Non-invertible SPT, gauging and symmetry fractionalization

TL;DR

This paper constructs explicit lattice models for non-invertible symmetry-protected topological phases, revealing complex interactions between various symmetries and their fractionalization in higher-dimensional bulk systems.

Contribution

It provides explicit realizations of non-invertible SPT states and explores their dualities and symmetry fractionalization in lattice models.

Findings

Constructed lattice models for Rep(Q_8) non-invertible SPT states.

Demonstrated duality relations involving non-invertible, non-abelian, and anomalous symmetries.

Linked symmetry fractionalization in 2+1d bulk SET to lattice model symmetries.

Abstract

We explicitly realize the Rep($Q_8$) non-invertible symmetry-protected topological (SPT) state as a 1+1d cluster state on a tensor product Hilbert space of qubits. Using the Kramers-Wannier operator, we construct the lattice models for the phases of all the symmetries in the Rep($Q_8$) duality web. We further show that we can construct a class of lattice models with Rep($G$) symmetry including non-invertible SPT phases if they have a dual anomalous abelian symmetry. Upon dualizing, there is a rich interplay between onsite symmetries, non-onsite symmetries, non-abelian symmetries, and non-invertible symmetries. We show that these interplay can be explained using the symmetry fractionalization in the 2+1d bulk SET.