Sampling su n gauge theory on 2d lattice from independent plaquettes
Sampling SU(N) gauge theory on a 2D lattice from independent plaquettes via holonomies and corner reweighting
Machine Learning
Summary
Simulating certain physical theories on a grid can be complicated because the pieces involved are linked together in tricky ways. The authors found a way to break down this problem for a special kind of gauge theory into sampling smaller independent parts, called plaquettes, while keeping everything consistent by using mathematical objects called holonomies. This approach simplifies the sampling process and achieves high acceptance rates in their tests, making simulations more feasible in two dimensions for groups like SU(2) and SU(3).
What this means in practice
- •For computational physicists: Simulate nonabelian gauge theories on 2D lattices more efficiently by sampling independent plaquettes with holonomy-based conditional weights.
- •For machine learning engineers: Design normalizing flow models for complicated group-valued distributions involving conditional constraints arising from lattice gauge theories.
Authors
Javad Komijani
Abstract
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.