Graph Neural Networks in the Wilson Loop Representation of Abelian Lattice Gauge Theories
This work provides a physically grounded framework for learning and simulating Abelian lattice gauge systems, which are central to condensed matter and quantum platforms.
The authors introduce a gauge-invariant graph neural network for Abelian lattice gauge theories that uses Wilson loops as inputs, achieving accurate predictions of global and spatially resolved observables in Z2 and U(1) models, and serving as an efficient surrogate for semiclassical dynamics in U(1) quantum link models without repeated fermionic diagonalization.
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.