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The Ginzburg criteria is a self-consistency check on the mean field solution - it does not explicitly check if mean theory is correct just that it produces a self-consistent answer. This therefore leads to the natural questions: Is there any system where the Ginzburg criterion is satisfied but mean field theory does not work? Or can it be shown that the Ginzburg criteria is actually more then a self-consistency check and does predict the validity of mean-field theory?

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I think the question is whether there are cases in which perturbative corrections to the mean field theory are small, but non-perturbative effects invalidate the mean field approximation. In general this sort of thing of course happens in many quantum field theories, but I am not aware of an example in the context of statistical field theories of second order phase transitions (which is where the Ginzburg criterion is introduced). Of course, it is always possible that the MFA is wrong because you picked the wrong order parameter, or the Landau-Ginzburg paradigm of local order parameters is not applicable (as in topological phase transitions).

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