Ergodic topology sampling toward the continuum with Stochastic Normalizing Flows

In the Wilson formulation of lattice gauge theory, the fundamental degrees of freedom are group-valued link variables, while the action is a sum over the trace of the plaquettes, the smallest Wilson loops. For generative sampling methods such as normalizing flows, this poses a challenge: the distribution of an individual plaquette is easy to model, but mapping sampled plaquettes to the links is the obstruction. We explore a statistical way around this in two dimensions for the $\mathrm{SU}(N)$ gauge theory with the Wilson plaquette action. The action can be written in terms of holonomy variables, which determine the links once a consistency condition at the four corners of an extended lattice is satisfied. Using a boundary condition that only affects the holonomies, we trade this consistency condition for a conditional sampling problem, which reduces to sampling $X,Y\in\mathrm{SU}(N)$ given the group commutator $Z=XYX^\dagger Y^\dagger$, where $Z\in\mathrm{SU}(N)$ depends on the four corners. Plaquettes are sampled independently, and the resulting configurations carry weights due to the additional conditional sampling. We obtain the density of $Z$, which determines these weights, in closed form for $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$. A normalizing flow models the single-plaquette distribution for $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$; for the group-commutator problem we use a closed-form sampler for $\mathrm{SU}(2)$ and a trained normalizing flow for $\mathrm{SU}(3)$. In our tests on $2\times2$ and $32\times32$ lattices at two couplings each, per-plaquette acceptance rates exceed $99\%$ and the effective sample size of the final weights is moderate, typically above one half.

In this review I summarize how machine learning can be used in lattice gauge theory simulations and what ap-proaches are currently available to improve the sampling of gauge field configurations, with a focus on applications in four-dimensional SU(3) gauge theories. These include approaches based on generative machine-learning models such as (stochastic) normalizing flows and diffusion processes, and an approach based on renormalization group (RG) transformations, more specifically the machine learning of RG-improved gauge actions using gauge-equivariant convolutional neural networks. In particular, I present scaling results for a machine-learned fixed-point action in four-dimensional SU(3) gauge theory towards the continuum limit. The results include observables based on the classically perfect gradient-flow scales, which are free of tree-level lattice artefacts to all orders, and quantities related to the static potential and the deconfinement transition.

We present a first study of a diffusion-based approach to accelerated sampling of the $N_f = 2$ lattice Schwinger model. Our work is inspired by recent and growing successes in developing such generative models for ensemble generation in LFT to overcome the well-known critical slowing down problem. We train a U(1)-equivariant score-based generative model to sample gauge link configurations from the marginal Schwinger model. By computing model likelihoods, we obtain unbiased estimates for observables that closely match those produced by MCMC simulations. We also demonstrate improvement over HMC as measured qualitatively by a reduction in topological freezing near critical parameters.