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arXiv 2005.11830cond-mat.stat-mechhep-thmath-phmath.MP
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长程关联随机曲面的拓扑效应与共形不变性

Topological effects and conformal invariance in long-range correlated random surfaces

Nina Javerzat, Sebastian Grijalva, Alberto Rosso, Raoul Santachiara

备注：
35 pages. Replacement takes into account the comments of the two SciPost referees, which can be found at https://scipost.org/submissions/2005.11830v1/#report_1
发表信息：
SciPost Phys. 9, 050 (2020)
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摘要：

本文研究了具有负Hurst指数的离散随机分形曲面$H<0$。通过激活表面高度大于给定水平$h$的位点来提供晶格的随机着色。激活位点的集合通常表示为偏移集合。这一组的连接组件，水平集群，定义一个参数（$H$）家庭的渗流模型与长期相关的网站占领。水平集群渗滤在一个有限的值$h=h_c$和$H\leq-\frac{3}{4}$的相变预计将保持在相同的普适性类的纯（即不相关）渗滤。相反，对于$-\frac{3}{4}<H< 0$，存在一条指数连续变化的临界点线。这些点的普适性类，特别是关于水平集群的共形不变性，是知之甚少。通过结合共形场论和数值方法，我们提供了这些阶段的新见解。我们专注于连接功能，定义为概率，两个网站属于同一级别的集群。在我们的模拟中，表面被定义在尺寸为$M\times N$的晶格环面上。我们证明了拓扑效应对连通性函数的影响使得所有的临界线$H<0$都具有共形不变性。特别是，利用各向异性的矩形环面（$M\neq N$），我们直接测试的存在下的两个组件的无痕应力能量张量。此外，我们探讨了共形场论的光谱和结构常数。最后，我们观察到，标度的修正清楚地指出了从纯渗流点到长程相关点的可积性的突破。

英文摘要：

We consider discrete random fractal surfaces with negative Hurst exponent $H<0$. A random colouring of the lattice is provided by activating the sites at which the surface height is greater than a given level $h$. The set of activated sites is usually denoted as the excursion set. The connected components of this set, the level clusters, define a one-parameter ($H$) family of percolation models with long-range correlation in the site occupation. The level clusters percolate at a finite value $h=h_c$ and for $H\leq-\frac{3}{4}$ the phase transition is expected to remain in the same universality class of the pure (i.e. uncorrelated) percolation. For $-\frac{3}{4}<H< 0$ instead, there is a line of critical points with continously varying exponents. The universality class of these points, in particular concerning the conformal invariance of the level clusters, is poorly understood. By combining the Conformal Field Theory and the numerical approach, we provide new insights on these phases. We focus on the connectivity function, defined as the probability that two sites belong to the same level cluster. In our simulations, the surfaces are defined on a lattice torus of size $M\times N$. We show that the topological effects on the connectivity function make manifest the conformal invariance for all the critical line $H<0$. In particular, exploiting the anisotropy of the rectangular torus ($M\neq N$), we directly test the presence of the two components of the traceless stress-energy tensor. Moreover, we probe the spectrum and the structure constants of the underlying Conformal Field Theory. Finally, we observed that the corrections to the scaling clearly point out a breaking of integrability moving from the pure percolation point to the long-range correlated one.

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