Solving Antiferromagnetic Ising Model on Square Lattice

In summary, the conversation discusses the difficulties in applying mean field theory to an antiferromagnetic Ising model on a square lattice. The participant is struggling with finding a suitable parameter and self-consistency requirement to solve the equation for the expectation value. Suggestions for choosing a parameter such as (-1)^r(m) are mentioned, but it is noted that this may only work in the absence of an external magnetic field. The use of a variational method is proposed, but the participant is unsure of how to choose the appropriate trial field and order parameter.
  • #1
coolbeets
2
0
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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  • #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...
 
  • #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.
 

1. What is the Antiferromagnetic Ising Model on Square Lattice?

The Antiferromagnetic Ising Model on Square Lattice is a mathematical model used to study the behavior of magnetic materials. It consists of a lattice made up of individual spins that can be in either an "up" or "down" state, representing the two possible orientations of a magnetic moment. The model takes into account the interactions between neighboring spins and the external magnetic field.

2. Why is it important to solve the Antiferromagnetic Ising Model on Square Lattice?

The Antiferromagnetic Ising Model on Square Lattice is important because it provides insights into the behavior of real-world magnetic materials. By solving the model, scientists can better understand the underlying principles that govern the properties of magnetic materials, such as their phase transitions and critical points.

3. What methods are used to solve the Antiferromagnetic Ising Model on Square Lattice?

There are several methods that can be used to solve the Antiferromagnetic Ising Model on Square Lattice. These include mean field theory, Monte Carlo simulations, and exact solutions using techniques such as transfer matrices and Bethe ansatz. Each method has its own strengths and limitations, and the choice of method depends on the specific research question.

4. How does the temperature affect the behavior of the Antiferromagnetic Ising Model on Square Lattice?

The temperature plays a crucial role in the behavior of the Antiferromagnetic Ising Model on Square Lattice. At low temperatures, the model exhibits ordered phases where neighboring spins align in opposite directions, resulting in antiferromagnetic order. As the temperature increases, the system undergoes a phase transition to a disordered phase where the spins are randomly oriented. The critical temperature at which this transition occurs is a key parameter in the model.

5. What are the applications of solving the Antiferromagnetic Ising Model on Square Lattice?

The Antiferromagnetic Ising Model on Square Lattice has numerous applications in physics, materials science, and engineering. It can be used to study the properties of various magnetic materials, such as metals, alloys, and thin films. The model has also been applied to other systems, such as social networks and neural networks, to understand their behavior and phase transitions. In addition, the insights gained from solving the model can aid in the development of new technologies, such as spintronics and quantum computing.

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