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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Supported in part by U.S.A.F.O.S.R. under contract F 44620-71-C-0013, P001.
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Lebowitz, J.L., Martin-Löf, A. On the uniqueness of the equilibrium state for Ising spin systems. Commun.Math. Phys. 25, 276–282 (1972). https://doi.org/10.1007/BF01877686
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DOI: https://doi.org/10.1007/BF01877686