Decimation of a triangular lattice

[CAF123]

Homework Statement

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 $J'$.

Homework Equations

Given below:

The Attempt at a Solution

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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!

 

[mark726]

Hello,

Thank you for sharing your thoughts on this problem. I think your approach using transfer matrix is a good one. However, I believe the size of your matrix would be 8x8, as there are 8 possible combinations of spins (since each spin can take on two values: up or down).

To find the recursion relation for the renormalised coupling $J'$, you could use the transfer matrix to calculate the partition function and then take the derivative with respect to $J'$. This would give you the average value of the bond action, which can be related to the renormalised coupling.

I hope this helps! Let me know if you have any further questions.