- #1
coolbeets
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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.
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.