Unsupervised learning of topological phase diagram using topological data analysis

TL;DR

This paper introduces an unsupervised machine learning method that uses topological data analysis to identify and visualize topological phases in quantum systems by analyzing the shape of wavefunction spaces.

Contribution

It proposes a novel approach combining topological data analysis with volume minimization to classify topological phases without prior labels.

Findings

Successfully distinguishes topologically trivial and nontrivial phases

Provides a convergent algorithm based on geodesic and minimal surface interpretations

Demonstrates effectiveness on various quantum models

Abstract

Topology and machine learning are two actively researched topics not only in condensed matter physics, but also in data science. Here, we propose the use of topological data analysis in unsupervised learning of the topological phase diagrams. This is possible because the quantum distance can capture the shape of the space formed by the Bloch wavefunctions as we sweep over the Brillouin zone. Therefore, if we minimize the volume of the space formed by the wavefunction through a continuous deformation, the wavefunctions will end up forming distinct spaces which depend on the topology of the wavefunctions. Combining this observation with the topological data analysis, which provides tools such as the persistence diagram to capture the topology of the space formed by the wavefunctions, we can cluster together Hamiltonians that give rise to similar persistence diagrams after the deformation. By examining these clusters as well as representative persistence diagrams in the clusters, we can draw the phase diagram as well as distinguish between topologically trivial and nontrivial phases. Our proposal to minimize the volume can be interpreted as finding geodesics in 1D Brillouin zone, and minimal surfaces in 2D and higher-dimensional Brillouin zones. Using this interpretation, we can guarantee the convergence of the minimization under certain conditions, which is an outstanding feature of our algorithm. We demonstrate the working principles of our machine learning algorithm using various models.