Critical behavior of weakly interacting bosons: A functional renormalization-group approach

Abstract

We present a detailed investigation of the momentum-dependent self-energy Σ(k) at zero frequency of weakly interacting bosons at the critical temperature Tc of Bose-Einstein condensation in dimensions 3 ≤ D < 4. Applying the functional renormalization group, we calculate the universal scaling function for the self-energy at zero frequency but at all wave vectors within an approximation which truncates the flow equations of the irreducible vertices at the four-point level. The self-energy interpolates between the critical regime k ≪ kc and the short-wavelength regime k ≫ kc, where kc is the crossover scale. In the critical regime, the self-energy correctly approaches the asymptotic behavior Σ(k) ∝ k 2−η , and in the short-wavelength regime the behavior is Σ(k) ∝ k 2(D−3) in D > 3. In D = 3, we recover the logarithmic divergence Σ(k) ∝ ln(k/kc) encountered in perturbation theory. Our approach yields the crossover scale kc as well as a reasonable estimate for the critical exponent η in D = 3. From our scaling function we find for the interaction-induced shift in Tc in three dimensions, ∆Tc/Tc = 1.23an 1/3 , where a is the s-wave scattering length and n is the density, in excellent agreement with other approaches. We also discuss the flow of marginal parameters in D = 3 and extend our truncation scheme of the renormalization group equations by including the six-and eight-point vertex, which yields an improved estimate for the anomalous dimension η ≈ 0.0513. We further calculate the constant lim k→0 Σ(k)/k 2−η and find good agreement with recent Monte-Carlo data.

Phase diagram for interacting Bose gases

“…The latter result can also be found by variational perturbation theory [19] yielding c 1 = 1.23 ± 0.12. The same coefficient has been obtained by exact renormalization group calculations [20,21] which provide the momentum dependence of the self energy at zero frequency as well. The usage of Ursell operators [22] has given an even smaller value c 1 = 0.7.…”

The nonperturbative functional renormalization group and its applications

“…2.4) gives c 1.37 [121,122,517] in reasonable agreement with lattice results (c = 1.32(2) [518] and c = 1.29(5) [519]) and seven-loop resummed calculations (c = 1.27 (10) [520]). Another FRG calculation, based on a vertex expansion, gives c = 1.23 [209,521]. In the case of a model with N-component real fields (Eq.…”

Interacting preformed Cooper pairs in resonant Fermi gases

“…To study interaction effects in an interacting Bose gas in a nonperturbative manner, the renormalization group (RG) approach is an established approach [42]. RG studies for interacting bosons on the verge of becoming su-perfluid have increased our understanding of the resulting phase transition [43][44][45][46][47]. In this article, we perform a renormalization-group study of interacting Cooper pairs in the unitarity limit.…”