It is often stated that the Ginzburg criterion for mean field theory and the Gaussian approximation are the same.

Goldenfeld, 1992; pg$\sim$170 tries to show the Ginzburg criterion for mean field theory. The author does this by first deriving a two-point correlator $G(\vec r,\vec r')$ ($= G(\vec r-\vec r')$ if translationally invariant) found as follows (my words):

  1. Differentiate the free energy w.r.t the order parameter $\eta(r)$ and set the result to zero.
  2. Differentiate the result w.r.t an inhomogeneous filed $H(r)$ and use that: $$\chi_T(\vec r,\vec r')= \frac{\delta \left< \eta(\vec r)\right>}{\delta H}=G(\vec r,\vec r')$$

The Ginzburg criterion is then calculated using the following condition: $$\frac{\left| \int_V d^d \vec r G(\vec r)\right|}{\int_V d^d \vec r \eta(\vec r)^2}$$

The same calculation is done using the Gaussian approximation with $G(\vec r)$ found using a functional integral. The volume here is that set by the correlation length (cf. my recent PSE question).

I understand the process for the Gaussian approximation what I don't understand is why the correlation function found using the above technique needs to be small compared to $\eta(\vec r)$. In what $G(\vec r)$ actually corresponds to in the mean field theory.

(Note this arxiv:1112.1375 article has a similar derivation (pg4) but with different notation).


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