Group-Equivariant Diffusion Models for Lattice Field Theory

Abstract: Near the critical point, Markov Chain Monte Carlo (MCMC) simulations of lattice quantum field theories (LQFT) become increasingly inefficient due to critical slowing down. In this work, we investigate score-based symmetry-preserving diffusion models as an alternative strategy to sample two-dimensional $\phi^4$ and ${\rm U}(1)$ lattice field theories. We develop score networks that are equivariant to a range of group transformations, including global $\mathbb{Z}_2$ reflections, local ${\rm U}(1)$ rotations, and periodic translations $\mathbb{T}$. The score networks are trained using an augmented training scheme, which significantly improves sample quality in the simulated field theories. We also demonstrate empirically that our symmetry-aware models outperform generic score networks in sample quality, expressivity, and effective sample size.