Unsupervised learning of topological phase transitions using the Calinski-Harabaz index

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, whose critical points can be difficult to be obtained by using unsupervised learning, such as the principal component analysis. Recently, the authors of [Nat. Phys. 15, 790 (2019)] employed the diffusion map method for identifying topological orders and were able to determine the Berezinskii-Kosterlitz-Thouless (BKT) phase transition of the XY model, specifically via the intersection of the average cluster distance (D) over bar and the within-cluster dispersion parameter (sigma) over bar (when the different clusters vary from separation to mixing together). However, sometimes it is not easy to find the intersection if (D) over bar or (sigma) over bar 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 (D) over bar / (sigma) over bar, to determine the critical points at which the ch index reaches a maximum or minimum value or jumps 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 and on the honeycomb lattices, our results of the ch index show the convergence of the peaks over a range of parameters epsilon/epsilon(0) in the Gaussian kernel. We also examine the generalized XY model with q = 2 and q = 8 and study the phase transition using the fractional 1/2-vortex or 1/8-vortex constraint, respectively. The global phase diagram can be obtained by our method, which does not use the label of configuration needed by supervised learning, nor a crossing from two curves (D) over bar and (sigma) over bar. Our method is, thus, useful to both topological and nontopological 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.