Spin exchange-enabled quantum simulator for large-scale non-Abelian gauge theories

A central requirement for the faithful implementation of large-scale lattice gauge theories (LGTs) on quantum simulators is the protection of the underlying gauge symmetry. Recent advancements in the experimental realizations of large-scale LGTs have been impressive, albeit mostly restricted to Abelian gauge groups. Guided by this requirement for gauge protection, we propose an experimentally feasible approach to implement large-scale non-Abelian $\mathrm{SU}(N)$ and $\mathrm{U}(N)$ LGTs with dynamical matter in $d+1$D, enabled by two-body spin-exchange interactions realizing local emergent gauge-symmetry stabilizer terms. We present two concrete proposals for $2+1$D $\mathrm{SU}(2)$ and $\mathrm{U}(2)$ LGTs, including dynamical bosonic matter and induced plaquette terms, that can be readily implemented in current ultracold-molecule and next-generation ultracold-atom platforms. We provide numerical benchmarks showcasing experimentally accessible dynamics, and demonstrate the stability of the underlying non-Abelian gauge invariance. We develop a method to obtain the effective gauge-invariant model featuring the relevant magnetic plaquette and minimal gauge-matter coupling terms. Our approach paves the way towards near-term realizations of large-scale non-Abelian quantum link models in analog quantum simulators.