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I have to prove, considering the limits of high and low temperatures, duality equation: $$ Z=q e^{2 N K} F\left(e^{-K}\right)=q^{-N}\left(e^{K}+q-1\right)^{2 N} F\left(\frac{e^{K}-1}{e^{K}+q-1}\right) $$ for square lattice with $$Z=\sum_{\sigma_{1}, \sigma_{N}} \prod_{<i j>} e^{K \delta_{\sigma_{i} \sigma_{j}}}$$ where $<ij>$ are neighboring atoms and $\sigma_{i} \epsilon$ $\{1,2, \ldots, q\}$. Also i have information that there is only one critical point.

I can try to make analog of low and high temperature Kramers–Wannier duality, let's assume we have s horizontally and r vertically pairs of different spins, then low temperature decomposition looks like: $$ \exp \left(K \sum_{<i j>} \delta_{\sigma_{i} \sigma_{j}}\right)=\exp ((s+r)K)$$ But I have problems with high temperature decomposition: $$ e^{K \delta_{\sigma_{i} \sigma_{j}}}=\cosh \left(\frac{1+\sigma \sigma^{\prime}}{2}K\right)+\frac{1+\sigma \sigma^{\prime}}{2}\sinh (K) $$

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  • $\begingroup$ "I have to prove..." Why do you have to? $\endgroup$
    – hft
    Apr 22 at 0:14
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    $\begingroup$ i forgot to add tag homework, sorry $\endgroup$ Apr 22 at 0:14
  • $\begingroup$ What textbook are you using for the class? $\endgroup$
    – hft
    Apr 22 at 0:15
  • $\begingroup$ lecture 2 from my university department slava.itp.ac.ru/intro-to-integrable-lattice-models but you will not understand this language $\endgroup$ Apr 22 at 0:16
  • $\begingroup$ I don't understand your high-temperature decomposition. I would have written the latter as $e^{K\delta_{\sigma_i\sigma_j}} = 1 + (e^K-1) \delta_{\sigma_i\sigma_j}$. $\endgroup$ Apr 22 at 11:57

1 Answer 1

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The duality equation can be proved using the Fortuin-Kasteleyn representation of the Potts model. Introduce link variables $b_{ij}\in\{0,1\}$ to rewrite the partition function as $$\eqalign{ Z&=\sum_{\{\sigma\}} \prod_{(i,j)} e^{K\delta_{\sigma_i,\sigma_j}}\cr &=\sum_{\{\sigma\}} \prod_{(i,j)} \big[e^K\delta_{\sigma_i,\sigma_j} +e^0(1-\delta_{\sigma_i,\sigma_j})\big]\cr &=\sum_{\{\sigma\}} \prod_{(i,j)}\Big[\sum_{b_{ij}=0}^1\big( (e^K-1)\delta_{\sigma_i,\sigma_j}\delta_{b_{ij},1}+\delta_{b_{ij},0}\Big] }$$ The variables $b_{ij}$ form a graph on the lattice. The sum over the spins $\sigma$ can now be performed leaving $$Z=\sum_{G} u^{b(G)} q^{C(G)}$$ where $u=e^K-1$, $b(G)=\sum b_{ij}$ is the number of edges in the graph $G$ and $C(G)$ is the number of clusters. The duality transformation consists in associating to each link $b_{ij}$ on the lattice a dual link $b_{ij}^*=1-b_{ij}$ on the dual lattice. Each loops on the dual lattice encloses a cluster on the lattice. Using Euler relations, one finally gets the duality relation. For more details, see Ref. 2.

References:

  1. Fortuin, C.M.; Kasteleyn, P.W. (1972). "On the random-cluster model : I. Introduction and relation to other models". Physica. 57 (4): 536–564. doi:10.1016/0031-8914(72)90045-6
  2. G. Grimmett (2006) The Random Cluster Model, Springer-Verlag
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