##### Department of Mathematics,

University of California San Diego

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### Math 243, Functional Analysis seminar

## David Penneys

#### Ohio State University

## An operator algebraic axiomatization of local topological order

##### Abstract:

Topological order is a notion in theoretical condensed matter physics describing new phases of matter beyond Landau's symmetry breaking paradigm. Bravyi, Hastings, and Michalakis introduced certain topological quantum order (TQO) axioms to ensure gap stability of a commuting projector local Hamiltonian and stabilize the ground state space with respect to local operators in a quantum spin system. In joint work with Corey Jones, Pieter Naaijkens, and Daniel Wallick (arXiv:2307.12552), we study nets of finite dimensional C*-algebras on a 2D $\mathbb{Z}^2$ lattice equipped with a net of projections as an abstract version of a quantum spin system equipped with a local Hamiltonian. We introduce a set of local topological order (LTO) axioms which imply the TQO conditions of Bravyi-Hastings-Michalakis in the frustration free commuting projector setting, and we show our LTO axioms are satisfied by known 2D examples, including Kitaev's toric code and Levin-Wen string net models associated to unitary fusion categories (UFCs). From the LTO axioms, we can produce a canonical net of algebras on a codimension 1 $\mathbb{Z}$ sublattice which we call the net of boundary algebras. We get a canonical state on the boundary net, and we calculate this canonical state for both the toric code and Levin-Wen string net models. Surprisingly, for the Levin-Wen model, this state is a trace on the boundary net exactly when the UFC is pointed, i.e., all quantum dimensions are equal to 1. Moreover, the boundary net for Levin-Wen is isomorphic to a fusion categorical net arising directly from the UFC. For these latter nets, Corey Jones' category of DHR bimodules recovers the Drinfeld center, leading to a bulk-boundary correspondence where the bulk topological order is described by representations of the boundary net.

Host: Priyanga Ganesan

### October 17, 2023

### 11:00 AM

Zoom ID 94246284235

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