Absence of Continuous Symmetry Breaking in a One-Dimensional n-Z Model

There has been considerable interest in long-range one-dimensional lattice gases, in part because of formal connections with the Kondo problem, and in part because of an analogy with higher-dimensional models: continuous variation of rate of falloff is somewhat akin to continuous variation of dimension. For pair-interacting ferromagnetic models with coupling J ( j ) = j ~ , it has been known for some time that a = 2 is the borderline. Ruelle (7) showed if ~ > 2, neither the Ising or multicomponent models have multiple phases; if a < 2, then Dyson (1) proved that the Ising model has multiple phases and Frohlieh et al. (3) proved the same thing for the multicomponent models. Naturally, interest has focused on the borderline case a = 2. Recently, Frohlich and Spencer (4) proved the existence of discrete symmetry breaking for the Ising model with this value of a. Our main goal here is to prove the absence of continuous symmetry breaking in models of the same type.