Phase transitions in three-dimensional loop models and the $CP^{n-1}$ sigma model
Phase transitions in three-dimensional loop models and the $CP^{n-1}$ sigma model
复制标题
三维环路模型和 $CP^{n-1}$ sigma 模型中的相变
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
A. Somoza
中科院分区：
文献类型：
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作者：
A. Nahum;J. Chalker;P. Serna;M. Ortuño;A. Somoza
本文研究了三维格点上一类具有逸度$n$的密排环模型的统计力学问题。随着耦合常数的变化，模型表现出两种类型的相位：一种是所有环路都是有限的，另一种是一些环路无限扩展。我们证明了循环模型是$C{P}^{nensuremath{-}1}$ $ensuremath{sigma}$模型的离散化。$ensuremath{sigma}$模型的有限相和无限相分别表示无序相和有序相，并讨论了环路性质与$ensuremath{sigma}$模型相关器之间的关系。在大尺度上，环在有序相中是布朗的，在临界点处具有非平凡的分形维数。我们对模型进行了模拟，发现$n=1,2,3$的两个阶段之间的连续过渡和$nensuremath{ge}4$的一阶过渡。我们还给出了$C{P}^{nensuremath{-}1}$模型的重整化群处理，该模型显示了尽管在朗道-金兹堡描述中存在三次不变量，但当$n$大于（但接近）2时，连续过渡如何存活。我们得到的结果与（2+1）维中的$ mathm {SU}(n)$量子磁体、对称类$C$中的Anderson局域化以及三维随机曲线的统计等问题具有更广泛的相关性。
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 $C{P}^{nensuremath{-}1}$ $ensuremath{sigma}$ models. The finite and infinite loop phases represent, respectively, disordered and ordered phases of the $ensuremath{sigma}$ model, and we discuss the relationship between loop properties and $ensuremath{sigma}$ 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 $nensuremath{ge}4$. We also give a renormalization-group treatment of the $C{P}^{nensuremath{-}1}$ 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 $mathrm{SU}(n)$ quantum magnets in (2+1) dimensions, Anderson localization in symmetry class $C$, and the statistics of random curves in three dimensions.