Absence and presence of phase transitions for Potts models on pseudo-lattices

Abstract

By considering a Potts model on a pseudo-lattice such as a tree or a Husimi tree, it is shown that the analyticity of the free energy in the absence of a field for such systems is due to a graph-theoretical property of these lattices called locality. It is proved that, if this property is destroyed by introducing next-nearest neighbour interactions, then the free energy in the absence of a field is nonanalytic as a function of the nearest-neighbour interaction energy below a critical temperature. An exact relation shows that this nonanalyticity is of the same type (phase transition of continuous order) as that found for the Ising model on a Cayley tree with external field.