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Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models

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Abstract

Results from percolation theory are used to study phase transitions in one-dimensional Ising andq-state Potts models with couplings of the asymptotic formJ x,y≈ const/¦x−y¦2. For translation-invariant systems with well-defined lim x→∞ x 2 J x =J + (possibly 0 or ∞) we establish: (1) There is no long-range order at inverse temperaturesβ withβJ +⩽1. (2) IfβJ +>q, then by sufficiently increasingJ 1 the spontaneous magnetizationM is made positive. (3) In models with 0<J +<∞ the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeysM(β c )⩾1/(β c J +)1/2. (4) For Ising (q=2) models withJ +<∞, it is noted that the correlation function decays as 〈σxσy〉(β)≈c(β)/|x−y|2 wheneverβ<β c . Points 1–3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values ofq.

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Authors and Affiliations

  1. Courant Institute for Mathematical Sciences, New York University, 10012, New York, New York

    M. Aizenman

  2. Department of Mathematics, University of California, 90024, Los Angeles, California

    J. T. Chayes & L. Chayes

  3. Department of Mathematics, University of Arizona, 85721, Tucson, Arizona

    C. M. Newman

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  1. M. Aizenman
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  2. J. T. Chayes
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  3. L. Chayes
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  4. C. M. Newman
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Aizenman, M., Chayes, J.T., Chayes, L. et al. Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models. J Stat Phys 50, 1–40 (1988). https://doi.org/10.1007/BF01022985

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