Machine learning of frustrated classical spin models (II): Kernel principal component analysis

Abstract

In this work, we apply a principal component analysis (PCA) method with a kernel trick to study the classification of phases and phase transitions in classical XY models of frustrated lattices. Compared to our previous work with the linear PCA method, the kernel PCA can capture nonlinear functions. In this case, the Z₂ chiral order of the classical spins in these lattices is indeed a nonlinear function of the input spin configurations. In addition to the principal component revealed by the linear PCA, the kernel PCA can find two more principal components using the data generated by Monte Carlo simulation for various temperatures as the input. One of them is related 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 second-order phase transition behavior.

Keywords

machine learning / classical XY model / kernel PCA / frustrated lattice

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