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2 x 2 Ising lattice problem

  1. May 22, 2004 #1
    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
     
  2. jcsd
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