Solving Antiferromagnetic Ising Model on Square Lattice

[coolbeets]

Hello,

I am trying to work out a mean field theory for an antiferromagnetic Ising model on a square lattice. The Hamiltonian is:

$H = + J \sum_{<i,j>} s_{i} s_{j} - B \sum_{i} s_{i}$
$J > 0$

I'm running into issues trying to use

$<s_{i}> = m$

together with the self-consistency requirement that $<s_{i}>$ also satisfies the definition of expectation value. I end up with

$m = -tanh(\beta(4mJ-B))$

which doesn't make much sense. No matter what, I get only one solution. I think the issue is arising from the fact that my parameter (m) is a bad one. When half the spins are up and half are down, there is zero magnetization.

I have seen some suggestions around about choosing the parameter to be something like

$(-1)^{r}(m)$,

but people also seem to claim that this only works in the case of no external magnetic field (due to some symmetry, which is broken by the filed).

What is a good way to think about this? What is a smarter choice of parameter in this case?

Any insight is appreciated. Thank you.

 

[UVCatastrophe]

I've actually seen that function before in my stat mech class, the $m = \tanh m$ one, so I think you're on the right track. I remembered me and my classmates were puzzled as well...I don't have anything else useful to say about this, sorry...

 

[coolbeets]

Thanks for your response! Yeah, the m=tanhm one is what you get for the regular (ferromagnetic) Ising model, which is already not so simple, as you said, but I think it's even more complicated in the antiferromagnetic case. In the latter case, nearest neighbors have opposite spins at low T, and the net magnetization is zero.

I believe I should use a variational method, in which I choose some trial, effective field and order parameter. I'm just not sure how to choose them.