The computation of the potential energy is based on Matsubara formalism, moving from Fetter-Walecka eq. 23.14 and transforming to imaginary frequency with the assumption of a local self-energy: $\Sigma(k,iω) = \Sigma({A,B},iω)$. This would simply give:
$$ E_\mathrm{pot} = \frac{2}{\beta} \sum_ω \mathrm{Tr}[\Sigma(iω)G(iω)] $$
but we also need a semi-analytic tail correction for that we cannot compute enough matsubara points to get an accurate summation.
The customary way is to assume the product $\Sigma(iω)G(iω)$ to have a $\frac{U^2}{4w^2}$ tail, but this won't work here being the self-energy just a constant. So instead we tried with
$$ \Sigma(iω)G(iω) = \Sigma_0 G(iω) \propto \frac{U}{2\omega}\times\Sigma_0$$
which unfortunately does not work. I believe this is the right idea, but some detail might be off. To be checked when I have time.
The computation of the potential energy is based on Matsubara formalism, moving from Fetter-Walecka eq. 23.14 and transforming to imaginary frequency with the assumption of a local self-energy:$\Sigma(k,iω) = \Sigma({A,B},iω)$ . This would simply give:
but we also need a semi-analytic tail correction for that we cannot compute enough matsubara points to get an accurate summation.
The customary way is to assume the product$\Sigma(iω)G(iω)$ to have a $\frac{U^2}{4w^2}$ tail, but this won't work here being the self-energy just a constant. So instead we tried with
which unfortunately does not work. I believe this is the right idea, but some detail might be off. To be checked when I have time.