Physics-Conditioned Diffusion Models for Lattice Gauge Theory

TL;DR

This work introduces physics-conditioned diffusion models to sample lattice gauge configurations by embedding stochastic quantization into the sampling process. A diffusion model trained on a 2D U(1) gauge theory at a small inverse coupling $\beta_0$ extrapolates to larger $\beta$ and to larger lattices without retraining, while preserving exactness through a Metropolis-adjusted annealed Langevin scheme. The approach yields improved sampling of topological observables compared with Hybrid Monte Carlo and Langevin dynamics, effectively mitigating topological freezing in the frozen regime. The method shows strong cross-volume generalization and paves the way for applying diffusion-based samplers to more complex gauge theories, including non-Abelian cases and fermionic systems, with potential impact on lattice QCD computations.

Abstract

We develop diffusion models for simulating lattice gauge theories, where stochastic quantization is explicitly incorporated as a physical condition for sampling. We demonstrate the applicability of this novel sampler to U(1) gauge theory in two spacetime dimensions and find that a model trained at a small inverse coupling constant can be extrapolated to larger inverse coupling regions without encountering the topological freezing problem. Additionally, the trained model can be employed to sample configurations on different lattice sizes without requiring further training. The exactness of the generated samples is ensured by incorporating Metropolis-adjusted Langevin dynamics into the generation process. Furthermore, we demonstrate that this approach enables more efficient sampling of topological quantities compared to traditional algorithms such as Hybrid Monte Carlo and Langevin simulations.

Figures (10)

• Figure 1: Monte Carlo evolution of the topological charge $Q$ on multi-Markov chains after thermalization with $L=16$ and $\beta=7$.
• Figure 2: The forward diffusion process gradually adds noise and the reverse denoising process tries to remove noise. The two stochastic processes are described by two stochastic differential equations. The target distribution is typically unknown but its log derivative is learned from the training data.
• Figure 3: U-Net architecture. The gray lines with arrows denote skip connections between the encoding and decoding paths. In this study, the default input shape is specified as $(\cdot,2,16,16)$, where the first dimension represents the batch size, the second corresponds to the channel dimension, and the last two denote the spatial dimensions of the field configuration.
• Figure 4: Metropolis-adjusted annealed Langevin sampler. The evolution along the horizontal axis represents the diffusion model running backwards in time $t$ from the naive distribution $\mathcal{N}({\bf0, I})$ to the target physical distribution. The evolution along the vertical axis corresponds to the annealed Langevin dynamics with time $\tau$ and a fixed drift term, where the first dashed arrows indicate the copy operation. To distinguish them, we use a different notation to label the field variables.
• Figure 5: Histograms of the Wilson loop (left) and the topological charge (right) at $\beta = 1, L=16$ from the testing data-set (HMC) and from the trained diffusion model (DM).