Identifying topological order through unsupervised machine learning

TL;DR

This paper introduces an unsupervised machine learning method using diffusion maps to identify topological phases directly from raw data, successfully classifying models like the XY model and Ising gauge theory without manual feature engineering.

Contribution

The paper presents a novel unsupervised approach based on diffusion maps for detecting topological order from raw data, bypassing the need for predefined features.

Findings

Successfully classifies XY model topological phases by winding number.

Captures Berezinskii-Kosterlitz-Thouless transition.

Applies to Ising gauge theory with topological order.

Abstract

The Landau description of phase transitions relies on the identification of a local order parameter that indicates the onset of a symmetry-breaking phase. In contrast, topological phase transitions evade this paradigm and, as a result, are harder to identify. Recently, machine learning techniques have been shown to be capable of characterizing topological order in the presence of human supervision. Here, we propose an unsupervised approach based on diffusion maps that learns topological phase transitions from raw data without the need of manual feature engineering. Using bare spin configurations as input, the approach is shown to be capable of classifying samples of the two-dimensional XY model by winding number and capture the Berezinskii-Kosterlitz-Thouless transition. We also demonstrate the success of the approach on the Ising gauge theory, another paradigmatic model with topological order. In addition, a connection between the output of diffusion maps and the eigenstates of a quantum-well Hamiltonian is derived. Topological classification via diffusion maps can therefore enable fully unsupervised studies of exotic phases of matter.