论文标题
带有远距离耦合的小人模型
Villain model with long-range couplings
论文作者
论文摘要
最近的邻居反派或周期性高斯模型是一种有用的工具，可以理解二维二维近距离邻居$ xy $模型的拓扑缺陷的物理，因为这两个模型共享相同的对称性并且在同一普遍性类别中。最近已证明二维$ xy $模型的远程对应物表现出一种非平凡的临界行为，其中一个复杂的相图包括耦合衰减的幂律指数的一系列值，$σ$，其中有一个磁性化，无序和一个关键阶段和一个关键阶段（arxiv：arxiv：arxiv：21044.132217）。在这里，我们解决了一个问题，即该模型的反派对应者是否可以描述具有远距离耦合的二维$ XY $模型的关键行为。在引入了具有远距离耦合的小人模型的合适概括之后，我们得出了一组涡旋 - 涡流电位的重新归一化组方程，这与远程$ xy $模型之一的不同，这表明在此制度中不再是自旋波和拓扑缺陷的解耦合。主要结果是，对于$σ<2 $，这两个模型不再共享相同的普遍性类别。值得注意的是，在其相图的大区域内，发现反派模型的行为与$ 1/r^2 $交互的一维iSing模型相似。
The nearest-neighbor Villain, or periodic Gaussian, model is a useful tool to understand the physics of the topological defects of the two-dimensional nearest-neighbor $XY$ model, as the two models share the same symmetries and are in the same universality class. The long-range counterpart of the two-dimensional $XY$ model has been recently shown to exhibit a non-trivial critical behavior, with a complex phase diagram including a range of values of the power-law exponent of the couplings decay, $σ$, in which there are a magnetized, a disordered and a critical phase (arXiv:2104.13217). Here we address the issue of whether the critical behavior of the two-dimensional $XY$ model with long-range couplings can be described by the Villain counterpart of the model. After introducing a suitable generalization of the Villain model with long-range couplings, we derive a set of renormalization-group equations for the vortex-vortex potential, which differs from the one of the long-range $XY$ model, signaling that the decoupling of spin-waves and topological defects is no longer justified in this regime. The main results are that for $σ<2$ the two models no longer share the same universality class. Remarkably, within a large region of its phase diagram, the Villain model is found to behave similarly to the one-dimensional Ising model with $1/r^2$ interactions.
