Department of Physics and Astronomy, San José State University, San José, California 95192, USA.
Phys Rev E. 2018 Jan;97(1-1):013306. doi: 10.1103/PhysRevE.97.013306.
We employ several unsupervised machine learning techniques, including autoencoders, random trees embedding, and t-distributed stochastic neighboring ensemble (t-SNE), to reduce the dimensionality of, and therefore classify, raw (auxiliary) spin configurations generated, through Monte Carlo simulations of small clusters, for the Ising and Fermi-Hubbard models at finite temperatures. Results from a convolutional autoencoder for the three-dimensional Ising model can be shown to produce the magnetization and the susceptibility as a function of temperature with a high degree of accuracy. Quantum fluctuations distort this picture and prevent us from making such connections between the output of the autoencoder and physical observables for the Hubbard model. However, we are able to define an indicator based on the output of the t-SNE algorithm that shows a near perfect agreement with the antiferromagnetic structure factor of the model in two and three spatial dimensions in the weak-coupling regime. t-SNE also predicts a transition to the canted antiferromagnetic phase for the three-dimensional model when a strong magnetic field is present. We show that these techniques cannot be expected to work away from half filling when the "sign problem" in quantum Monte Carlo simulations is present.
我们采用了几种无监督机器学习技术，包括自动编码器、随机树嵌入和 t 分布随机近邻集成（t-SNE），以降低通过小团簇的蒙特卡罗模拟生成的原始（辅助）自旋构型的维数，并对有限温度下的伊辛和费米-哈伯德模型进行分类。可以证明，三维伊辛模型的卷积自动编码器的结果可以高精度地生成磁化强度和磁化率作为温度的函数。量子涨落扭曲了这一图景，使我们无法在自动编码器的输出和哈伯德模型的物理可观测量之间建立这种联系。然而，我们能够基于 t-SNE 算法的输出定义一个指标，该指标在弱耦合情况下与模型在二维和三维空间中的反铁磁结构因子非常吻合。t-SNE 还预测当存在强磁场时，三维模型将过渡到倾斜反铁磁相。我们表明，当量子蒙特卡罗模拟中存在“符号问题”时，这些技术不能期望在远离半填充的情况下发挥作用。
