Ising Model using renormalisation group

[CAF123]

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.

 

[Asim Riaz]

This implies that $$w'(\sigma_1, \sigma_4) = w(\sigma_1, \sigma_2) + w(\sigma_2, \sigma_3) + w(\sigma_3, \sigma_4)$$Now for the Hamiltonian, we have $$\mathcal H = − \sum_{ i=1}^N K\sigma_i \sigma_{i+1},$$ where $K ≥ 0$ and $\sigma_i$ are the Ising spins (i.e. $\sigma_i= \pm 1$).So, $$w(\sigma_1, \sigma_2) = - K \sigma_1 \sigma_2$$and $$w'(\sigma_1, \sigma_4) = - K' \sigma_1 \sigma_4$$Therefore the recursion relation is $$K' = K + K + K = 3K$$