We present a new approximation scheme to solve the Non-Perturbative Renormalization Group equations and obtain the full momentum dependence of n -point functions. The core of the method consists in deriving approximate flow equations that are then solved exactly. The results thus obtained can next be improved through an iteration procedure. The approximations proposed in this paper lead to approximate equations that are accurate both in the perturbative and the scaling regimes. The scheme is tested with the calculation of the shift Δ T c in the transition temperature of the weakly repulsive Bose gas, a quantity which is very sensitive to all momentum scales. The leading-order result is in agreement with lattice calculations, albeit with a theoretical uncertainty of about 25%. The next-to-leading order differs by about 10% from the best accepted result.
In a companion paper [Blaizot, Phys. Rev. E 74, 051116 (2006)], we have presented an approximation scheme to solve the nonperturbative renormalization group equations that allows the calculation of the n-point functions for arbitrary values of the external momenta. The method was applied in its leading order to the calculation of the self-energy of the O(N) model in the critical regime. The purpose of the present paper is to extend this study to the next-to-leading order of the approximation scheme. This involves the calculation of the four-point function at leading order, where interesting features arise, related to the occurrence of exceptional configurations of momenta in the flow equations. These require a special treatment, inviting us to improve the straightforward iteration scheme that we originally proposed. The final result for the self-energy at next-to-leading order exhibits a remarkable improvement as compared to the leading order calculation. This is demonstrated by the calculation of the shift DeltaTc, caused by weak interactions, in the temperature of Bose-Einstein condensation. This quantity depends on the self-energy at all momentum scales and can be used as a benchmark of the approximation. The improved next-to-leading order calculation of the self-energy presented in this paper leads to excellent agreement with lattice data and is within 4% of the exact large N result.
Using the exact renormalization group we calculate the momentum-dependent self-energy $\ensuremath{\Sigma}(k)$ at zero frequency of weakly interacting bosons at the critical temperature
Tc of Bose-Einstein condensation in dimensions $3\ensuremath{\leqslant}D<4$. We obtain the complete crossover function interpolating between the critical regime
k⪡kc, where $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}$, and the short-wavelength regime
k⪢kc, where $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2(D\ensuremath{-}3)}$ in
D>3 and $\ensuremath{\Sigma}(k)\ensuremath{\propto}\mathrm{ln}(k∕{k}_{c})$ in
D=3. Our approach yields the crossover scale
kc on the same footing with a reasonable estimate for the critical exponent $\ensuremath{\eta}$ in
D=3. From our $\ensuremath{\Sigma}(k)$ we find for the interaction-induced shift of
Tc in three dimensions $\ensuremath{\Delta}{T}_{c}∕{T}_{c}\ensuremath{\approx}1.23a{n}^{1∕3}$, where
a is the
s-wave scattering length and
n is the density.