Jones, Corey, Naaijkens, Pieter  ORCID: https://orcid.org/0000-0001-5670-243X, Penneys, David, Wallick, Daniel and Izumi, Masaki
2025.
Local topological order and boundary algebras.
Forum of Mathematics, Sigma
13
, e135.
10.1017/fms.2025.16
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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 Zk, we define a local net of boundary algebras on Zk−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 II, and we show that Levin-Wen models can have cone algebras of type 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.
| Item Type: | Article |
|---|---|
| Date Type: | Published Online |
| Status: | Published |
| Schools: | Schools > Mathematics |
| Publisher: | Cambridge University Press |
| ISSN: | 2050-5094 |
| Date of First Compliant Deposit: | 14 February 2025 |
| Date of Acceptance: | 23 January 2025 |
| Last Modified: | 18 Aug 2025 10:22 |
| URI: | https://orca.cardiff.ac.uk/id/eprint/176203 |
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