Can Renormalization Simplify the 2 x 2 Ising Lattice Problem?

[shetland]

Hey all,

I have a problem I'm working on. A 2 x 2 ising lattice,

[tex]
\ H = K_1\sum_{nn}\sigma_i\sigma_j \ + \ K_2\sum_{nnn}\sigma_i\sigma_j \ + \ K_3\sum_{sg}\sigma_i\sigma_j\sigma_k\sigma_l
[/tex]

Were to find H as an explicit function the sigma's,

[tex] H(\sigma_1,\sigma_2,\sigma_3,\sigma_4)[/tex]

Its the typical spin lattice situation, with [tex]\sigma = =\pm1[/tex]

For those in the know, one standard way is to use renormalization. But wouldn't this arrive at a function of H only in terms of the new K? Another way would be to set up a transfer matrix method...but I haven't been exposed to this before.

Also, if I understand the process of re-normalization, I guess in a finite example like the 2x2, is the goal to reduce the degrees of freedom to one, meaning some ultimate K for site 1?

Any help/suggestions would be greatly appreciated.

Shelley

 

[Aaron Bex]

Hi Shelley,

It sounds like you're trying to find an explicit function for the Hamiltonian of a 2x2 Ising lattice in terms of the sigma variables. Have you considered using Monte Carlo simulations to solve this problem? This approach has been used a lot in the study of spin systems and may be able to help you find the explicit function you're looking for. Good luck!

 

[GSXR750]

Hi Shelley,

The 2 x 2 Ising lattice problem is a classic problem in statistical mechanics and has been extensively studied. As you mentioned, one way to approach this problem is through renormalization, which involves reducing the degrees of freedom in the system to obtain a simpler model.

In the case of a 2 x 2 lattice, there are four spins (\sigma_1, \sigma_2, \sigma_3, \sigma_4) and three coupling constants (K_1, K_2, K_3) in the Hamiltonian. Using renormalization, we can reduce the degrees of freedom to one by grouping the spins into two pairs and taking the average of their values. This results in a new Hamiltonian with two spins and two coupling constants, which can be further simplified using the transfer matrix method.

The goal of this approach is to obtain a simpler model that still captures the essential features of the original system. It is not necessarily to reduce the degrees of freedom to one, but rather to find a more manageable model that can still provide insight into the behavior of the system.

I hope this helps and good luck with your problem!