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Phase transitions in three-dimensional loop models and the CP~(n-1) sigma model

机译：三维回路模型和CP〜（n-1）sigma模型中的相变

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摘要

We consider the statistical mechanics of a class of models involving close-packed loops with fugacity n on three-dimensional lattices. The models exhibit phases of two types as a coupling constant is varied: in one, all loops are finite, and in the other, some loops are infinitely extended. We show that the loop models are discretizations of CP~(n-1) a models. The finite and infinite loop phases represent, respectively, disordered and ordered phases of the a model, and we discuss the relationship between loop properties and a model correlators. On large scales, loops are Brownian in an ordered phase and have a nontrivial fractal dimension at a critical point. We simulate the models, finding continuous transitions between the two phases for n = 1,2,3 and first order transitions for n ≥ 4. We also give a renormalization-group treatment of the CP~(n-1) model that shows how a continuous transition can survive for values of n larger than (but close to) 2, despite the presence of a cubic invariant in the Landau-Ginzburg description. The results we obtain are of broader relevance to a variety of problems, including SU(n) quantum magnets in (2 + 1) dimensions, Anderson localization in symmetry class C, and the statistics of random curves in three dimensions.
机译：我们考虑一类模型的统计力学，该模型涉及在三维晶格上具有逸度为n的密排循环。随着耦合常数的变化，模型显示出两种类型的相位：一种是所有循环都是有限的，而另一种是某些循环是无限扩展的。我们证明了循环模型是CP〜（n-1）a模型的离散化。有限和无限循环阶段分别表示模型的无序和有序阶段，我们讨论了循环属性与模型相关器之间的关系。大规模而言，环是有序相中的布朗，在临界点上具有不平凡的分形维数。我们对模型进行仿真，找到n = 1,2,3的两个相之间的连续过渡和n≥4的一阶过渡。我们还给出了CP〜（n-1）模型的重归一化组处理，该处理显示了如何尽管在Landau-Ginzburg描述中存在三次不变式，但连续过渡可以保留大于（但接近）n的n值。我们获得的结果与各种问题具有更广泛的相关性，包括（2 + 1）维的SU（n）量子磁体，对称性C类的安德森局部化以及三个维的随机曲线的统计量。

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