Realization of anyonic Hubbard and lattice gauge models with Rydberg atoms

We propose a scheme to realize a one-dimensional Hubbard model for anyons with tunable statistical exchange phase. The scheme utilizes the density-dependent Peierl's phase in the hopping amplitude of excitations of Rydberg atoms in a zig-zag lattice, as was recently realized experimentally in [1]. The obtained Hamiltonian for hard-core anyons contains nearest-neighbor hopping as well as next nearest neighbor density-dependent hopping that results from the combination of the direct and second-order dipole-dipole exchange interactions between the atoms in the Rydberg $ns$ and $np$ states. We show how the effective anyons in the lattice can be braided to reveal their exotic exchange statistics. As a second application of the same setup, we propose the realization of a lattice gauge theory using the density-dependent second-order hopping of the Rydberg excitations.\\ [1] V. Lienhard \textit{et al.}, \textit{"Realization of a density-dependent Peierls phase in a synthetic, spin-orbit coupled Rydberg system"}, arxiv:2001.10357