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Abstract We introduce a set of axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections, and we show they are satisfied by Kitaev’s Toric Code and Levin-Wen type models. For a locally topologically ordered spin system on$$\mathbb {Z}^{k}$$, we define a local net of boundary algebras on$$\mathbb {Z}^{k-1}$$, which provides a mathematically precise algebraic description of the holographic dual of the bulk topological order. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian. As a corollary, we obtain a new proof of a recent result of Ogata [Oga24] that the bulk cone von Neumann algebra in the Toric Code is of type$$\mathrm {II}$$, and we show that Levin-Wen models can have cone algebras of type$$\mathrm {III}$$. Finally, we argue that the braided tensor category of DHR bimodules for the net of boundary algebras characterizes the bulk topological order in (2+1)D, and can also be used to characterize the topological order of boundary states.
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Boundary algebras of the Kitaev quantum double model
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 (Hilb(G),C[G]) or (Rep(G),CG) 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.
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- PAR ID:
- 10628216
- Publisher / Repository:
- AIP Publishing
- Date Published:
- Journal Name:
- Journal of Mathematical Physics
- Volume:
- 65
- Issue:
- 10
- ISSN:
- 0022-2488
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0212164
