Classification of intrinsically mixed $1+1$D non-invertible Rep$(G) \times G$ SPT phases

TL;DR

This paper classifies 1+1D bosonic SPT phases with non-invertible symmetry structures, revealing their parametrization by endomorphisms of G and providing explicit lattice models for realization.

Contribution

It introduces a complete classification of intrinsically mixed SPT phases with non-invertible symmetries using fiber functors and algebraic structures, and constructs explicit lattice models.

Findings

Phases are parametrized by endomorphisms of G.

Identification of condensable algebras in the bulk.

Explicit lattice models with domain walls and boundary conditions.

Abstract

We classify $1+1$d bosonic SPT phases with non-invertible symmetry $\mathrm{Rep}(G)\times G$, equivalently the fusion-category symmetry $\mathcal{H}=\mathrm{Rep}(G)\times\mathrm{Vec}_G$. Focusing on \emph{intrinsically mixed} phases (trivial under either factor alone), we use the correspondence between $\mathcal{H}$-SPTs, $\mathcal{H}$-modules over $\mathrm{Vec}$, and fiber functors $\mathcal{H}\to\mathrm{Vec}$ to obtain a complete classification: such phases are parametrized by $\phi\in\operatorname{End}(G)$. For each $\phi$ we identify the associated condensable (Lagrangian) algebra $\mathcal{A}_\phi$ in the bulk $\mathcal{Z}(\mathcal{H})\simeq\mathcal{D}_G^2$. We further provide an explicit lattice realization by modifying Kitaev's quantum double model with a domain wall $\mathcal{B}_\phi$ and smooth/rough boundaries, and then contracting to a 1D chain, yielding a (possibly twisted) group-based cluster state whose ribbon-generated symmetry operators encode the same $\phi$.