arXiv 1811.033582018-11-08cond-mat.stat-mech

伊辛模型的Fortuin-Kasteleyn表示的几何性质

Geometric properties of the Fortuin-Kasteleyn representation of the Ising model

Pengcheng Hou, Sheng Fang, Junfeng Wang, Hao Hu, Youjin Deng

备注：: 12 pages, 10 figures, 11 tables
发表信息：: Phys. Rev. E 99, 042150 (2019)
DOI：: 10.1103/PhysRevE.99.042150

摘要：

本文用Monte Carlo方法研究了正方形（2D）和简单立方（3D）晶格上的伊辛模型的Fortuin-Kasteleyn（FK）团簇.包裹概率，一个无量纲的量表征的拓扑结构的FK集群上的一个环面，被发现遭受较小的有限尺寸的修正比著名的Binder比，并产生一个高精度的临界耦合$K_c(3\rm D)=0.221\,654\,631(8)$。然后，我们研究了临界状态下FK团簇的几何性质。结果表明，最大团簇尺寸$C_1$的分布服从单变量函数$P(C_1,L){\rm d}C_1=\tilde P(x){\rm d}x$和$x\equiv C_1/L^{d_{\rm F}}$（$L$为线性尺寸），分维$d_{\rm F}$与磁性指数相同.一个有趣的双峰特征，观察到分布$\tilde P(x)$在3D中，并归因于不同的接近行为的$K \to K_c+0^\pm$。对于一个临界的FK配置，每个站点的大小$s$的集群数$n(s,L)$被确认为遵守标准的标度形式$n(s,L)\sim s^{-τ}\tilde n(s/L^{d_{\rm F}})$，超标度关系$τ=1+d/d_{\rm F}$和空间维度$d$。为了进一步刻画FK簇的紧凑性，我们测量它们的图距离，并确定最短路径指数为$d_{\rm min}(3\rm D)=1.259\,4(2)$和$d_{\rm min}(2\rm D)=1.094\,0(3)$。进一步，通过从占据的键中排除所有的桥，我们得到无桥构型，并确定骨架指数为$d_{\rm B}(3\rm D)=2.167\,3(15)$和$d_{\rm B}(2\rm D)=1.732\,1(4)$。3D伊辛模型的通用包装概率和几何临界指数$d_{\rm min}$和$d_{\rm B}$的估计要么改善现有的结果或尚未报道。

英文摘要：

We present a Monte Carlo study of the Fortuin-Kasteleyn (FK) clusters of the Ising model on the square (2D) and simple-cubic (3D) lattices. The wrapping probability, a dimensionless quantity characterizing the topology of the FK clusters on a torus, is found to suffer from smaller finite-size corrections than the well-known Binder ratio, and yields a high-precision critical coupling as $K_c(3\rm D)=0.221\,654\,631(8)$. We then study geometric properties of the FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size $C_1$ follows a single-variable function as $P(C_1,L){\rm d}C_1=\tilde P(x){\rm d}x$ with $x\equiv C_1/L^{d_{\rm F}}$ ($L$ is the linear size), and that the fractal dimension $d_{\rm F}$ is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution $\tilde P(x)$ in 3D, and attributed to the different approaching behaviors for $K \to K_c+0^\pm$. For a critical FK configuration, the cluster number per site $n(s,L)$ of size $s$ is confirmed to obey the standard scaling form $n(s,L)\sim s^{-τ}\tilde n(s/L^{d_{\rm F}})$, with hyper-scaling relation $τ=1+d/d_{\rm F}$ and the spatial dimension $d$. To further characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as $d_{\rm min}(3\rm D)=1.259\,4(2)$ and $d_{\rm min}(2\rm D)=1.094\,0(3)$. Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as $d_{\rm B}(3\rm D)=2.167\,3(15)$ and $d_{\rm B}(2\rm D)=1.732\,1(4)$. The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents $d_{\rm min}$ and $d_{\rm B}$ either improve over the existing results or have not been reported yet.

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英文摘要：

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