Non-invertible SPTs: an on-site realization of (1+1)d anomaly-free fusion category symmetry

TL;DR

This paper develops a systematic framework for classifying and constructing (1+1)d SPT phases with fusion category symmetries, including explicit lattice models and dualities, advancing understanding of non-invertible symmetries.

Contribution

It introduces an onsite MPO framework for non-invertible symmetry operators and classifies SPTs via Q-systems in the charge category, with explicit examples.

Findings

Constructed onsite MPO symmetry operators for fusion categories.

Classified SPT phases using Q-systems in the charge category.

Demonstrated S_3-duality among three explicit SPT phases.

Abstract

We investigate (1+1)d symmetry-protected topological (SPT) phases with fusion category symmetries. We emphasize that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase. We construct an ``onsite'' matrix-product-operator (MPO) version of the Hopf algebra symmetry operators in a lattice model with tensor-product Hilbert space. In particular, we propose a systematic framework for classifying and constructing SPTs with non-invertible symmetries. An SPT phase corresponds to a Q-system in the charge category, such that the Q-system becomes a matrix algebra when the symmetry is forgotten. As an example, we provide an explicit microscopic realization of all three $\mathsf{Rep}^\dagger(D_8)$ SPT phases, including a trivial phase, and further demonstrate the $S_3$-duality among these three SPT phases.