Local topological order and boundary algebras
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910823215005696 |
|---|---|
| author | Jones, Corey Naaijkens, Pieter Penneys, David Wallick, Daniel |
| author_facet | Jones, Corey Naaijkens, Pieter Penneys, David Wallick, Daniel |
| contents | 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 [Ann. H. Poincaré 25, 2024] that the bulk cone von Neumann algebra in the Toric Code is of type $\rm{II}$, and we show that Levin-Wen models can have cone algebras of type $\rm{III}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12552 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local topological order and boundary algebras Jones, Corey Naaijkens, Pieter Penneys, David Wallick, Daniel Mathematical Physics Strongly Correlated Electrons Operator Algebras Quantum Algebra Quantum Physics 81T05, 81T25 (primary), 18M20, 46L37, 46L60, 81V27 (secondary) 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 [Ann. H. Poincaré 25, 2024] that the bulk cone von Neumann algebra in the Toric Code is of type $\rm{II}$, and we show that Levin-Wen models can have cone algebras of type $\rm{III}$. 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. |
| title | Local topological order and boundary algebras |
| topic | Mathematical Physics Strongly Correlated Electrons Operator Algebras Quantum Algebra Quantum Physics 81T05, 81T25 (primary), 18M20, 46L37, 46L60, 81V27 (secondary) |
| url | https://arxiv.org/abs/2307.12552 |