长程相互作用系统中平均场驱动的一级相变
Mean-field driven first-order phase transitions in systems with long-range interactions
摘要：
我们考虑一类具有矢量值自旋$(\bS_x)$的$\Z^d$上的自旋系统，它们通过对势$J_{x,y} \bS_x\cdot\bS_y$相互作用。在$J_{x,y}$表现出指数或幂律衰减的意义上，相互作用通常是分散的。在反射正性的技术条件下，充分分散的相互作用，我们证明了该模型表现出一级相变时，相关的平均场论信号这样的过渡。因此，例如，在$d\ge3$维中，我们最终可以提供具有扩展的、指数衰减的相互作用的三态Potts模型的例子，该模型随着温度的变化而经历一阶相变。类似的过渡建立在尺寸$d=1,2$的幂律衰减相互作用和下一个最近邻耦合的高维度。此外，我们还研究了无限扩展相互作用的极限。具体地说，我们表明，一旦平均场论是在一个独特的“状态”，然后在任何序列的双折射不变吉布斯状态各种可观的收敛到他们的平均场值和状态本身收敛到一个产品的措施。
英文摘要：
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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