Mean-field theory : variational approach versus self-consistency

Question

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.

Answer 1

Your question: 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)?

Well, indeed, the fundamental approach for variational method is minimizing free-energy with respect to the variational action. If we define \begin{equation} F = - \frac{1}{\beta} \ln Z \qquad \quad Z = \int {\cal D}\phi e^{-S/\hbar} \end{equation} where $Z$ is a partition function, then if $S_{var}$ is a variational action with variational parameter (e.g. Ising Hamiltonian whose interaction is replaced by mean-field parameter m as you mentioned), we need to minimized \begin{equation} F \leq F^* = F_{var} - \frac{1}{\beta \hbar} \langle S-S_{var} \rangle_{var} \end{equation} where $F_{var} = -\beta \ln Z_{var} $ and the expectation value is evaluated with respect to the variational action. If you expand minimizing condition for $F^*$, you will obtain self-consistency equation for Ising model. Therefore, self-consistency equation $m = <S_i>$ is in fact 'physical intuition' which can be rigorously derived from minimizing variational action.

Answer 2

As Nikos M. had stated, at equilibrium both should lead to the same result.

This could also be visualized from the basic formalism of mean-field theories, which states that Free energy of the system has an upper-bound.

$$ F \le \bar{H_0} - TS $$

And this function (Bogoliubov inequality) is minimized to determine observables.