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Homework Statement
Compute correlation functions ##<\sigma_r \sigma_{r+l}>## for the 1D Ising model of length L with the follow BD conditions
(i) Periodic
(ii) Anti-Periodic
(iii) ##\sigma_1 = \sigma_{L+1}=1##
(iv) ##\sigma_1= -\sigma_{L+1}=1##
Homework Equations
##<\sigma_r \sigma_{r+l}> = \displaystyle\frac{1}{Z}\sum_{\sigma_l=\pm 1}^{L-1}e^{K(\sum_{k=1}^{L-1}\sigma_k \sigma_{k+1} +\sigma_1 \sigma_{L+1})} \sigma_r \sigma_{r+l}##
The Attempt at a Solution
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I know how to compute the partition function for the periodic case as its fairly common and I have a solution to computing the correlation function to http://www.colorado.edu/physics/phys7240/phys7240_fa14/notes/Week1.pdf although I don't understand how he goes from 1.11 to 1.12.But these solutions use the Trace of the transfer matrix which I am pretty sure is unique to the periodic BD conditions. Any help on how to compute these in general would be appreciated as I missed the lecture on it.