Machine Learning of Frustrated Classical Spin Models. II. Kernel Principal Component Analysis

In this work we apply the principal component analysis (PCA) method with kernel trick to study classification of phases and phase transition in classical XY models in frustrated lattices. Comparing to our previous work with linear PCA method, the kernel PCA can capture non-linear function. In this case, the Z2 chiral order of classical spins in these lattices are indeed a non-linear function of the input spin configurations. In addition to the principal component revealed by linear PCA, the kernel PCA can find out two more principal components using data generated by Monte Carlo simulation at various temperatures at input. One of them relates to the strength of the U(1) order parameter and the other directly manifests the chiral order parameter that characterizes the Z2 symmetry breaking. For a temperature resolved study, the temperature dependence of the principal eigenvalue associated with the Z2 symmetry breaking clearly shows a second order phase transition behavior.