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arXiv:2602.07548

Capturing the Topological Phase Transition and Thermodynamics of the 2D XY Model via Manifold-Aware Score-Based Generative Modeling

Pratyush Jha

Published Feb 7, 2026• 0 citations
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Abstract

The application of generative modeling to many-body physics offers a promising pathway for analyzing high-dimensional state spaces of spin systems. However, unlike computer vision tasks where visual fidelity suffices, physical systems require the rigorous reproduction of higher-order statistical moments and thermodynamic quantities. While Score-Based Generative Models (SGMs) have emerged as a powerful tool, their standard formulation on Euclidean embedding space is ill-suited for continuous spin systems, where variables inherently reside on a manifold. In this work, we demonstrate that training on the Euclidean space compromises the model's ability to learn the target distribution as it prioritizes to learn the manifold constraints. We address this limitation by proposing the use of Manifold-Aware Score-Based Generative Modeling framework applied to the 64x64 2D XY model (a 4096-dimensional torus). We show that our method estimates the theoretical Boltzmann score with superior precision compared to standard diffusion models. Consequently, we successfully capture the Berezinskii-Kosterlitz Thouless (BKT) phase transition and accurately reproduce second-moment quantities, such as heat capacity without explicit feature engineering. Furthermore, we demonstrate zero-shot generalization to unseen lattice sizes, accurately recovering the physics of variable system scales without retraining. Since this approach bypasses domain-specific feature engineering, it remains intrinsically generalizable to other continuous spin systems.

