Department of Physics, University of California Davis, Davis, California 95616, USA.
Department of Computer Science, University of California Davis, Davis, California 95616, USA.
Phys Rev E. 2017 Jun;95(6-1):062122. doi: 10.1103/PhysRevE.95.062122. Epub 2017 Jun 19.
We apply unsupervised machine learning techniques, mainly principal component analysis (PCA), to compare and contrast the phase behavior and phase transitions in several classical spin models-the square- and triangular-lattice Ising models, the Blume-Capel model, a highly degenerate biquadratic-exchange spin-1 Ising (BSI) model, and the two-dimensional XY model-and we examine critically what machine learning is teaching us. We find that quantified principal components from PCA not only allow the exploration of different phases and symmetry-breaking, but they can distinguish phase-transition types and locate critical points. We show that the corresponding weight vectors have a clear physical interpretation, which is particularly interesting in the frustrated models such as the triangular antiferromagnet, where they can point to incipient orders. Unlike the other well-studied models, the properties of the BSI model are less well known. Using both PCA and conventional Monte Carlo analysis, we demonstrate that the BSI model shows an absence of phase transition and macroscopic ground-state degeneracy. The failure to capture the "charge" correlations (vorticity) in the BSI model (XY model) from raw spin configurations points to some of the limitations of PCA. Finally, we employ a nonlinear unsupervised machine learning procedure, the "autoencoder method," and we demonstrate that it too can be trained to capture phase transitions and critical points.
我们应用无监督机器学习技术，主要是主成分分析（PCA），来比较和对比几种经典的自旋模型的相行为和相变——正方形和三角形晶格伊辛模型、布吕姆-卡佩尔模型、高度简并的双二次交换自旋-1 伊辛（BSI）模型和二维 XY 模型——并批判性地研究机器学习正在向我们传授什么。我们发现，PCA 的量化主成分不仅允许探索不同的相和对称破缺，而且可以区分相变类型和定位临界点。我们表明，相应的权向量具有明确的物理解释，在三角形反铁磁体等受挫模型中尤为有趣，它们可以指向初始的有序。与其他研究充分的模型不同，BSI 模型的性质知之甚少。我们使用 PCA 和传统的蒙特卡罗分析，证明 BSI 模型没有相变和宏观基态简并。从原始的自旋构型中无法捕捉到 BSI 模型（XY 模型）中的“电荷”相关性（涡度），这表明 PCA 存在一些局限性。最后，我们采用了一种非线性的无监督机器学习过程，即“自动编码器方法”，并证明它也可以被训练来捕捉相变和临界点。
