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

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 $β_0= 2.0$ on an $8\times 8$ lattice, augmented to 20,000 samples via random gauge transformations. Through physics-conditioned sampling exploiting the linear $β$-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 against exact analytical predictions. At the training lattice size ($8\times 8$), the model reproduces the exact plaquette with biases $|Δ| \leq 0.001$ for $β\in [1.5, 2.5]$ and $|Δ| < 0.06$ across $β\in [1, 4]$. For lattices sharing the training extent $L=8$ in at least one direction, biases remain below $\sim 0.003$ for $β\in [1.5, 2.5]$, 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.