyzes how repulsive interactions affect the critical temperature of a dilute Bose gas, highlighting differences between trapped and homogeneous systems and emphasizing the need for non-perturbative calculations.

Contribution

It provides a detailed theoretical analysis of the interaction-induced shifts in Bose-Einstein transition temperature, including non-perturbative methods for homogeneous gases.

Findings

Repulsive interactions increase the critical temperature proportionally to the scattering length.

Mean field theory captures temperature shifts in trapped gases via density profile changes.

Non-perturbative calculations are necessary for accurate predictions in homogeneous gases.

Abstract

We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory - in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k. These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.