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A while a go I asked this question on the difference between mean field theory and the Gaussian approximation. This question is related to that.

The mean field critical exponents for the Ising model are given by: $$\alpha_\pm=0,\quad\beta=\frac{1}{2},\quad \gamma_\pm =1, \quad \delta =3, \quad \nu = \frac{1}{2},\quad \eta=0$$ Let us focus on $\nu$. This cannot be calculated using the saddle point approximation and is most often calculated via the Gaussian approximation. Given that it is the saddle point approximation which corresponds to mean field theory why is $\nu=\frac{1}{2}$ taken as a mean field critical exponent when it is calculated in an approximation 'beyond' mean field theory.

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