Solving Antiferromagnetic Ising Model on Square Lattice

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The discussion focuses on developing a mean field theory for the antiferromagnetic Ising model on a square lattice, specifically addressing challenges with the self-consistency condition for the magnetization parameter, m. The equation derived, m = -tanh(β(4mJ-B)), leads to confusion as it yields only one solution, highlighting the issue of m being a poor choice for representing the system's behavior. Suggestions to use an alternating parameter like (-1)^(r)(m) are mentioned, but concerns arise regarding its effectiveness in the presence of an external magnetic field. Participants acknowledge the complexity of the antiferromagnetic case compared to the ferromagnetic model, noting that nearest neighbors have opposite spins at low temperatures, resulting in zero net magnetization. The discussion concludes with a suggestion to employ a variational method to select an appropriate trial effective field and order parameter.
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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.
 
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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...
 
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.
 
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