Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology

Abstract

We use persistent homology and persistence images as an observable of three different variants of the two-dimensional XY model in order to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and k-nearest neighbours models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.

Emergent strongly coupled ultraviolet fixed point in four dimensions with 8 Kähler-Dirac fermions

“…In the 8-flavor system I have concentrated on fitting the leading terms only. More sophisticated curve collapse fitting, better statistics, and larger volumes might allow the determination of the exponent ν in the future [60].…”

Persistent Homology of $\mathbb{Z}_2$ Gauge Theories

“…Ref. [16], which studies various XY models, was the first to show that accurate estimates for critical exponents of the correlation length can be obtained from finite-scaling analysis of persistent homology observables.…”

“…[15], phase transitions in the mean-field XY model and classical φ 4 model were detected by directly computing the persistent homology of the configuration space. A more recent perspective promotes persistent homology as an observable [16]. Here, configurations are sampled from a physical model, and persistence diagrams of each of these configurations are computed.…”

“…In Refs. [16,19], persistence diagrams are instead transformed into persistence images, and classified using logistic regression and k-nearest neighbours methods. Ref.…”