Group-Equivariant Diffusion Models for Lattice Field Theory

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 $\mathrm{U}(1)$ lattice field theories. We develop score networks that are equivariant to a range of group transformations, including global ${\mathbb{Z}}_2$ reflections, local $\mathrm{U}(1)$ 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.