# Discovering Phases, Phase Transitions and Crossovers through Unsupervised Machine Learning: A critical examination

## TL;DR

This paper critically examines how unsupervised machine learning, especially PCA and autoencoders, can identify phases, phase transitions, and critical points in various classical spin models, revealing both capabilities and limitations.

## Contribution

It demonstrates that PCA can effectively explore phases and symmetry-breaking, interpret physical features, and locate critical points, while also highlighting PCA's limitations in capturing certain correlations.

## Key findings

- PCA identifies phases and symmetry-breaking in classical spin models.
- PCA can distinguish phase transition types and locate critical points.
- Autoencoder methods can also detect phase transitions and critical points.

## Abstract

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-one Ising (BSI) model, and the 2D XY model, and examine critically what machine learning is teaching us. We find that quantified principal components from PCA not only allow exploration of different phases and symmetry-breaking, but 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 `antoencoder method', and demonstrate that it too can be trained to capture phase transitions and critical points.