Abstract
We show that for an Ising spin system of arbitrary spin with a ferromagnetic pair interaction and a “periodic” external magnetic field there is a unique equilibrium state if and only if the magnetization is continuous with respect to a uniform change in the external field. Hence, if the critical temperatureT c is defined as the temperature where the spontaneous magnetization (which is a non-increasing function of the temperature) becomes positive, then the equilibrium state is unique forT>T c and is non-unique forT<T c (when the external field is zero). This implies that the correlation functions have a cluster property forT>T c .We also show that for an anti-ferromagnet consisting of two sublattices there is a unique equilibrium state if and only if the staggered magnetization is continuous with respect to a change in the staggered field.
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