Roman Orus, Tzu-Chieh Wei, Oliver Buerschaper, Maarten Van den Nest
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 large block sizes the geometric entanglement is, up to possible subleading corrections, the sum of two contributions: a non-universal bulk contribution obeying a boundary law times the number of blocks, and a universal 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, and constitutes a novel characterization of topological order 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 [H. Bombin, G. Duclos-Cianci, D. Poulin, New J. Phys. 14 (2012) 073048]. Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyze quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Finally, in the Appendix we relate our results to the behavior of the relative entropy of entanglement in topologically-ordered systems.
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http://arxiv.org/abs/1304.1339
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