The star-triangle transformation for self-dual spin models

A necessary condition for the existence of a star-triangle transformation for a general, self-dual spin model is found. It is shown that this condition is, by symmetry, also sufficient for the Potts model and for the symmetric and general Ashkin-Teller models. For other models, explicit calculations show the sufficiency of the condition as well. The symmetry groups of the models considered are: L(M) ≈ L(M), L(2) ⊗ L(M), L(M) ⊗ L(M), D(5), D(7), L(2) ≈ L(2) ≈ L(2) and G(G9). These include all self-dual models with M⩽10 states, which have at most three independent energy parameters. The results are discussed. In particular, the “extraspecial” points found in the phase diagrams of models which contain a regular Z(M) subgroup for M⪖5 are tentatively related to the existence of Kosterlitz-Thouless phases for these models.