Difference in calculation of Ginzburg Criterion in Gaussian and Mean-Field theory?

Question

In the past I asked this question asking about a subtlety in the Ginzburg criterion for mean field theory and the Gaussian approximation.

I am now having a hard time determining in general what is different between the calculations of the Ginzburg criterion for these two approximations.

Most books only seem to focus on the mean field theory calculation and barley mention the Gaussian.

My question is therefore what is the difference between the derivation of the Ginzburg criterion for the MFA and the Gaussian approximation and why.

(for simplity I am considering a general $\phi^n$ theory)

Answer 1

I believe this to be explained in (Dimo, 1993; sections 7.5.2 and 7.11)^$1$.

Saddle Point Approximation

In the saddle point approximation we have a thermodynamic potential given by$^2$: $$\Phi_G=-V \frac{r_0^2}{16 u} -\ln \int \mathcal{D} \delta \phi e^{-\mathcal{H}_G}$$ In the mean field approximation we assume that the second term does not affect the thermodynamics of the system and thus require it to be small compared to the first term. The Gaussian approximation allows us to evaluate the second term.

Gaussian Approximation

The criterion set by Gaussian approximation is determined by the integral $I_1(r)$ were: $$I_1(r)=I_1(0)+\Delta I_1(r)$$ and $$\Delta I_1(r)=-r \int_k \frac{1}{ck^2(r+c k^2)}$$ this second term comes from some sort of 4-point interaction$^3$. We then need this second term to not contribute much to the value of $r$. This gives the Ginzburg criterion for the Gaussian approximation.

_{Footnotes
$^1$ Unfortunately I can't get access to the full book so what I say in this answer comes from the Google preview - which was fairly limited.
$^2$ I am using the (approximately) notation of the above source. Since I feel it is self evident and can't actually verify it I will not define symbols.
$^3$ As far as I can tell.}