Chiral Potts model

Example

Let $N \ge 2$ be an integer, and let $G = (V, E)$ be a finite piece of the square lattice whose edges are oriented to the right and upward. A configuration is a map $\sigma \colon V \to \Z/N\Z$. Let $W, \overline{W} \colon \Z/N\Z \to (0, \infty)$ be Boltzmann weights for horizontal and vertical edges. The chiral Potts model is the probability measure on configurations with weight proportional to $$\prod_{(x, y) \text{ horizontal}} W(\sigma_x - \sigma_y) \prod_{(x, y) \text{ vertical}} \overline{W}(\sigma_x - \sigma_y).$$ The products run over oriented edges $(x, y)$, and in general $W(k) \ne W(-k)$. The weights are thus invariant under the cyclic shift $\sigma \mapsto \sigma + 1$ but not under $\sigma \mapsto -\sigma$; this lack of mirror symmetry is the chirality.