Quasi-particle Statistics and Braiding from Ground State Entanglement

Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the Topological Entanglement Entropy (TEE) on partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasi-particle excitations of the topological phase. The transformation of these states allows for a determination of the modular S and U matrices which encode quasi-particle properties. We demonstrate our method by extracting the modular S matrix of an SU(2) spin symmetric chiral spin liquid phase using a Monte Carlo scheme to calculate TEE, and prove that the quasi-particles obey semionic statistics. This method offers a route to a nearly complete determination of the topological order in certain cases.