# Mean-field theory : variational approach versus self-consistency

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.

• at equilibrium, both approaches should lead to same result (by thermodynamic argument) – Nikos M. Jun 14 '14 at 0:18

## 2 Answers

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.

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.