Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology

TL;DR

This paper introduces a novel approach using persistent homology to identify and interpret phase transitions in lattice spin models, providing a multiscale topological perspective and interpretable order parameters.

Contribution

The paper applies persistent homology and persistence images to characterize phase transitions, offering a new, interpretable topological method for analyzing complex spin models.

Findings

Successfully distinguishes phases using logistic regression on persistence images

Identifies key features like magnetization, frustration, and vortices as relevant for phase transitions

Defines persistence-based critical exponents related to traditional critical exponents

Abstract

We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four two-dimensional lattice spin models: the Ising, square ice, XY, and fully-frustrated XY models. In particular, we use persistent homology, which computes the births and deaths of individual topological features as a coarse-graining scale or sublevel threshold is increased, to summarize multiscale and high-point correlations in a spin configuration. We employ vector representations of this information called persistence images to formulate and perform the statistical task of distinguishing phases. For the models we consider, a simple logistic regression on these images is sufficient to identify the phase transition. Interpretable order parameters are then read from the weights of the regression. This method suffices to identify magnetization, frustration, and vortex-antivortex structure as relevant features for phase transitions in our models. We also define "persistence" critical exponents and study how they are related to those critical exponents usually considered.