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

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.

💡 Research Summary

This paper introduces a manifold‑aware score‑based generative modeling framework (MASGM) for the two‑dimensional XY model, a prototypical continuous‑spin system whose configuration space is a high‑dimensional torus (T^{L²}). The authors first demonstrate that conventional score‑based generative models (SGMs) trained in Euclidean space suffer from a fundamental mismatch: Gaussian noise pushes samples off the torus, causing the normal component of the score to diverge as 1/σ when the noise scale σ → 0. Consequently, the network spends a large portion of its capacity learning the manifold constraint rather than the true Boltzmann gradient ∇_θ log p_T(θ).

To resolve this, the paper proposes three key technical innovations. (1) A forward diffusion process based on the Wrapped Normal distribution, which respects the 2π periodicity of each spin angle and keeps all noisy intermediates on the torus. (2) A conditional Noise‑Conditional Score Network (NCSN) that takes both the noise level (time t) and the physical temperature T as inputs, thereby learning the temperature‑dependent Boltzmann score directly. (3) A U‑Net architecture equipped with circular padding and Riemannian‑metric‑aware layers, ensuring that convolutional operations honor periodic boundary conditions intrinsic to the XY lattice.

Training proceeds on a dataset generated by overdamped Langevin dynamics for a 64 × 64 lattice (4096 spins) across a temperature range that spans the Berezinskii‑Kosterlitz‑Thouless (BKT) transition. After learning the score, sampling is performed via an ODE‑based reverse diffusion, followed by a Metropolis‑Adjusted Langevin Algorithm (MALA) step that uses the learned score as a proposal drift and corrects any residual bias through Metropolis acceptance. This hybrid sampling scheme yields samples that faithfully follow the target Boltzmann distribution without requiring any post‑hoc reweighting.

Empirical evaluation focuses on four aspects: (i) direct comparison of the learned score with the analytically computed Boltzmann score, showing a >30 % reduction in L2 error relative to a Euclidean baseline; (ii) reconstruction of the helicity modulus Υ(T), whose finite‑size scaling curve matches the theoretical 2/π T line and accurately locates the BKT transition temperature (T_{BKT}≈0.893±0.012); (iii) reproduction of the specific heat C_v(T), where the peak position and height agree with exact Monte‑Carlo results within 2 %; and (iv) zero‑shot generalization to lattices of different sizes (32 × 32, 128 × 128) without retraining, with both Υ(T) and C_v(T) scaling correctly according to known finite‑size formulas.

The authors argue that manifold‑aware training eliminates the need for ad‑hoc regularization terms previously introduced to improve second‑moment observables. By embedding the periodic geometry directly into the diffusion process and network architecture, the model learns the true physical score and consequently reproduces higher‑order thermodynamic quantities automatically.

Beyond the XY model, the methodology is readily extensible to other continuous‑spin manifolds such as the Heisenberg S², O(N) vector models, or even gauge‑field configurations where the underlying space is a product of compact Lie groups. The paper concludes that manifold‑aware diffusion models constitute a principled, generalizable tool for many‑body physics, capable of capturing topological phase transitions, thermodynamic scaling, and system‑size generalization in a single training run. Future work is suggested on incorporating long‑range interactions, external fields, and quantum fluctuations, thereby broadening the impact of score‑based generative modeling across condensed‑matter and statistical‑physics research.