Ising Model using renormalisation group

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CAF123
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Homework Statement


The Hamiltonian of the 1D Ising model without a magnetic field, is defined via: $$\mathcal H = − \sum_{ i=1}^N K\sigma_i \sigma_{i+1},$$ where ##K ≥ 0## and ##\sigma_i## are the Ising spins (i.e. ##\sigma_i= \pm 1##).

A) Set up a decimation procedure with decimation parameter, λ, equal to 3, and find out the recursion relation for the coupling K. In order to do so, you will have to include a trivial field in H.

Homework Equations


partition function ##Z_N = \sum_{\sigma} \exp(-\beta \mathcal H (\sigma)) = \sum_{\sigma} \prod_i \exp(w(\sigma_i, \sigma_{i+1})##

The Attempt at a Solution


Just a quick check on the notation, does ##\lambda=3## mean that we keep ##\sigma_i## and ##\sigma_{i+3}## and ##\sigma_{i+1}, \sigma_{i+2}## are decimated? So the partiiton function listed in relevant equations becomes $$\sum_{\sigma_1, \sigma_4,\sigma_7 \dots} \left[ \sum_{\sigma_2, \sigma_3} e^{w(\sigma_1, \sigma_2)}e^{w(\sigma_2, \sigma_3)}e^{w(\sigma_3, \sigma_4)}\right] \left[\sum_{\sigma_5, \sigma_6} \dots\right]$$ and ##e^{w(\sigma_1, \sigma_2)}e^{w(\sigma_2, \sigma_3)}e^{w(\sigma_3, \sigma_4)}= e^{w'(\sigma_1, \sigma_4)}## and the prime denotes this constitutes a term in the Hamiltonian describing the scaled system.
 

Answers and Replies

  • #2
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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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