Geometric Entanglement in Topologically Ordered States

Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically-ordered systems such as the toric code, double semion, color code, and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks, and a contribution quantifying the underlying pattern of long-range entanglement of the topologically-ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterisation of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the 2D color code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments. Motivated by this, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian symmetry, and which we use to analyse quantum double models. Furthermore, we also analyse the robustness of the topological contribution using renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of our results rely on the ability to disentangle single sites from the system, which is always possible in our framework. Aditionally, we relate our results to the relative entropy of entanglement in topological systems, and discuss some tensor network numerical approaches that could be used to extract the topological contribution for large systems beyond exactly-solvable models.