1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Ising Model using renormalisation group

  1. Oct 25, 2015 #1

    CAF123

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    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.

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

    3. 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.
     
  2. jcsd
  3. Oct 30, 2015 #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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Ising Model using renormalisation group
  1. Ising model (Replies: 5)

Loading...