An operator-algebraic framework proves that boundary conditions in (1+1)D gapped phases with categorical symmetry are classified by objects of the module category M_Q^op via an equivalence of categories, yielding a bulk-boundary correspondence as the enriched center.
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The paper classifies one-dimensional Abelian translationally covariant modulated symmetries via Jordan normal forms and derives their Goldstone actions, which modify the conventional theorem by type of symmetry.
String order parameters in 1D gapped phases with invertible or non-invertible symmetries organize into Lagrangian algebras in the Drinfel'd centre via tensor-network module categories.
Non-invertible symmetry-breaking phases are characterized by long-range order parameters obeying generalized algebra, with certain transitions dual to beyond-Landau points of invertible symmetries under precise conditions established via generalized gauging.
Spatially modulated symmetries arise from gauging ordinary symmetries under generalized LSM anomalies, with explicit lattice models in 2D and 3D plus field-theoretic descriptions in arbitrary dimensions that connect to higher-group structures.
Applying a small-system entropy criticality detector to holographic lattice transfer matrices efficiently identifies critical boundary conditions, estimates central charges, and maps multi-dimensional phase diagrams.
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Bulk-boundary correspondence of (1+1)D symmetric gapped phases
An operator-algebraic framework proves that boundary conditions in (1+1)D gapped phases with categorical symmetry are classified by objects of the module category M_Q^op via an equivalence of categories, yielding a bulk-boundary correspondence as the enriched center.