DOI
10.48550/arxiv.2602.07548

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The application of generative modeling to many-body physics offers a promising pathway for analyzing high-dimensional state spaces of spin systems. However, unlike computer vision tasks where visual fidelity suffices, physical systems require the rigorous reproduction of higher-order statistical moments and thermodynamic quantities. While Score-Based Generative Models (SGMs) have emerged as a powerful tool, their standard formulation on Euclidean embedding space is ill-suited for continuous spin systems, where variables inherently reside on a manifold. In this work, we demonstrate that training on the Euclidean space compromises the model's ability to learn the target distribution as it prioritizes to learn the manifold constraints. We address this limitation by proposing the use of Manifold-Aware Score-Based Generative Modeling framework applied to the 64x64 2D XY model (a 4096-dimensional torus). We show that our method estimates the theoretical Boltzmann score with superior precision compared to standard diffusion models. Consequently, we successfully capture the Berezinskii-Kosterlitz Thouless (BKT) phase transition and accurately reproduce second-moment quantities, such as heat capacity without explicit feature engineering. Furthermore, we demonstrate zero-shot generalization to unseen lattice sizes, accurately recovering the physics of variable system scales without retraining. Since this approach bypasses domain-specific feature engineering, it remains intrinsically generalizable to other continuous spin systems.
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Detecting the topological Kosterlitz-Thouless (KT) transition in the prototypical 2D XY model using unsupervised machine learning methods has long been a challenging problem due to the lack of suitable order parameters. To address this issue, we begin with a conventional real-valued RBM (RBM-xy), which uses exponential conditional probabilities to generate visible units. We then develop a novel real-valued RBM (RBM-CosSin) featuring nonlinear cos/sin activation, whose visible units follow the von Mises distribution. Our findings reveal that RBM-CosSin effectively learns the underlying Boltzmann distribution of 2D XY systems and generate authentic XY configurations that accurately capture both thermodynamics and topological order (vortex). Furthermore, we demonstrate that it is possible to extract phase transition information, including the KT transition, from the weight matrices without relying on prior physics knowledge.
Abstract Machine learning (ML) has become a useful tool for studying phase transitions in statistical systems. For the two-dimensional classical XY model, however, the topological character of the Berezinskii–Kosterlitz–Thouless (BKT) transition and pronounced finite-size effects make it nontrivial to extract reproducible size-dependent pseudo-critical temperatures from configuration data. Existing studies often stop at phase classification, leaving open how standard neural-network outputs can be turned into quantitatively testable observables. Here we develop a ML-assisted framework for the 2D XY model that uses standard network outputs to extract a size-dependent sequence of operational pseudo-critical temperatures T ∗ ( L ) . Specifically, we generate Monte Carlo configurations using embedded cluster updates, train a standard ResNet18 only on samples from the Quasi-ordered Phase and the Disordered Phase, and determine T ∗ ( L ) from bootstrap-averaged probability curves using the 50% crossing criterion. We then analyze the finite-size drift of this temperature sequence using BKT-motivated scaling and compare it with susceptibility-peak temperatures. The selected T ∗ ( L ) values display an overall downward finite-size drift compatible with BKT-motivated behavior and lie within the conventional susceptibility fluctuation regime. We further characterize and assess the extracted scale through threshold-sensitivity tests, window-variation analysis, probability calibration, comparison with a simple-physics baseline, vortex-density measurement, input-encoding comparison, and saliency mapping. These diagnostics reveal that while the network response is dominated by macroscopic physical observables, the extracted T ∗ ( L ) is conditional on representation, augmentation, and model-selection protocols. These results demonstrate that standard classifier outputs can be converted into an operational finite-size temperature scale on a well-characterized benchmark model, while also revealing the dependence of that scale on the adopted inference protocol.
It is a long-standing challenge to accurately and efficiently compute thermodynamic quantities of many-body systems at thermal equilibrium. The conventional methods, e.g., Markov chain Monte Carlo, require many steps to equilibrate. The recently developed deep learning methods can perform direct sampling, but only work at a single trained temperature point and risk biased sampling. Here, we propose a variational method for canonical ensembles with differentiable temperatures, which gives thermodynamic quantities as continuous functions of temperature akin to an analytical solution. The proposed method is a general framework that works with any tractable density generative model. At optimal, the model is theoretically guaranteed to be the unbiased Boltzmann distribution. We validated our method by calculating phase transitions in the Ising and XY models, demonstrating that our direct-sampling simulations are as accurate as Markov chain Monte Carlo methods but more efficient. Moreover, our differentiable free energy aligns closely with the exact one to the second-order derivative, indicating that the variational model captures the subtle thermal transitions at the phase transitions. This functional dependence on external parameters is a fundamental advancement in combining the exceptional fitting ability of deep learning with rigorous physical analysis.
We develop deep autoregressive networks with multi channels to compute many-body systems with continuous spin degrees of freedom directly. As a concrete example, we demonstrate the two-dimensional XY model with the continuous-mixture networks and rediscover the Kosterlitz–Thouless (KT) phase transition on a periodic square lattice. Vortices characterizing the quasi-long range order are accurately detected by the generative model. By learning the microscopic probability distributions from the macroscopic thermal distribution, the networks are trained as an efficient physical sampler which can approximate the free energy and estimate thermodynamic observables unbiasedly with importance sampling. As a more precise evaluation, we compute the helicity modulus to determine the KT transition temperature. Although the training process becomes more time-consuming with larger lattice sizes, the training time remains unchanged around the KT transition temperature. The continuous-mixture autoregressive networks we developed thus can be potentially used to study other many-body systems with continuous degrees of freedom.
We introduce a novel technique for scalable sampling of spin-system states with continuous symmetries using diffusion models. By applying our approach to the XY model, a fundamental continuous-spin model in condensed matter physics, we show that our technique addresses the shortfalls of the Markov chain Monte Carlo (MCMC) in generalization to varying system sizes. More specifically, we show that training a temperature-conditioned diffusion model on smaller-size XY model lattices enables the generation of accurate samples in larger lattice sizes. By tracking physically important observables of the model, such as spin correlations, our experiments demonstrate that diffusion sampling followed by a few MCMC steps reduces the thermalization time by an order of magnitude relative to the standard MCMC with random initialization. Our study provides valuable insight as to how generative models can be used to study continuous-state condensed matter systems at scale.