Local topological order and boundary algebras

TL;DR

This paper develops an algebraic framework for understanding topological order in quantum spin systems, introducing boundary algebras and channels, and classifying their types to characterize bulk topological phases.

Contribution

It introduces axioms for local topological order, constructs boundary algebras, and links boundary categories to bulk topological order, providing new proofs and classifications.

Findings

Bulk cone von Neumann algebra in the Toric Code is type II.

Levin-Wen models can have cone algebras of type III.

Boundary braided tensor categories characterize bulk topological order.

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 [Ann. H. Poincar\'e 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.