Unsupervised learning of topological phase transitions using Calinski-Harabaz index

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, 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 $\bar{\sigma }$ (when the different clusters vary from separation to mixing together). However, sometimes it is not easy to find the intersection if $\bar{D}$ or $\bar{\sigma }$ 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 $\bar{D}/\bar{\sigma }$, 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 $\epsilon /{\epsilon }_0$ 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.

arXiv comment15 pages, 17 figures

References in corpus (43)

• Machine learning and the physical sciences
• Machine learning phases of matter
• Learning phase transitions by confusion
• Discovering Phase Transitions with Unsupervised Learning
• Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders
• Machine learning quantum phases of matter beyond the fermion sign problem
• Discovering Phases, Phase Transitions and Crossovers through Unsupervised Machine Learning: A critical examination
• Identifying topological order through unsupervised machine learning
• Quantum Loop Topography for Machine Learning
• Machine Learning Topological Invariants with Neural Networks
• Approximating quantum many-body wave-functions using artificial neural networks
• Machine learning vortices at the Kosterlitz-Thouless transition
• Unsupervised machine learning and band topology
• Unsupervised machine learning account of magnetic transitions in the Hubbard model
• Machine learning out-of-equilibrium phases of matter
• Kernel methods for interpretable machine learning of order parameters
• Identifying Quantum Phase Transitions with Adversarial Neural Networks
• Machine Learning ${\mathbb{Z}}_2$ Quantum Spin Liquids with Quasi-particle Statistics
• Classifying Snapshots of the Doped Hubbard Model with Machine Learning
• Machine learning of phase transitions in the percolation and XY models
• Discriminative Cooperative Networks for Detecting Phase Transitions
• Machine learning of quantum phase transitions
• Unsupervised Manifold Clustering of Topological Phononics
• Topological quantum phase transitions retrieved through unsupervised machine learning
• Machine Learning Many-Body Localization: Search for the Elusive Nonergodic Metal
• Finite-size scaling method for the Berezinskii-Kosterlitz-Thouless transition
• Parameter diagnostics of phases and phase transition learning by neural networks
• Probing hidden spin order with interpretable machine learning
• Machine Learning Topological Phases with a Solid-state Quantum Simulator
• Extrapolating quantum observables with machine learning: Inferring multiple phase transitions from properties of a single phase
• Super-resolving the Ising model with convolutional neural networks
• Learning phase transitions from dynamics
• Smallest Neural Network to Learn the Ising Criticality
• Supervised learning magnetic skyrmion phases
• Applications of neural networks to the studies of phase transitions of two-dimensional Potts models
• Machine Learning as a universal tool for quantitative investigations of phase transition
• Interpretable machine learning for inferring the phase boundaries in a nonequilibrium system
• Competing nematic interactions in a generalized XY model in two and three dimensions
• Cluster simulations of loop models on two-dimensional lattices
• Machine Learning Application to Two-Dimensional Dzyaloshinskii-Moriya Ferromagnets
• Restricted Boltzmann Machines for the Long Range Ising Models
• Extracting Critical Exponent by Finite-Size Scaling with Convolutional Neural Networks
• Self-organizing maps as a method for detecting phase transitions and phase identification

Cited by in corpus (19)

• Intrinsic dimension of path integrals: data mining quantum criticality and emergent simplicity
• Visualizing Quantum Phases And Identifying Quantum Phase Transitions By Nonlinear Dimensionality Reduction
• Transfer learning of phase transitions in percolation and directed percolation
• Machine-learning detection of the Berezinskii-Kosterlitz-Thouless transition and the second-order phase transition in the XXZ models
• Machine learning for percolation utilizing auxiliary Ising variables
• Quantum diffusion map for nonlinear dimensionality reduction
• Machine learning of XY model on a spherical Fibonacci lattice
• Unsupervised learning of phase transitions via modified anomaly detection with autoencoders
• Study of phase transition of Potts model with Domain Adversarial Neural Network
• Phase determination with and without deep learning
• Machine learning of phases and structures for model systems in physics
• BKT transitions of the XY and six-state clock models on the various two-dimensional lattices
• A simple framework for contrastive learning phases of matter
• Self-Supervised Ensemble Learning: A Universal Method for Phase Transition Classification of Many-Body Systems
• Learning phase transitions by siamese neural network
• Machine-Learning Detection of the Berezinskii-Kosterlitz-Thouless Transitions
• A Unsupervised Framework for Identifying Diverse Quantum Phase Transitions Using Classical Shadow Tomography
• Snake net and balloon force with a neural network for detecting multiple phases
• Autonomous Discovery of the Ising Model's Critical Parameters with Reinforcement Learning