Self-energy and critical temperature of weakly interacting bosons

TL;DR

This paper uses the exact renormalization group to analyze the momentum-dependent self-energy of weakly interacting bosons at the Bose-Einstein condensation critical temperature, providing a detailed crossover function and estimating the shift in T_c.

Contribution

It introduces a comprehensive crossover function for the self-energy of weakly interacting bosons near T_c using the exact renormalization group, including a scale for the crossover and an estimate for the critical exponent eta.

Findings

Derived the crossover function for Sigma(k) across different regimes.

Estimated the critical exponent eta in three dimensions.

Calculated the interaction-induced shift of T_c as approximately 1.23 a n^{1/3}.

Abstract

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.