# 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 $$Z=\sum_{\sigma_{1}, \sigma_{N}} \prod_{} e^{K \delta_{\sigma_{i} \sigma_{j}}}$$ where $$$$ are neighboring atoms and $$\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: $$\exp \left(K \sum_{} \delta_{\sigma_{i} \sigma_{j}}\right)=\exp ((s+r)K)$$ But I have problems with high temperature decomposition: $$e^{K \delta_{\sigma_{i} \sigma_{j}}}=\cosh \left(\frac{1+\sigma \sigma^{\prime}}{2}K\right)+\frac{1+\sigma \sigma^{\prime}}{2}\sinh (K)$$

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

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.