- #1

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.