Critical point in 2D Potts model and duality equation 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:
$$
e^{K \delta_{\sigma_{i} \sigma_{j}}}=\cosh \left(\frac{1+\sigma \sigma^{\prime}}{2}K\right)+\frac{1+\sigma \sigma^{\prime}}{2}\sinh (K)
$$
 A: 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:

*

*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

*G. Grimmett (2006) The Random Cluster Model, Springer-Verlag

