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Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions

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Abstract

We consider a class of spin systems on ℤd with vector valued spins (S x ) that interact via the pair-potentials J x,y S x ⋅S y . The interactions are generally spread-out in the sense that the J x,y 's exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions d≥3, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions d = 1,2 for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique “state,” then in any sequence of translation-invariant Gibbs states various observables converge to their mean-field values and the states themselves converge to a product measure.

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  1. Department of Mathematics, UCLA, Los Angeles, CA, 90095-1555, USA

    Marek Biskup, Lincoln Chayes & Nicholas Crawford

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  1. Marek Biskup
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  2. Lincoln Chayes
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  3. Nicholas Crawford
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Biskup, M., Chayes, L. & Crawford, N. Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions. J Stat Phys 122, 1139–1193 (2006). https://doi.org/10.1007/s10955-005-8072-0

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