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Loop models with crossings

机译：交叉路口模型

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The universal behavior of two-dimensional loop models can change dramatically when loops are allowed to cross.We study models with crossings both analytically and with extensive Monte Carlo simulations.Our main focus (the "completely packed loop model with crossings") is a simple generalization of well-known models that shows an interesting phase diagram with continuous phase transitions of a new kind.These separate the unusual "Goldstone" phase observed previously from phases with short loops.Using mappings to Z_2 lattice gauge theory, we show that the continuum description of the model is a replica limit of the σ model on real projective space (RP~(n-1)).This field theory sustains Z_2 point defects, which proliferate at the transition.In addition to studying the new critical points, we characterize the universal properties of the Goldstone phase in detail, comparing renormalization group (RG) calculations with numerical data on systems of linear size up to L = 10~6 at loop fugacity n = 1.(Very large sizes are necessary because of the logarithmic form of correlation functions and other observables.) The model is relevant to polymers on the verge of collapse, and a particular point in parameter space maps to self-avoiding trails at their Θ point; we use the RG treatment of a perturbed σ model to resolve some perplexing features in the previous literature on trails.Finally, one of the phase transitions considered here is a close analog of those in disordered electronic systems-specifically, Anderson metal-insulator transitions-and provides a simpler context in which to study the properties of these poorly understood (central-charge-zero)critical points.
机译：允许循环通过时，二维循环模型的通用行为会发生巨大变化。我们通过分析和广泛的蒙特卡洛模拟研究具有交叉的模型。我们的主要研究重点（“具有交叉的完全填充的循环模型”）众所周知的模型的一般化，它显示了一种有趣的，具有新型连续相变的相图。这些相图将以前观察到的异常“金石”相与短环相分离开来。使用Z_2晶格规理论的映射，我们可以看到连续体该模型的描述是σ模型在实际射影空间（RP〜（n-1））上的一个复制极限。该场论维持Z_2点缺陷，该缺陷在过渡时扩散。除了研究新的临界点外，我们详细描述戈德斯通相的通用特性，将重归一化组（RG）的计算与回路松散度n为L = 10〜6的线性系统的数值数据进行比较= 1（由于相关函数和其他可观察物的对数形式，因此需要非常大的尺寸。）该模型与处于崩溃边缘的聚合物相关，并且参数空间中的特定点映射到其Θ处的自回避轨迹点;我们使用σ模型的RG处理来解决先前文献中关于路径的一些令人困惑的特征。最后，这里考虑的相变之一与无序电子系统中的相变非常相似，具体来说，是安德森金属-绝缘体转变-并提供了一个更简单的上下文来研究这些知之甚少的（中心电荷零）临界点的性质。

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