Observation of Four-body Ring-exchange Interactions and Anyonic Fractional Statistics

Ring exchange is an elementary interaction for modeling unconventional topological matters which hold promise for efficient quantum information processing. 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 by engineering a Hubbard Hamiltonian that describes atomic spins in disconnected plaquette arrays formed by two orthogonal superlattices. The ring-exchange interactions were resolved from the dynamical evolutions in the spin orders, 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 $\frac{1}{2}$-anynoic statistics. This work represents an essential step towards studying topological matters with many-body systems and the applications in quantum computation and simulation.