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

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 $\phi$.