Geometric entanglement in topologically ordered states

Orus, Roman, Wei, Tzu-Chieh, Buerschaper, Oliver and Nest, Maarten Van den (2014) Geometric entanglement in topologically ordered states. New Journal of Physics, 16 . doi:10.1088/1367-2630/16/1/013015

Author Orus, Roman
Wei, Tzu-Chieh
Buerschaper, Oliver
Nest, Maarten Van den
Title Geometric entanglement in topologically ordered states
Journal name New Journal of Physics   Check publisher's open access policy
ISSN 1367-2630
Publication date 2014-01-01
Year available 2014
Sub-type Article (original research)
DOI 10.1088/1367-2630/16/1/013015
Open Access Status DOI
Volume 16
Total pages 35
Place of publication Bristol, United Kingdom
Publisher Institute of Physics Publishing
Language eng
Formatted abstract
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, colour 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 characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 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 analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. 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. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models.
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status UQ
Additional Notes Article number 013015.

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
Official 2015 Collection
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Citation counts: TR Web of Science Citation Count  Cited 11 times in Thomson Reuters Web of Science Article | Citations
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