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On the uniqueness of the equilibrium state for Ising spin systems

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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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Author notes

  1. Anders Martin-Löf

    Present address: Department of Mathematics, Royal Institute of Technology, Stockholm 70, Sweden

Authors and Affiliations

  1. Belfer Graduate School of Science, Yeshiva University, New York, N.Y.

    Joel L. Lebowitz

  2. Rockefeller University, New York, N.Y.

    Anders Martin-Löf

Authors
  1. Joel L. Lebowitz
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  2. Anders Martin-Löf
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Additional information

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

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