Unsupervised machine learning account of magnetic transitions in the Hubbard model

TL;DR

This paper explores the use of unsupervised machine learning techniques to classify magnetic phases in the Hubbard model and related spin systems, successfully identifying phase transitions and magnetic order parameters.

Contribution

It demonstrates that autoencoders and t-SNE can effectively classify magnetic phases and detect phase transitions in classical and quantum spin models, even under quantum fluctuations.

Findings

Autoencoders accurately reproduce magnetization and susceptibility in the 3D Ising model.

t-SNE-based indicators align with the antiferromagnetic structure factor in the Hubbard model.

Machine learning techniques struggle away from half filling due to the sign problem.

Abstract

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.