Symmetries of spin models on bipartite lattices

The symmetries of the partition function of a class of spin models with a transitive symmetry group on a bipartite graph or lattice are shown to be generated by a set of automorphisms of this permutation group and by the involutions of its center. The latter are generalized “antiferromagnetic” symmetries. If these models contain an Abelian, regular subgroup and are considered on the square lattice, the duality transformation induces an extra symmetry. The total group induced by duality, automorphism and center symmetries is studied. These results are applied to obtain information on the phase diagrams of the symmetric Ashkin–Teller model as well as of a dual pair of Zamolodchikov–Monastyrskii models.