Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian

Ring exchange is an elementary interaction for modelling unconventional topological matter. Here, we report the observation of four-body ring-exchange interactions and the topological properties of anyonic excitations within an ultracold atom system. A minimum toric-code Hamiltonian, in which the ring exchange is the dominant term, was implemented in disconnected four-spin plaquette arrays formed by two orthogonal superlattices. The ring-exchange interactions were resolved from the dynamical evolutions of the spin orders in each plaquette, matching well with the predicted energy gaps between two anyonic excitations of the spin system. A braiding operation was applied to the spins in the plaquettes and an induced phase 1.00(3)pi in the four-spin state was observed, confirming 1/2 mutual statistics. This work offers new prospects for the quantum simulation of topological phases by engineering many-body interactions.