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Non-invertible duality and symmetry topological order of one-dimensional lattice models with spatially modulated symmetry
Strongly Correlated Electrons · Submitted 6 Nov 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.89 | 0.85 | 0.05 | asserted 0.86 |
| s1 | The spatially modulated symmetry of the model becomes trivial when the model's parameters satisfy a specific number-theoretic relation. | result 1.00 | 0.61 | 0.08 | asserted 1.00 |
| s2 | 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. | result 0.97 | 0.96 | 0.74 | asserted 1.00 |
| s3 | 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. | result 0.59 | 0.78 | 0.12 | asserted 1.00 |
| s4 | 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. | interpretation 1.00 | 0.77 | 0.24 | asserted 0.75 |
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The paper
Original 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.
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How the literature uses it
2 arXiv papers cite this in 2 sentences
Papers citing it
Cites it as background 2 papers
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Note that explorations of noninvertible operators via gauging spatially modulated symmetries have recently started (see e.g., [ref] ).
Noninvertible operators in one, two, and three dimensions via gauging spatially modulated symmetry · Introduction · 0.98
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See [ref] for more discussions on the mixing between non-invertible symmetries and lattice translations.
Gauging non-invertible symmetries on the lattice · F-symbols · 0.92
What it cites · 8 sentences without a confident label
Cites as background 5 papers
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Recent studies however show growing interest in more generic UV/IR mixing within lattice models [ref] .
Unveiling UV/IR Mixing via Symmetry Defects: A View from Topological Entanglement Entropy · Introduction · 0.83
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Although non-invertible symmetries [ref] have only recently come into focus, they appear surprisingly frequently in the context of duality.
What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries · Introduction · 0.87
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Interest in spatially modulated symmetries, whose symmetry generators vary across space, has been recently growing [ref] .
2306.17121 · Introduction · 0.86
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