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Decimation of a triangular lattice

  1. Apr 15, 2016 #1


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    1. The problem statement, all variables and given/known data
    Consider the network of spins shown below. The Hamiltonian is given by $$H = - \sum_{\langle i j k \rangle} [J \sigma_i \sigma_j \sigma_k + J_0]$$ with ##J,J_o \geq 0## and ##\langle i j k \rangle## denoting spin in the same triangle (the triangles under consideration are each highlighted by a looping arrow). Decimate over the crosses and find a recursion relation for the renormalised coupling ##J'##.

    2. Relevant equations

    Given below:

    3. The attempt at a solution

    I was going to proceed by the method of transfer matrix but then I am not sure what size my matrix would be in this case. The bond action is $$e^{w(\sigma, \sigma' \sigma'')} = e^{J_o} e^{J \sigma \sigma' \sigma''}$$ and I want to find a new (renormalised) bond action $$e^{w'(\mu \mu' \mu'')} = e^{J_o'} e^{J' \mu \mu' \mu''}$$

    I think $$e^{w'(\mu \mu' \mu'')} = \sum_{\sigma \sigma' \sigma''} = e^{\mu \sigma \sigma'} e^{\sigma' \mu'' \sigma''} e^{\sigma'' \sigma \mu'}$$ if I understood the set up correctly. I would proceed by writing down the matrix representation of these quantities but I am not sure what size to make them. There are ##2 \times 2 \times 2## possible combinations of spins.

    Thanks for any tips!
  2. jcsd
  3. Apr 20, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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