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Mean Field Theory of Antiferromagnetic Ising Model (Square Lattice) with Magnetic Field

  1. Nov 20, 2014 #1
    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.
     
  2. jcsd
  3. Nov 21, 2014 #2
    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...
     
  4. Nov 21, 2014 #3
    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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