Graph Neural Networks in the Wilson Loop Representation of Abelian Lattice Gauge Theories

TL;DR

This paper presents a gauge-invariant graph neural network architecture for Abelian lattice gauge theories, enabling accurate predictions and efficient simulations while preserving gauge symmetry explicitly.

Contribution

The authors introduce a novel GNN framework that enforces gauge invariance through local inputs like Wilson loops, improving modeling of Abelian lattice gauge systems.

Findings

Accurately predicts global and local observables in $ ext{Z}_2$ and U(1) models.

Serves as an efficient surrogate for semiclassical dynamics in U(1) quantum link models.

Enables stable, scalable time evolution without fermionic diagonalization.

Abstract

Local gauge structures play a central role in a wide range of condensed matter systems and synthetic quantum platforms, where they emerge as effective descriptions of strongly correlated phases and engineered dynamics. We introduce a gauge-invariant graph neural network (GNN) architecture for Abelian lattice gauge models, in which symmetry is enforced explicitly through local gauge-invariant inputs, such as Wilson loops, and preserved throughout message passing, eliminating redundant gauge degrees of freedom while retaining expressive power. We benchmark the approach on both $\mathbb{Z}_2$ and $\mathrm{U}(1)$ lattice gauge models, achieving accurate predictions of global observables and spatially resolved quantities despite the nonlocal correlations induced by gauge-matter coupling. We further demonstrate that the learned model serves as an efficient surrogate for semiclassical dynamics in $\mathrm{U}(1)$ quantum link models, enabling stable and scalable time evolution without repeated fermionic diagonalization, while faithfully reproducing both local dynamics and statistical correlations. These results establish gauge-invariant message passing as a compact and physically grounded framework for learning and simulating Abelian lattice gauge systems.