Ashkin–Teller model as a vertex problem

It is shown that the Ashkin–Teller model on any planar lattice is equivalent to an eight-vertex model on a related lattice. The exact equivalence is given for finite lattices with a boundary. We show, in particular, that the AT model on the triangular or honeycomb lattice is related to an eight-vertex model on a Kagomé lattice. The occurrence of two phase transitions in the AT model in general is also discussed.