I am bit confused by how to approach this concept with mean field theory. As I understand a symmetry break (like a acquiring a finite magnetization) can happen if at low enough temperatures the Free energy decreases when breaking the symmetry. Normally this temperature is found by calculating a thermal average and solving a self-consistent equation for the critical temperature. But is it obvious, that this is the same as finding the critical temperature at which the free energy begins to decrease when breaking the symmetry?
As far as I remember, it is usually simpler to calculate the critical temperature as that temperature where some correlation functions calculated for the normal state diverge.