Universal fusion category symmetries on tensor products of infinite-dimensional Hilbert spaces

TL;DR

This paper demonstrates that unitary fusion categories can be realized as symmetries on tensor products of infinite-dimensional Hilbert spaces, establishing a stable classification of anyon chain symmetries and boundary algebras.

Contribution

It introduces a method to realize any fusion category as symmetries on infinite-dimensional Hilbert spaces and classifies symmetry realizations up to stable equivalence.

Findings

Anyon chains factorize into tensor products after stabilization.

Any two chains with the same symmetry are related by a symmetry-preserving unitary.

Boundary algebras are isomorphic if and only if bulk topological order matches.

Abstract

We show that anyon chains, after stabilizing with infinite-dimensional ancilla spaces, factorize locally as tensor products of infinite-dimensional Hilbert spaces. This implies that any unitary fusion category can be realized as symmetries on a tensor product of infinite-dimensional Hilbert spaces. We then show that any two anyon chains with the same symmetry category are related by a symmetry-compatible locality-preserving unitary after stabilizing with infinite-dimensional ancilla, showing that for a fixed fusion category, there is a single stable equivalence class of symmetry realizations on the lattice via anyon chains. As a corollary of our proof, we show that the physical boundary algebras of Levin-Wen type models are bounded spread isomorphic after stabilization if and only if they have the same bulk topological order.