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

TL;DR

This paper employs kernel PCA to analyze phase classification and transitions in frustrated classical XY spin models, capturing non-linear features and revealing detailed order parameters and phase transition behavior.

Contribution

It introduces the use of kernel PCA to identify non-linear order parameters in frustrated XY models, extending previous linear PCA approaches.

Findings

Kernel PCA captures non-linear order parameters.

Identifies principal components related to U(1) and Z2 symmetries.

Reveals second order phase transition through eigenvalue analysis.

Abstract

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.