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How much of persistent homology is topology? Density decomposition and finite-size scaling of the topological signal in critical spin models

Matthew Loftus

Published Mar 31, 2026• 0 citations
Publisher

Abstract

We introduce a quantitative decomposition that separates the persistent homology (PH) signal into density-driven and genuinely topological components, analogous to complete spatial randomness testing in spatial statistics. Applying this to the 2D Ising (L=16-1024) and Potts q=3 (L=32-128) models, we find H0 persistence is 94-96% density-driven, while H1 has a topological fraction growing with system size via a saturation model (delta-BIC=21 over power law). The raw topological excess Delta = TP_real - TP_shuf exhibits excellent finite-size scaling collapse, with the below-T_c scaling function consistent with G(x) ~ (1+x/x0)^{-(1+beta/nu)} (F-test p=0.996 for Ising gamma=9/8). At L~1024, topological and density contributions to H1 feature count are approximately equal. We identify a density degeneracy in the f_topology metric and show that published claims of topological phase detection are largely explained by density effects.

DOI
10.5281/zenodo.19343927

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We introduce a quantitative decomposition that separates the persistent homology (PH) signal into density-driven and genuinely topological components, analogous to complete spatial randomness testing in spatial statistics. Applying this to the 2D Ising (L=16-1024) and Potts q=3 (L=32-128) models, we find H0 persistence is 94-96% density-driven, while H1 has a topological fraction growing with system size via a saturation model (delta-BIC=21 over power law). The raw topological excess Delta = TP_real - TP_shuf exhibits excellent finite-size scaling collapse, with the below-T_c scaling function consistent with G(x) ~ (1+x/x0)^{-(1+beta/nu)} (F-test p=0.996 for Ising gamma=9/8). At L~1024, topological and density contributions to H1 feature count are approximately equal. We identify a density degeneracy in the f_topology metric and show that published claims of topological phase detection are largely explained by density effects.
Point-cloud persistent homology (PH) -- computing alpha or Rips complexes on spin-position point clouds -- has been widely applied to detect phase transitions in classical spin models since Donato et al. (2016), with subsequent studies attributing the detection to the topological content of the persistence diagram. We ask a simple question that has not been posed: what fraction of the PH signal is genuinely topological? We introduce f_topo, a quantitative decomposition that separates the density-driven and topological contributions to any PH statistic by comparing real spin configurations against density-matched shuffled null models. Across the 2D Ising model (system sizes L = 16-128, ten temperatures) and Potts models (q = 3, 5), we find that H_0 statistics -- total persistence, persistence entropy, feature count -- are 94-100% density-driven (f_topo < 0.07). The density-matched shuffled null detects T_c at the identical location and with comparable peak height as real configurations, showing that density alone is sufficient for phase transition detection. However, H_1 statistics are partially topological: the topological fraction grows with system size as delta(TP_{H_1}) ~ L^{0.53} and follows a finite-size scaling collapse delta(T, L) = L^{0.53} g(tL^{1/nu}) with collapse quality CV = 0.27. The longest persistence bar is strongly topological (f_topo > 1) and scales with the correlation length. A scale-resolved analysis reveals that the topological excess shifts from large-scale to small-scale features as L increases. We propose that the TDA-for-phase-transitions community adopt shuffled null models as standard practice, and that H_1 rather than H_0 statistics be used when genuine topological information is sought.
This thesis motivates and examines the use of methods from topological data analysis in detecting and analysing topological features relevant to models from sta-tistical physics and particle physics.In statistical physics, we use persistent homology as an observable of three dif-ferent variants of the two-dimensional XY model in order to identify relevant topo-logical features and study their relation to the phase transitions undergone by each model. 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 demonstrate its use in detecting topological defects called vortices. By considering the fluctuations in the output of logistic regression and k-nearest neighbours mod-els 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.In particle physics, we investigate the use of persistent homology as a means to detect and quantitatively describe center vortices in SU(2) lattice gauge theory in a gauge-invariant manner. The sensitivity of our method to vortices in the deconfined phase is confirmed by using twisted boundary conditions which inspires the definition of a new phase indicator for the deconfinement phase transition. We also construct a phase indicator without reference to twisted boundary conditions using a k-nearest neighbours classifier. Finite-size scaling analyses of both persistence-based indicators yield accurate estimates of the critical β and critical exponent of correlation length for the deconfinement phase transition. We also use persistent homology to study the stability of vortices under gradient flow and the classification of different vortex surface geometries.
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.
We use persistent homology and persistence images as an observable of three variants of the two-dimensional XY model 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 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 neighbor 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 behavior 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.
We use persistent homology and persistence images as an observable of three variants of the two-dimensional XY model 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 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 neighbor 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 behavior 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.
Topologically ordered phases of matter display a number of unique characteristics, including ground states that can be interpreted as patterns of closed strings in the case of general Z2{\mathbb{Z}}_{2}Z2​ string liquids. In this paper, we consider the problem of detecting and distinguishing closed strings in Ising spin configurations sampled from the classical Z2{\mathbb{Z}}_{2}Z2​ gauge theory. We address this using the framework of persistent homology, which computes the size and frequency of general loop structures in spin configurations via the formation of geometric complexes. Implemented numerically on finite-size lattices, we show that the first Betti number of the Vietoris-Rips complexes achieves a high density at low temperatures in the Z2{\mathbb{Z}}_{2}Z2​ gauge theory. In addition, it displays a clear signal at the finite-temperature deconfinement transition of the three-dimensional theory. We argue that persistent homology should be capable of interpreting prominent loop structures that occur in a variety of systems, making it a useful tool in theoretical and experimental searches for topological order.