  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.
