Modulated SPT phases in 1D are classified by H²(G, U(1)_s) and obey LSM-type theorems forbidding symmetric short-range entangled ground states.
Defect networks for topological phases pr otected by modulated symmetries
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Z_N cluster and dipolar-cluster SPT wavefunctions admit closed-form projector, neural, and tensor-product representations that generalize Z_2 constructions and yield a TPS benchmarked against MPS via DMRG.
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.
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Projector, Neural, and Tensor-Network Representations of $\mathbb{Z}_N$ Cluster and Dipolar-cluster SPT States
Z_N cluster and dipolar-cluster SPT wavefunctions admit closed-form projector, neural, and tensor-product representations that generalize Z_2 constructions and yield a TPS benchmarked against MPS via DMRG.