Diffusion Models for SU(2) Lattice Gauge Theory in Two Dimensions

TL;DR

This paper demonstrates the application of score-based diffusion models to generate SU(2) lattice gauge configurations in two dimensions, achieving accurate results and flexibility across parameters and lattice sizes, advancing non-Abelian gauge theory simulations.

Contribution

It introduces a novel diffusion model approach for SU(2) lattice gauge theories, handling manifold structure and enabling physics-conditioned sampling without retraining.

Findings

Accurately reproduces plaquette and Wilson action density within small biases

Generates configurations at different couplings without retraining

Works across various lattice sizes with minimal bias

Abstract

We apply score-based diffusion models to two-dimensional SU(2) lattice pure gauge theory with the Wilson action, extending recent work on U(1) gauge theories. The SU(2) manifold structure is handled through a quaternion parameterization. The model is trained on 10,000 configurations generated via Hybrid Monte Carlo at a fixed coupling $\beta_0= 2.0$ on an $8\times 8$ lattice, augmented to 20,000 samples via random gauge transformations. Through physics-conditioned sampling exploiting the linear $\beta$-dependence of the score function, we generate configurations at different values of the coupling without retraining; through the fully convolutional U-Net architecture with periodic boundary conditions, we generate configurations on lattices of different spatial extents. We validate our approach by comparing the average plaquette and Wilson action density against exact analytical predictions. At the training lattice size ($8\times 8$), the model reproduces the exact plaquette with biases $|\Delta| \leq 0.001$ for $\beta \in [1.5, 2.5]$ and $|\Delta| < 0.06$ across $\beta \in [1, 4]$. For lattices sharing the training extent $L=8$ in at least one direction, biases remain below $\sim 0.003$ for $\beta \in [1.5, 2.5]$, with larger deviations at higher couplings. This work demonstrates that diffusion models are a promising tool for non-Abelian gauge field generation and motivates further investigation toward higher-dimensional theories.