论文标题
使用Calinski-Harabaz指数无监督学习拓扑相变
Unsupervised learning of topological phase transitions using Calinski-Harabaz index
论文作者
论文摘要
机器学习方法最近已应用于物质的学习阶段和它们之间的过渡阶段。特别令人感兴趣的是拓扑相变，例如XY模型中，对于无监督学习，例如主要成分分析可能很难。 Recently, authors of [Nature Physics \textbf{15},790 (2019)] employed the diffusion-map method for identifying topological order and were able to determine the BKT phase transition of the XY model, specifically via the intersection of the average cluster distance $\bar{D}$ and the within cluster dispersion $\barσ$ (when the different clusters vary from separation to mixing together).但是，有时候，如果$ \ bar {d} $或$ \barσ$由于拓扑约束而不会太大变化，那么找到交叉点并不容易。在本文中，我们建议使用Calinski-Harabaz（$ CH $）索引，该指数大致定义为比率$ \ bar d/\barσ$，以确定$ CH $索引达到最大值或最小值的关键点，或者大量跳高。我们检查了几种统计模型中的$ CH $索引，其中包括包含BKT相过渡的模型。对于ISING模型，数量$ CH $或其组件的峰与特定热量最大的位置一致。对于方格和蜂窝晶格上的XY模型，我们的$ CH $索引的结果显示了高斯内核中$ \ varepsilon/\ varepsilon_0 $的峰值峰的收敛性。我们还使用$ Q = 2 $和$ Q = 8 $的广义XY模型，并以纯XY限制为单位。因此，我们的方法对拓扑和非血域过渡都有用，并且可以与先前在这些模型中使用的监督学习方法一样良好，并且可以用于从实验数据中搜索阶段。
Machine learning methods have been recently applied to learning phases of matter and transitions between them. Of particular interest is the topological phase transition, such as in the XY model, which can be difficult for unsupervised learning such as the principal component analysis. Recently, authors of [Nature Physics \textbf{15},790 (2019)] employed the diffusion-map method for identifying topological order and were able to determine the BKT phase transition of the XY model, specifically via the intersection of the average cluster distance $\bar{D}$ and the within cluster dispersion $\barσ$ (when the different clusters vary from separation to mixing together). However, sometimes it is not easy to find the intersection if $\bar{D}$ or $\barσ$ does not change too much due to topological constraint. In this paper, we propose to use the Calinski-Harabaz ($ch$) index, defined roughly as the ratio $\bar D/\bar σ$, to determine the critical points, at which the $ch$ index reaches a maximum or minimum value, or jump sharply. We examine the $ch$ index in several statistical models, including ones that contain a BKT phase transition. For the Ising model, the peaks of the quantity $ch$ or its components are consistent with the position of the specific heat maximum. For the XY model both on the square lattices and honeycomb lattices, our results of the $ch$ index show the convergence of the peaks over a range of the parameters $\varepsilon/\varepsilon_0$ in the Gaussian kernel. We also examine the generalized XY model with $q=2$ and $q=8$ and at the value away from the pure XY limit. Our method is thus useful to both topological and non-topological phase transitions and can achieve accuracy as good as supervised learning methods previously used in these models, and may be used for searching phases from experimental data.
