Generalizable Equivariant Diffusion Models for Non-Abelian Lattice Gauge Theory

TL;DR

This work tackles the sampling bottleneck in non-Abelian lattice gauge theory by introducing gauge-equivariant diffusion models built on lattice gauge equivariant CNNs (L-CNNs) and a score-based diffusion framework. It employs Metropolis-adjusted annealed Langevin dynamics (MAALA) with score rescaling to enable reverse sampling at arbitrary couplings and lattice sizes, trained on a single ensemble (2D $U(2)$ and $SU(2)$ cases). The approach yields accurate Wilson loops of various sizes and topological susceptibility measurements across a broad range of couplings up to about $14$, including out-of-domain extrapolations, while maintaining reasonable Metropolis acceptance rates and reducing topological freezing relative to stochastic quantization. Results indicate potential speedups over traditional HMC, with future directions toward 4D $SU(3)$ QCD, fermionic extensions, and fixed-point action applications. $S_E$ and observables are connected to the traditional lattice-QCD framework through gauge-invariant formulations, and the method generalizes to higher dimensions and other gauge groups.

Abstract

We demonstrate that gauge equivariant diffusion models can accurately model the physics of non-Abelian lattice gauge theory using the Metropolis-adjusted annealed Langevin algorithm (MAALA), as exemplified by computations in two-dimensional U(2) and SU(2) gauge theories. Our network architecture is based on lattice gauge equivariant convolutional neural networks (L-CNNs), which respect local and global symmetries on the lattice. Models are trained on a single ensemble generated using a traditional Monte Carlo method. By studying Wilson loops of various size as well as the topological susceptibility, we find that the diffusion approach generalizes remarkably well to larger inverse couplings and lattice sizes with negligible loss of accuracy while retaining moderately high acceptance rates.

Figures (5)

• Figure 1: Schematic overview of the diffusion and generative process using a lattice of gauge links (edges) forming plaquettes (colorful faces). Training samples from the target distribution at time $t=0$ undergo diffusion until they resemble uncorrelated noise at later time $T$. We train an L-CNN with diffused samples at a given inverse coupling $\beta_0$ and lattice size $L_0$. After training, we solve the reversed diffusion process using the Metropolis-adjusted annealed Langevin algorithm (MAALA) to generate new independent samples at arbitrary $\beta$ and $L$.
• Figure 2: Comparison of the learned generative (backward) process with the forward diffusion process defined in Eq. \ref{['eq:diffused_link']} for $\beta=2$ on a $L=16$ lattice. Close agreement (up to statistical precision) between the DM and the analytical diffusion process indicates that our equivariant DMs can successfully perform denoising. At $t=1$, the forward process leads to the strong-coupling limit where the plaquette vanishes and $\chi_\mathrm{top} = 1/12$.
• Figure 3: Predictions for two-dimensional $U(2)$ gauge theory on an $L=16$ lattice using a DM trained at $\beta=2$ (indicated in bold), $L=16$. The top panel shows predictions for $n \times n$ Wilson loops (circles with error bars) compared to their analytic values (solid lines) as a function of the inverse coupling $\beta$. The lower panel shows the topological susceptibility $\chi_\mathrm{top}$. Absolute errors are presented underneath. Significant deviations only occur for the largest $\beta$ values.
• Figure 4: Average Metropolis acceptance rates for extrapolations to different $\beta$ and lattice sizes. DMs are trained at $\beta=2$ and $L=16$.
• Figure 5: Evolution of the topological charge during the generative process of one field configuration for $\beta=14$ on a $L=16$ lattice. The top panel shows the evolution for the MAALA along (reverse) diffusion time $t$. In the bottom panel, the evolution along a Langevin trajectory using stochastic quantization with the Wilson action is shown. To enable a fair comparison, we use the same total number of Langevin steps and the same step size for both processes. While the Langevin trajectory always remains trapped in a few topological sectors for extended periods, MAALA initially exhibits frequent transitions.