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

We consider a class of spin systems on ℤ ^d 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.