I have a general question concerning mean-field approaches applied to quantum or classical statistical mechanics.
Does determining the mean-field by a variational approach always imply that the self-consistency is satisfied ? Moreover are there some instances where it is physically justified to look for saddle points and where a variational approach is misleading (say the energy is unbounded from below with respect to the mean-field parameters for instance) ?
For example, consider the simple case of the ferromagnetic Ising model. There, one introduces the magnetization ($m$) as the mean-field parameter with $m=\left<s_i^z\right>$ where the expectation value is taken with respect to the mean-field Hamiltonian that depends on $m$ (and $s_i^z$ is the spin variable $s_i^z=\pm 1$). In this particular case, when one finds a solution for $m=\left<s_i^z\right>$, this is equivalent to finding an extremum for the energy or the free-energy. Thus, my question is simply : instead of solving for the self-consistency can one instead look for the global energy (or free-energy) minimum with respect to the mean-field parameters (and does this approach always makes sense physically) ? Here I give the example of the Ising model, but my question also applies to any model (fermionic, bosonic, spin models, etc.).
I am more or less looking for counterexamples here, if there exist any.