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arXiv:2603.24289

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

Youxuan Wang

Published Mar 25, 2026• 0 citations
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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 $ϕ\in\operatorname{End}(G)$. For each $ϕ$ we identify the associated condensable (Lagrangian) algebra $\mathcal{A}_ϕ$ 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 $\mathcal{B}_ϕ$ 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 $ϕ$.

DOI
10.48550/arxiv.2603.24289

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