Automatic Learning of Topological Phase Boundaries

TL;DR

This paper introduces a hyperparameter-free heuristic for diffusion map-based machine learning of topological phase boundaries, enabling automatic and accurate phase diagram generation in various physical models.

Contribution

A novel heuristic method that eliminates the need for hyperparameter tuning in diffusion map analysis of topological phase transitions.

Findings

Successfully applied to three physical models: Haldane, SSH, and quantum ring.

Generated accurate phase diagrams without human intervention.

Demonstrated robustness and effectiveness of the heuristic.

Abstract

Topological phase transitions, which do not adhere to Landau's phenomenological model (i.e. a spontaneous symmetry breaking process and vanishing local order parameters) have been actively researched in condensed matter physics. Machine learning of topological phase transitions has generally proved difficult due to the global nature of the topological indices. Only recently has the method of diffusion maps been shown to be effective at identifying changes in topological order. However, previous diffusion map results required adjustments of two hyperparameters: a data length-scale and the number of phase boundaries. In this article we introduce a heuristic that requires no such tuning. This heuristic allows computer programs to locate appropriate hyperparameters without user input. We demonstrate this method's efficacy by drawing remarkably accurate phase diagrams in three physical models: the Haldane model of graphene, a generalization of the Su-Schreiffer-Haeger (SSH) model, and a model for a quantum ring with tunnel junctions. These diagrams are drawn, without human intervention, from a supplied range of model parameters.