I am getting confused about the distinction between Mean-field theory (MFT) and the Gaussian approximation (GA). I have being told on a number of occasions (in the context of the Ising model) that the Gaussian approximation is at the same level as MFT.
I think this refers to the mean-field theory associated with $$ H = -J \sum_{\langle i,j \rangle} \big( (\sigma_i-M) + M \big) \big( (\sigma_j - M) + M \big) ~, \tag{1} $$ where $M$ is the mean-field, and $(\sigma - M)$ represent the fluctuations. but then I have also read that the GA is the lowest order correction to the MFA — i.e., found with the saddle-point method; consult Kopietz et al. “Introduction to the functional renormalization group”. Springer (2013) [wcat].
How do these two agree? Is the MFT associated with (1) really on the same level as the GA which is at a higher level than the MFT associated with the saddle-point approximation? Or am I missing something?
Edit
The answers as they currently stand (23/03/2018) simply give (exactly or almost exactly) what is in the reference that I gave in this question - and as such this is clearly something I have seen before. This reference does explain (as I have said in my question) that the GA is a correction to the saddle point MFA. But does not mention anything about the MFA in (1) and how it relates to the GA - which is basically the crux of my question.
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