Unsupervised learning of topological phase transitions using the Calinski-Harabaz index
Open Access
- 22 January 2021
- journal article
- research article
- Published by American Physical Society (APS) in Physical Review Research
- Vol. 3 (1), 013074
- https://doi.org/10.1103/physrevresearch.3.013074
Abstract
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 and the within-cluster dispersion parameter (when the different clusters vary from separation to mixing together). However, sometimes it is not easy to find the intersection if or 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 , 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 in the Gaussian kernel. We also examine the generalized XY model with and 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 and . 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.
Funding Information
- National Science Foundation (PHY 1915165)
- National Natural Science Foundation of China (51901152)
- Shanxi Province Science Foundation for Youths (201901D211082)
This publication has 57 references indexed in Scilit:
- Finite-size scaling method for the Berezinskii–Kosterlitz–Thouless transitionJournal of Statistical Mechanics: Theory and Experiment, 2013
- Stiffness jump in the generalizedmodel on the square latticePhysical Review E, 2013
- Cluster Simulations of Loop Models on Two-Dimensional LatticesPhysical Review Letters, 2007
- Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion mapsProceedings of the National Academy of Sciences of the United States of America, 2005
- Collective Monte Carlo Updating for Spin SystemsPhysical Review Letters, 1989
- Silhouettes: A graphical aid to the interpretation and validation of cluster analysisJournal of Computational and Applied Mathematics, 1987
- Nonuniversal critical dynamics in Monte Carlo simulationsPhysical Review Letters, 1987
- Exact Critical Point and Critical Exponents ofModels in Two DimensionsPhysical Review Letters, 1982
- A dendrite method for cluster analysisCommunications in Statistics - Theory and Methods, 1974
- Ordering, metastability and phase transitions in two-dimensional systemsJournal of Physics C: Solid State Physics, 1973