Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology

• Main Authors: Cole, Alex, Loges, Gregory J, Shiu, Gary
• Format: Journal Article
• Language: English
• Published: 29-09-2020
• Subjects: Mathematics - Algebraic Topology; Physics - Statistical Mechanics
• DOI: 10.48550/arxiv.2009.14231

Phys. Rev. B 104, 104426 (2021) 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.