The Transition Temperature of the Dilute Interacting Bose Gas

Abstract

We show that the critical temperature of a uniform dilute Bose gas increases linearly with the s-wave scattering length describing the repulsion between the particles. Because of infrared divergences, the magnitude of the shift cannot be obtained from perturbation theory, even in the weak coupling regime; rather, it is proportional to the size of the critical region in momentum space. By means of a self-consistent calculation of the quasiparticle spectrum at low momenta at the transition, we find an estimate of the effect in reasonable agreement with numerical simulations.Determination of the effect of repulsive interactions on the transition temperature of a homogeneous dilute Bose gas at fixed density has had a long and controversial history [1][2][3][4][5]. While [1] predicted that the first change in the transition temperature, T c , is of order the scattering length a for the interaction between the particles, neither the sign of the effect nor its dependence on a has been obvious. Recent renormalization group studies [4] predict an increase of the critical temperature. Numerical calculations by Grüter, Ceperley, and Laloë [6], and more recently by Holzmann and Krauth [7], of the effect of interactions on the Bose-Einstein condensation transition in a uniform gas of hard sphere bosons, and approximate analytic calculations by Holzmann, Grüter, and Laloë of the dilute limit [8], have shown that the transition temperature, T c , initially rises linearly with a. The effect arises physically from the change in the energy of low momentum particles near T c [8]. Here we analyze the leading order behavior of diagrammatic perturbation theory, and argue that T c increases linearly with a. We then construct an approximate self-consistent solution of the single particle spectrum at T c which demonstrates the change in the low momentum spectrum, and which enables us to calculate the change in T c quantitatively.We consider a uniform system of identical bosons of mass m, at temperature T close to T c and use finite temperature quantum many-body perturbation theory. We assume that the range of the two-body potential is small compared to the interparticle distance n −1/3 , so that the potential can be taken to act locally and be characterized entirely by the s-wave scattering length a. Thus we work in the limit a ≪ λ, where λ = 2πh 2 /mk B T 1/2 is the thermal wavelength. (We generally use units h = k B = 1.)To compute the effects of the interactions on T c , we write the density n as a sum over Matsubara frequencies ω ν = 2πiνT (ν = 0, ±1, ±2, . . .) of the single particle Green's function, G(k,z):wherewith µ the chemical potential. The Bose-Einstein condensation transition is determined by the point where G −1 (0, 0) = 0, i.e., where Σ(0, 0) = µ. The first effect of interactions on Σ is a mean field term Σ mf = 2gn, where g = 4πh 2 a/m; the factor of two comes from including the exchange term. Such a contribution, independent of k and z has no effect on the transition temperature, as it can be simply absorbed in a redef...

Functional renormalization for the Bardeen–Cooper–Schrieffer to Bose–Einstein condensation crossover

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Bose-Einstein transition in a dilute interacting gas

“…Here we reconsider the simple analytical model calculation of Ref. [16] which provides an estimate for the scale k c , and acts as a reference for the numerical results presented later. In this analytical model we construct a self-consistent energy spectrum at the critical point:…”

“…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.…”

“…[21] provides c ∼ 1, and an estimate in Ref. [16] of an exact formula for ∆T c /T 0 c predicts c ∼ 2−3. In the limit of a large number N of components [22], c = 8π/3ζ(3/2) 4/3 ≃ 2.33; interestingly, this exact result for N → ∞ agrees with the numerical result c = 2.33 ± 0.25 of Ref.…”

Functional renormalization group in the broken symmetry phase: momentum dependence and two-parameter scaling of the self-energy

“…For models with weak interactions, a universal regime, covering the complete crossover from the vicinity of the Gaussian fixed point to the vicinity of the Wilson-Fisher-fixed point, exists which can be described completely within a two parameter scaling theory [21,22,4]. For the weakly interacting Bose gas at criticality the one-parameter scaling function σ * (k/k c ) = σ − (∞, k/k c ) has been calculated in [26,27,17,18]. However, the full implications of a finite scale k c at criticality and the resulting two-parameter scaling theory of the correlation function away from criticality has only recently been examined [20].…”