Abstract
Topological entanglement entropy (TEE), the sub-leading term in the entanglement entropy of topological order, is the direct evidence of the long-range entanglement. While effective in characterizing topological orders on closed manifolds, TEE is model-dependent when entanglement cuts intersect with physical gapped boundaries. In this paper, we study the origin of this model-dependence by introducing a model-independent picture of partitioning the topological orders with gapped boundaries. In our picture, the entanglement boundaries (EBs), i.e. the virtual boundaries of each subsystem induced by the entanglement cuts, are assumed to be gapped boundaries with boundary defects. At this model-independent stage, there are two choices one has to make manually in defining the bi-partition: the boundary condition on the EBs, and the coherence between certain boundary states. We show that TEE appears because of a constraint on the defect configurations on the EBs, which is choice-dependent in the cases where the EBs touch gapped boundaries. This choice-dependence is known as the ambiguity in entanglement entropy. Different models intrinsically employ different choices, rendering TEE model-dependent. For D(ℤ2) topological order, the ambiguity can be fully characterized by two parameters that respectively quantifies the EB condition and the coherence. In particular, calculations compatible with the folding trick naturally choose EB conditions that respect electric-magnetic duality and set specific parameter values.
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Acknowledgments
YL thanks Ling-Yan Hung and Yidun Wan for insightful comments, thanks Yanyan Chen for providing intuitions based on the calculations in the lattice models, and thanks Hongyu Wang and Yu Zhao for helpful discussions. The author appreciates his supervisor Prof. Yidun Wan for supporting him via the General Program of Science and Technology of Shanghai No. 21ZR1406700, and Shanghai Municipal Science and Technology Major Project (Grant No.2019SHZDZX01).
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Li, Y. Characterizing the ambiguity in topological entanglement entropy. J. High Energ. Phys. 2024, 74 (2024). https://doi.org/10.1007/JHEP03(2024)074
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DOI: https://doi.org/10.1007/JHEP03(2024)074