Communications in Mathematical Physics

, Volume 59, Issue 2, pp 97–108 | Cite as

On the equivalence of different order parameters and coexistence of phases for Ising ferromagnet. II

  • C. Gruber
  • J. L. Lebowitz


We investigate Ising spin systems with general ferromagnetic, translation invariant interactions,H=−ΣJBσB,JB≧0. We show that the critical temperatureTi for the order parameterpi defined as the temperature below whichpi>0, is independent of the way in which the symmetry breaking interactions approach zero from above. Furthermore, all the “equivalent” correlation functions have the same critical exponents asT ↗ Ti from below, e.g. for pair interactions all the odd correlations have the same critical index as the spontaneous magnetization. The number of fluid and crystalline phases (periodic equilibrium states) coexisting at a temperatureT at which the energy is continuous is shown to be related to the number of symmetries of the interactions. This generalizes previous results for Ising spins with even (and non-vanishing nearest-neighbour) ferromagnetic interactions. We discuss some applications of these results to the triangular lattice with three body interactions and to the Ashkin-Teller model. Our results give the answer to the question raised by R.J. Baxter et al. concerning the equality of some critical exponents.


Correlation Function Symmetry Breaking Spin System Critical Exponent Pair Interaction 
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Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • C. Gruber
    • 1
  • J. L. Lebowitz
    • 2
  1. 1.Ecole Polytechnique FédéraleLausanneSwitzerland
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA

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