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We introduce a set of axioms for locally topologically ordered quantum spinsystems in terms of nets of local ground state projections, and we show theyare satisfied by Kitaev's Toric Code and Levin-Wen type models. Then for alocally topologically ordered spin system on $\mathbb{Z}^{k}$, we define alocal net of boundary algebras on $\mathbb{Z}^{k-1}$, which gives a newoperator algebraic framework for studying topological spin systems. Weconstruct a canonical quantum channel so that states on the boundaryquasi-local algebra parameterize bulk-boundary states without reference to aboundary Hamiltonian. As a corollary, we obtain a new proof of a recent resultof Ogata [arXiv:2212.09036] that the bulk cone von Neumann algebra in the ToricCode is of type $\rm{II}$, and we show that Levin-Wen models can have conealgebras of type $\rm{III}$. Finally, we argue that the braided tensor categoryof DHR bimodules for the net of boundary algebras characterizes the bulktopological order in (2+1)D, and can also be used to characterize thetopological order of boundary states.

Local topological order and boundary algebras