The change in the number of phase transitions for perturbations of finite range interactions is studied.

A Monte-Carlo simulation was performed for a translation invariant spin 1/2 ferromagnetic model in Z² with fundamental bonds

A = {(0,0);(0,1)}

B = {(0,0);(2,0)}

C = {(0,0);(0,1);(1,1);(1,0)}

The model exhibits one phase transition if the coupling constant J(A) is zero, but two phase transitions were found when J(A) is non zero and small enough.

The generalization of this situation is provided by a construction, due to J. Slawny, which through a sequence of progressively smaller perturbations yields models with an arbitrary minimum number of phase transitions. However, such construction requires the existence of interactions with one fundamental bond such that for all values of the coupling constants the Gibbs state is unique even when the interaction is perturbed by an arbitrary finite range perturbation of small enough norm. In this work it is proven that such property is exhibited by some translation invariant systems in Z^ν with finite state space at each point. The proof applies to models with real interactions and whose fundamental bonds are all multiple of a single bond which is of prime order and which is obtained as the product—in the group ring structure of the dual space—of one dimensional bonds whose non trivial projections at each lattice site are unique. The proof is based on the Dobrushin-Pecherski criterion concerning the uniqueness of Gibbs states under perturbations. Such criterion is restated so that only transition functions on sets of simple geometry are involved.

In addition, an algebraic characterization is presented for the set of Gibbs states for ferromagnetic systems for which the state space at each lattice site is a compact abelian group. This is a generalization of the theory originally introduced by Slawny for spin 1/2 ferromagnetic models and later extended by Pfister to ferromagnetic models for which the state space at each point is a finite product of tori and finite abelian groups.