Using the exact renormalization group we calculate the momentum-dependent self-energy Sigma (k) at zero frequency of weakly interacting bosons at the critical temperature T_c of Bose-Einstein condensation in dimensions 3 <= D < 4. We obtain the complete crossover function interpolating between the critical regime k << k_c, where Sigma (k) propto k^{2 - eta}, and the short-wavelength regime k >> k_c, where Sigma (k) propto k^{2 (D-3)} in D> 3 and Sigma (k) \propto ln (k/k_c) in D=3. Our approach yields the crossover scale k_c on the same footing with a reasonable estimate for the critical exponent eta in D=3. From our Sigma (k) we find for the interaction-induced shift of T_c in three dimensions Delta T_c / T_c approx 1.23 a n^{1/3}, where a is the s-wave scattering length and n is the density.Comment: 4 pages,1 figur

Using functional renormalization group methods, we present a self-consistent calculation of the true Fermi momenta k 2 ≫ 1 even weak interachain backscattering leads to a strong reduction of the distance between the Fermi momenta.

The true Fermi surface of a fermionic many-body system can be viewed as a fixed point manifold of the renormalization group (RG). Within the framework of the exact functional RG we show that the fixed point condition implies an exact integral equation for the counterterm which is needed for a self-consistent calculation of the Fermi surface. In the simplest approximation, our integral equation reduces to the self-consistent Hartree-Fock equation for the counterterm.

Using a nonperturbative functional renormalization group approach involving both fermionic and bosonic fields we calculate the interaction-induced change of the Fermi surface of spinless fermions moving on two chains connected by weak interchain hopping t Ќ . For a model containing interband backward scattering only we show that the distance ⌬ between the Fermi momenta associated with the bonding and the antibonding band can be strongly reduced, corresponding to a large reduction of the effective interchain hopping t Ќ * ϰ⌬. A self-consistent one-loop approximation neglecting marginal vertex corrections and wave-function renormalizations predicts a confinement transition for sufficiently large interchain backscattering, where the renormalized t Ќ * vanishes. However, a more accurate calculation taking vertex corrections and wave-function renormalizations into account predicts only weak confinement in the sense that 0 Ͻ ͉t Ќ * ͉ Ӷ ͉t Ќ ͉. Our method can be applied to other strong-coupling problems where the dominant scattering channel is known.

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.