二维伊辛模型的主成分分析结果的理论推导

Theoretical derivation of principal component analysis results for the two-dimensional Ising model

在伊辛模型中,主成分分析(PCA)作为一种无监督学习方法被用来鉴别相变.我们理论推导了具有周期边界条件的二维方晶格伊辛模型的PCA结果.根据伊辛模型的哈密顿量和晶格结构可确定样本协方差矩阵的结构,进而可通过解决样本协方差矩阵的特征值问题实现PCA.晶格的所有对称性都隐含在样本协方差矩阵中,其中平移对称性决定了与傅里叶模式对应的特征向量,并且在平移对称性的基础上其他的对称性决定了特征值重数.进而结合伊辛模型的知识,明确了PCA结果的含义.我们的理论推导解释了先前研究中伊辛模型的PCA结果.在理论推导的基础上,我们还进一步挖掘了在伊辛模型中PCA鉴别相变的能力.从机器学习(ML)应用的角度,对PCA结果的理论推导揭示了PCA鉴别相变的机制,这有利于去除ML中的不透明,进而增加对ML在物理学领域的应用的信任.

In the Ising model,principal component analysis(PCA)serves as an unsupervised learning method for identifying phase transitions.The PCA results of the two-dimensional square-lattice Ising model with periodic boundary conditions are theoretically deduced.By utilizing the Hamiltonian and lattice structure of the Ising model,the structure of the sample covariance matrix can be determined,enabling the realization of PCA by solving the eigenvalue problem of the sample covariance matrix.All symmetries of the lattice are implicit in the sample covariance matrix,with translational symmetry determining the eigenvector corresponding to the Fourier mode.Moreover,based on translational symmetry,other symme-tries determine the degeneracy of eigenvalues.When combined with knowledge regarding the Ising model,the meaning of the PCA results becomes clarified.Our theoretical derivation elucidates the PCA results from previous studies on the Ising model.Based on theoretical derivation,we also investigated the ability of PCA to identify phase transitions in the Ising model.From the machine learning(ML)perspective,the theoretical derivation of PCA results sheds light on the mecha-nism through which PCA identifies phase transitions.This clarity contributes to removing opacity in ML and increasing trust in the application of ML in physics.