Non-invertible duality and symmetry topological order of one-dimensional lattice models with spatially modulated symmetry

Abstract

We investigate the interplay between self-duality and spatially modulated symmetry of generalized $N$-state clock models, which include the transverse-field Ising model and ordinary $N$-state clock models as special cases. The spatially modulated symmetry of the model becomes trivial when the model's parameters satisfy a specific number-theoretic relation. We find that the duality is non-invertible when the spatially modulated symmetry remains nontrivial, and show that this non-invertibility is resolved by introducing a generalized ${\mathbb{Z}}_N$ toric code, which manifests ultraviolet/infrared mixing, as the bulk topological order. In this framework, the boundary duality transformation corresponds to the boundary action of a bulk symmetry transformation, with the endpoint of the bulk symmetry defect realizing the boundary duality defect. Our results illuminate not only a holographic perspective on dualities but also a relationship between spatially modulated symmetry and ultraviolet/infrared mixing in one higher dimension.

7 pages, 2 figures. The introduction has been slightly modified, and additional references have been included

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• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Higher-order topological phases protected by noninvertible and subsystem symmetries
• Matrix product state classification of 1D multipole symmetry protected topological phases