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Topological Data Analysis and the Berezinskii–Kosterlitz–Thouless Transition: A Cautionary Cross-Validation Study

Luiz PUODZIUS

Published Aug 5, 2026• 0 citations
Publisher

Abstract

Topological data analysis (TDA), particularly persistent homology, has been proposed as a tool for detecting topological phase transitions in systems lacking conventional order parameters, such as the Berezinskii–Kosterlitz–Thouless (BKT) transition in the 2D XY model. We subject this proposal to rigorous empirical scrutiny, identifying and correcting two compounding numerical artifacts. First, allowing the spin-angle field to undergo an unbounded random walk during Metropolis updates introduces artificial scalar gradients that corrupt the sublevel-set filtration; this is corrected by explicitly wrapping the phase to (-π, π]. Second, and more critically, we identify that single-snapshot measurements at low temperatures capture transient, non-equilibrium vortex populations, systematically inflating the vortex baseline and distorting the inferred onset temperature and dynamic range. When both artifacts are corrected through proper measurement-phase averaging (30 snapshots per seed, n=15 independent seeds, L=32, 19-point temperature sweep over T∈[0.60,1.50]), the TDA observable is revealed to be not merely a weaker proxy for the BKT transition, but one that lacks the transition's defining signature entirely. The corrected analysis reverses the sign of our two principal quantitative claims. The onset temperatures, previously reported as statistically indistinguishable (ΔT₀ ≈ 0.01±0.05), are in fact separated by a 7σ gap: vortex counting detects the transition significantly earlier than TDA (T₀^Vortex = 1.196±0.002 vs. T₀^TDA = 1.294±0.014). The parametric curvature of TDA relative to vortex density, previously reported as statistically insignificant (1.62σ), is in fact a 6.3σ effect (+0.034±0.005). The dynamic-range disparity is widened from the originally reported ~75× vs. ~2.4× to ~228× vs. ~1.66×. We conclude that this specific TDA observable does not detect the BKT transition and cannot serve as a viable diagnostic for topological defect unbinding. These results constitute a definitive negative benchmark for future TDA applications in statistical mechanics.

