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

We present a detailed investigation of the momentum-dependent self-energy Σ(k) at zero frequency of weakly interacting bosons at the critical temperature T_c 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⪡k_c and the short-wavelength regime k⪢k_c, where k_c 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∕k_c) encountered in perturbation theory. Our approach yields the crossover scale k_c 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 T_c in three dimensions, ΔT_c∕T_c=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.