# 2 x 2 Ising lattice problem

1. May 22, 2004

### shetland

Hey all,

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

$$\ H = K_1\sum_{nn}\sigma_i\sigma_j \ + \ K_2\sum_{nnn}\sigma_i\sigma_j \ + \ K_3\sum_{sg}\sigma_i\sigma_j\sigma_k\sigma_l$$

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

$$H(\sigma_1,\sigma_2,\sigma_3,\sigma_4)$$

Its the typical spin lattice situation, with $$\sigma = =\pm1$$

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