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

We consider a class of spin systems on Z with vector valued spins (Sx) that interact via the pair-potentials Jx,y Sx ·Sy . The interactions are generally spreadout in the sense that the Jx,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 meanfield values and the states themselves converge to a product measure.