How to deal with mean field method in antiferromagnetism? There are lots of ways to apply the mean field method to deal with the Ising model whose ground state is a ferromagnetic state. Hence, it is easy to find the order parameter named magnetization to describe the mean behavior of spin-spin interaction.
But, in the antiferromagnetic Ising model, the ground state is an antiferromagnetic state, and I realize it is difficult to find a parameter which effectively describes the mean interaction. If I choose magnetization similar to what we did in Ising model, then, I find it is zero, the mean field is always zero. I get nothing from this mean field.
I cannot find a direct physical parameter describing the system to replace the spin-spin interaction. 
Does this imply that the mean behavior of spin-spin interaction is always zero? Is there a mean field method to deal with the antiferromagnetic Ising model?
I hope you can help me, thanks!
 A: Redefine your "field" to be multiplied by +1 on even sites and by -1 on odd sites. This way, the mean field will have opposite signs between the two states of antiferromagnetic order.
From what I recall, the zero-magnetic-field, nearest-neighbor ferromagnetic model (interaction J < 0) maps exactly to the zero-magnetic-field, nearest-neighbor antiferromagnetic model (interaction J > 0) through this approach.
When magnetic fields are applied, of course, things become much different.
A: The order parameter in the case of an antiferromagnet is the staggered magnetization. On a bipartite lattice, it is given by 
$$\tilde m_{\mathbf r}=(-1)^{\mathbf r} m$$ 
where $m$ is a constant (it would be the magnetization in the case of a ferromagnet), and $(-1)^{\mathbf r}$ just means that it is equal to $1$ on one sublattice and $-1$ on the other.
In the simple cases of bipartite lattices, one can get most of the properties of an antiferromagnet using the ferromagnet results by using the transformation $\hat S_{\mathbf r}=(-1)^{\mathbf r} S_{\mathbf r}$, which transform the Hamiltonian
$$ H=-J\sum_{\langle{\mathbf r} {\mathbf r'} \rangle}S_{\mathbf r}S_{\mathbf r'}$$
(with $J<0$) into
$$ \hat H=J\sum_{\langle{\mathbf r} {\mathbf r'} \rangle}\hat S_{\mathbf r}\hat S_{\mathbf r'}$$
which is the Hamiltonian of a ferromagnet. One can then use all the properties of the ferromagnet to compute the correlation functions of the $\hat S_{\mathbf r}$ to compute that of  $ S_{\mathbf r}$ using the inverse transformation. One then gets easily the result about the staggered magnetization quoted above.
In the case of a non-bipartite lattice (say, the triangular lattice), the transformation does not work, and the physics is much more complicated (and interesting). That's the physics of the geometric frustration, which is a very active subject of research.