DOI
10.5281/zenodo.21800575

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Topological data analysis (TDA), particularly persistent homology, has been proposed as a tool for detecting topological phase transitions in systems lacking conventional order parameters, such as the Berezinskii–Kosterlitz–Thouless (BKT) transition in the 2D XY model. We subject this proposal to rigorous empirical scrutiny, identifying and correcting two compounding numerical artifacts. First, allowing the spin-angle field to undergo an unbounded random walk during Metropolis updates introduces artificial scalar gradients that corrupt the sublevel-set filtration; this is corrected by explicitly wrapping the phase to (-π, π]. Second, and more critically, we identify that single-snapshot measurements at low temperatures capture transient, non-equilibrium vortex populations, systematically inflating the vortex baseline and distorting the inferred onset temperature and dynamic range. When both artifacts are corrected through proper measurement-phase averaging (30 snapshots per seed, n=15 independent seeds, L=32, 19-point temperature sweep over T∈[0.60,1.50]), the TDA observable is revealed to be not merely a weaker proxy for the BKT transition, but one that lacks the transition's defining signature entirely. The corrected analysis reverses the sign of our two principal quantitative claims. The onset temperatures, previously reported as statistically indistinguishable (ΔT₀ ≈ 0.01±0.05), are in fact separated by a 7σ gap: vortex counting detects the transition significantly earlier than TDA (T₀^Vortex = 1.196±0.002 vs. T₀^TDA = 1.294±0.014). The parametric curvature of TDA relative to vortex density, previously reported as statistically insignificant (1.62σ), is in fact a 6.3σ effect (+0.034±0.005). The dynamic-range disparity is widened from the originally reported ~75× vs. ~2.4× to ~228× vs. ~1.66×. We conclude that this specific TDA observable does not detect the BKT transition and cannot serve as a viable diagnostic for topological defect unbinding. These results constitute a definitive negative benchmark for future TDA applications in statistical mechanics.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is an archetypal example of a topological phase transition, which is driven by the proliferation of vortices. In this Letter, we analyze the persistence of the BKT transition in the XY model under the influence of long-range algebraically decaying interactions of the form ∼1/r^{2+σ}. The model hosts a magnetized low temperature phase for sufficiently small σ. Crucially, in the presence of long-range interactions, spin waves renormalize the interaction between vortices, which stabilizes the BKT transition. As a result, we find that there is no direct transition from the magnetized to the disordered phase and that the BKT transition persists for arbitrary long-range exponents, which is distinct from previous results. The key methodological advance is the inclusion of coupling between spin wave and topological excitations. We do this by developing Landau-Peierls-type arguments and using higher-order renormalization group calculations, obtaining compatible results. We emphasize that Landau-Peierls-type arguments are a powerful tool for analyzing continuous spin models. We discuss the relevance of our findings for current Rydberg atom experiments, and highlight the importance of long-range couplings for other types of topological defects.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature T_{BKT}. In this Letter, we consider the effect of long-range decaying couplings ∼r^{-2-σ} on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features-for 7/4<σ<2-a quasiordered phase in a finite temperature range T_{c}<T<T_{BKT}, which occurs between a symmetry broken phase for T<T_{c} and a disordered phase for T>T_{BKT}. The transition temperature T_{c} displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems.
We consider the 2d XY Model with topological lattice actions, which are invariant against small deformations of the field configuration. These actions constrain the angle between neighbouring spins by an upper bound, or they explicitly suppress vortices (and anti-vortices). Although topological actions do not have a classical limit, they still lead to the universal behaviour of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition — at least up to moderate vortex suppression. In the massive phase, the analytically known Step Scaling Function (SSF) is reproduced in numerical simulations. However, deviations from the expected universal behaviour of the lattice artifacts are observed. In the massless phase, the BKT value of the critical exponent ηc is confirmed. Hence, even though for some topological actions vortices cost zero energy, they still drive the standard BKT transition. In addition we identify a vortex-free transition point, which deviates from the BKT behaviour.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is a typical topological phase transition defined between binding and unbinding states of vortices and antivortices, which is not accompanied by spontaneous symmetry breaking. It is known that the BKT transition is difficult to detect from thermodynamic quantities such as specific heat and magnetic susceptibility because of the absence of anomaly in free energy and significant finite-size effects. Therefore, methods based on statistical mechanics which are commonly used to discuss phase transitions cannot be easily applied to the BKT transition. In recent years, several attempts to detect the BKT transition using machine-learning methods based on image recognition techniques have been reported. However, it has turned out that the detection is difficult even for machine learning methods because of the absence of trivial order parameters and symmetry breaking. Most of the methods proposed so far require prior knowledge about the models and/or preprocessing of input data for feature engineering, which is problematic in terms of the general applicability. In this article, we introduce recent development of the machine-learning methods to detect the BKT transitions in several spin models. Specifically, we demonstrate the success of two new methods named temperature-identification method and phase-classification method for detecting the BKT transitions in the q-state clock model and the XXZ model. This progress is expected to sublimate the machine-learning-based study of spin models for exploring new physics beyond simple benchmark test.
The 2D XY model exhibits an essential phase transition, which was predicted long ago—by Berezinskii, Kosterlitz and Thouless (BKT)—to be driven by the (un)binding of vortex–anti-vortex pairs. This transition has been confirmed for the standard lattice action, and for actions with distinct couplings, in agreement with universality. Here we study a highly unconventional formulation of this model, which belongs to the class of topological lattice actions: it does not have any couplings at all, but just a constraint for the relative angles between nearest neighbour spins. By means of dynamical boundary conditions we measure the helicity modulus ϒ, which shows that this formulation performs a BKT phase transition as well. Its finite size effects are amazingly mild, in contrast to other lattice actions. This provides one of the most precise numerical confirmations ever of a BKT transition in this model. On the other hand, up to the lattice sizes that we explored, there are deviations from the spin wave approximation, for instance for the Binder cumulant U 4 and for the leading finite size correction to ϒ. Finally we observe that the (un)binding mechanism follows the usual pattern, although free vortices do not require any energy in this formulation. Due to this observation, one should reconsider an aspect of the established picture, which estimates the critical temperature based on this energy requirement.
The Berezinskii-Kosterlitz-Thouless (BKT) phase transition drives the unbinding of topological defects in many two-dimensional systems. In the two-dimensional Coulomb gas, it corresponds to an insulator-conductor transition driven by charge deconfinement. We investigate the global topological properties of this transition, both analytically and by numerical simulation, using a lattice-field description of the two-dimensional Coulomb gas on a torus. The BKT transition is shown to be an ergodicity breaking between the topological sectors of the electric field, which implies a definition of topological order in terms of broken ergodicity. The breakdown of local topological order at the BKT transition leads to the excitation of global topological defects in the electric field, corresponding to different topological sectors. The quantized nature of these classical excitations, and their strict suppression by ergodicity breaking in the low-temperature phase, afford striking global signatures of topological-sector fluctuations at the BKT transition. We discuss how these signatures could be detected in experiments on, for example, magnetic films and cold-atom systems.