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

TL;DR

This work tackles the challenge of learning the Boltzmann distribution for a continuous-spin system—the 2D XY model—by employing a manifold-aware score-based diffusion framework that operates directly on the torus $\mathcal{M}=\mathbb{T}^{L^2}$. By using Wrapped Normal perturbations, circular padding, and a torus-respecting input representation, the method learns the Boltzmann score $\nabla_x \log p(x)$ with high fidelity and reproduces key thermodynamic and topological signatures, including the BKT transition and the helicity modulus jump. A two-stage sampling pipeline (PF-ODE followed by Metropolis-adjusted Langevin polish) is essential to accurately recover higher-order quantities such as the heat capacity across lattice sizes, achieving zero-shot generalization to unseen lattices. The framework is proposed as a general, geometry-respecting approach for continuous-spin systems, with potential extensions to more complex order parameters and other manifolds, reducing reliance on domain-specific feature engineering.

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.

Figures (12)

• Figure 1: Comparison of Score Norm Scaling. (a) In the standard Euclidean embedding, the score magnitude (red) explodes as $\mathcal{O}(1/\sigma)$ as $\sigma \to 0$. (b) In the proposed manifold-aware framework, the score magnitude (blue) remains $\mathcal{O}(1)$ and bounded.
• Figure 2: Specific Heat Capacity ($C_v$) vs. Temperature. The Manifold-Aware model (green) reproduces the ground truth curve (black) and the heat capacity peak. The Manifold-Agnostic model (light blue) increases significantly at low temperatures.
• Figure 3: Generated Spin Configurations. (a) At $T=0.8$, the model generates bound vortex-antivortex pairs. (b) Above the critical temperature ($T=1.3$), the model correctly generates free vortices.
• Figure 4: Helicity Modulus ($\Upsilon$) vs. Temperature. The Manifold-Aware model (green) accurately captures the stiffness jump at the critical temperature. The Manifold-Agnostic model (light blue) shows a lower critical temperature estimate.
• Figure 5: Score Analysis. Comparison of learned score components (solid) vs. theoretical values (dashed): (a.) Manifold-agnostic training of diffusion model, (b.) Manifold-aware training (our approach). This analysis has been done using a lattice at temperature$(T)=0.1$