Edwards-Anderson Hamiltonian of a Hopf link

[omarshehab]

Hi,

I was calculating the Edwards-Anderson Hamiltonian of a Hopf link. A hopf link is like attachment 1. I have drawn the Seifert surface of that link. The surface is shown in attachment 2. It also contains the Boltzmann weight. So, this is an Ising model. I am confused as there are more than one interaction between a pair of sites. How will I keep that consideration when I calculate the Edwards-Anderson Hamiltonian.

Thanks in advance for your answers.

 

[omarshehab]

A quick supplementary question: should I model it as a frustrated Ising system?

 

[omarshehab]

I have another question. Can there be both ferromagnetic and anti-ferromagnetic interactions between two sites of an Ising model at the same time? In the attached picture you can see that two Ising sites have two interactions, one ferromagnetic and one anti-ferromagnetic.

 

[omarshehab]

I talked with Professor David P. Landau on this issue.

Let me quote his suggestion: "I think that you are attempting to produce an extension of the original Edwards-Anderson model, and I imagine that retention of the frustration through symmetric delta-function couplings is all that is needed.".

 

[omarshehab]

This is my reply to Professor David P. Landau:

"
Hi Dr. Landau,

Thank you very much for your reply. A Hopf link is a standard topological construct. To derive the equivalent spin model from it, we have to draw its Seifert surface first. The attached hopf_link.jpg is the Hopf link and hopf-seifert-pattern-1-oriented-Boltzmann.pdf shows how a spin model can be derived from its surface. To do this, we have to mark the Hopf link in a checker board pattern. Then the marked parts are considered as sites of a spin model and the oriented crossings are considered interactions among them. That's how we get a spin model from a link or knot. Here I have two sites with more than one interactions between them. The same pair of sites have one ferromagnetic and one anti-ferromagnetic interactions between them.

If the assumption of the original Edwards-Anderson model is that any two sites will have only one interaction between them, you are right that I am trying to extend it. I am not exactly sure what you meant by 'retention of the frustration through symmetric delta-function couplings'.

It looks, by 'retention of frustration', you indicated that I should expect degeneracy in finding the ground state of the system. Do you think that the degeneracy will give us two separate spin systems one with a ferromagnetic interaction and one with anti-ferromagnetic interaction?

I am not familiar with the term 'symmetric delta-function couplings'. I have tried to look up introductory resources for this term. Do you indicate that I have to calculate a superposition of these two different interactions and only consider the single resultant interaction? I have found another paper using non-linear Ising model for social science. Can I consider the spin system in hopf-seifert-pattern-1-oriented-Boltzmann.pdf as a non-linear Ising model?

I understand that you should be very busy. I would sincerely appreciate if you could give me some hint about my issues. Thanks in advance for your time.
"