The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to or depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group.
REFERENCES
Since this article was published to the arXiv, two articles related to Ref. 18 have appeared. Kim et al. [I. H. Kim, T.-C. Lin, D. Ranard, and B. Shi, “Strict area law implies commuting parent Hamiltonian,” arXiv:2404.05867 (2024)] show that if the entanglement entropy of a quantum state satisfies a strict area law, it can be obtained as the ground state of a commuting projector Hamiltonian, and the work by Kim and Ranard [I. H. Kim and D. Ranard, “Classifying 2D topological phases: mapping ground states to string-nets,” arXiv:2405.17379 (2024)] constructs a finite depth quantum circuit taking such a state to a string net ground state. The approach of this latter article uses boundary excitations, and not just the bulk theory, to reproduce the desired fusion category.
Reference 28 uses a slightly different definition. In her paper, she only considers edges that are completely contained in the region enclosed by the two rays. However, she uses a dual convention to ours for the Quantum Double model, so this definition more closely aligns with hers when the differing conventions are taken into account.
Ogata only claims this result in the case when the group is abelian. However, her work in Ref. 28, combined with an argument from Ref. 27, can be used to prove the more general case. In addition, after this article was posted to the arXiv, Bols and Vadnekar [A. Bols and S. Vadnerkar, “Classification of the anyon sectors of Kitaev's quantum double model,” arXiv:2310.19661 (2023)] proved the necessary result for the proof in Ref. 28 to hold for nonabelian groups.
