Extracting Wilson loops and fractional statistics from a ground state wave function on a disk

An essential aspect of topological phases of matter is the existence of Wilson loop operators which keep the ground state subspace invariant. Here we present and implement an unbiased numerical optimization scheme to systematically find the Wilson loop operators given a single ground state wave function of a gapped, translationally invariant Hamiltonian on a disk. We then show how these Wilson loop operators can be cut and glued through further optimization to give operators that can create, move, and annihilate anyon excitations. We then use these operators to determine the braiding statistics and topological twists of the anyons, yielding a way to fully extract topological order from a single wave function. We apply our method to the ground state of the perturbed toric code and doubled semion models with a Zeeman field that is up to a third of the critical value. From a contemporary perspective, this can be thought of as a machine learning approach to discover emergent 1-form symmetries of a ground state wave function.