Mean-field driven first-order phase transitions in systems with long-range interactions

Details Mean-field driven first-order phase transitions in systems with long-range interactions Mean-field driven first-order phase transitions in systems with long-range interactions

2005-01-27

arxiv.org/abs/math-ph/0501067v2

J. Statist. Phys. 122 (2006), no. 6, 1139-1193

Physics

Mathematical Physics

57 pages; uses a (modified) jstatphys class file

Scientific paper

10.1007/s10955-005-8072-0

We consider a class of spin systems on $\Z^d$ with vector valued spins $(\bS_x)$ that interact via the pair-potentials $J_{x,y} \bS_x\cdot\bS_y$. 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 $d\ge3$, 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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