Consider the Potts model with three states , $\sigma (x) \in \{ 1, e^{2 \pi i/3}, e^{4 \pi i/3} \}$. I wanted to make sure that the following definition for two-point correlation function is correct:

$$K_2(x,y;n) \overset{?}{=} \frac{1}{Z_n} \sum_{\sigma} \sigma(x)\sigma(y) e^{-\beta H_n(\sigma)}$$ Where $Z_n = \displaystyle\sum_{\sigma} e^{-\beta H_n(\sigma)} $ is the partition function, and $$H_n(\sigma)= -J \sum^{n-1}_{j=1} \langle \sigma_j | \sigma_{j+1} \rangle -J(\langle b_1|\sigma_1 \rangle + \langle b_2|\sigma_n \rangle)$$ is the Hamiltonian associated to the configuration $\sigma$ (without magnetic field). Here, $ \langle a|b\rangle $ means the inner product.

  • $\begingroup$ or $$K_2(x,y;n) \overset{?}{=} \frac{1}{Z_n} \sum_{\sigma} \langle\sigma(x)|\sigma(y)\rangle e^{-\beta H_n(\sigma)}$$ $\endgroup$ – Nirvanacs Jan 30 '15 at 16:58

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