Unsupervised Learning Of Topological Phase Transitions Using The Calinski-Harabaz Index

In this paper,machine learning method is used to study the phase transition of statistical physical model,in which Principal Component Analysis Algorithm and its variant Kernel Principal Component Analysis and Diffusion Map are used to find the phase transition point of physical model.Firstly,the dimension reduction algorithm is used to reduce the dimension of the physical model data with large dimensions.Then K-Means clustering algorithm and Calinski-Harabaz are used to classify and evaluate the clustering situation.The critical line between different phases is found through the evaluation results,and the phase transition point is obtained.In recent years,machine learning methods have been used to identify the states of phase transition and the transitions between them.However,in topological phase transition(for example,in the classical XY model),it is difficult to obtain the phase transition points by using unsupervised learning(for example,PCA).In 2019,researchers from Harvard University published in Nature Physics used the diffusion map algorithm in the dimension reduction algorithm to identify the topological structure,and was able to determine the Berezinski-Kosterlitz-Thouless phase transition of the classical XY model by crossing the average distance of different clustersand the discrete distance within clusters.However,due to topological constraints,ifordoes not change much,it is not easy to find the intersection.In this paper,we use the ch index to determine the critical point,which is similar to the ratio ofand,but the ch index will reach the peak at the phase transition point,so as to help us determine the phase transition point.We tested the Calinski-Harabaz index in several statistical physical models,including those containing Berezinski-Kosterlitz-Thouless phase transition.For Ising model,the extreme value of Calinski-Harabaz index or its components is consistent with the extreme value of specific heat.For the classical XY model on square and hexagonal lattices,our results of ch index show that in the kernel parameter?/?₀ The convergence of the peak in the range of 0.We also study the generalized XY model with q=2 and q=8,and study the phase transition with 1/2 vortex or 1/8 vortex respectively,and obtain the global phase diagram.The main contents of this paper are as follows:In the first chapter,we introduce the background and significance of machine learning research on physical phase transition,the motivation of this paper,the data of the physical models and topological phase transition,and the specific research contents.In the second chapter,the dimensionality reduction algorithms used in this paper are described,which are principal component analysis,kernel principal component analysis and diffusion mapping,as well as the way to determine the super parameters of diffusion mapping,and the application scope of the three dimensionality reduction algorithms.In the third chapter,Calinski-Harabaz index is introduced to reduce the dimension of the data by using the three dimensionality reduction methods mentioned above,and Calinski-Harabaz index is used to predict the phase transition point.The experimental results are compared with the results of our method.In the fourth chapter,the accuracy of Calinski-Harabaz index for finding phase transition point is verified by experiments.The machine learning resutls are shown,by using the Ising model,classical XY model of square lattice and hexagonal lattice,generalized XY model of q=2 and q=8.In the fifth chapter,summary and prospect are described.In the appendix,we introduce the topological structure,kernel function of XY model,the calculation formula of common clustering indexes,and the comparison of the results between Diffusion Mapping and Kernel Principal Component Analysis when using Ising model data.