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

TL;DR

This paper demonstrates that in certain long-range interacting spin systems, mean-field theory accurately predicts first-order phase transitions, with rigorous proofs provided for various decay types and dimensions.

Contribution

It establishes rigorous links between mean-field predictions and actual phase transitions in long-range spin systems with reflection positivity.

Findings

First-order phase transitions occur in models with spread-out interactions when mean-field theory indicates such transitions.

In high dimensions, spread-out, exponentially decaying interactions lead to first-order transitions, exemplified by the 3-state Potts model.

In low dimensions, power-law decaying interactions also exhibit similar phase transition behavior.

Abstract

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.