Noninvertible symmetry and topological holography for modulated SPT in one dimension

The authors study one-dimensional models with various modulated symmetries and demonstrate that these models possess additional non-invertible symmetries. By applying the KT transformation, they map SPT models to SSB models and subsequently construct new SSB models with the same set of symmetries. Through dualization of these SSB models, they obtain SPT models whose distinctions from the original SPT models can be identified via boundary symmetry fractionalizations. Moreover, their bulk Hamiltonians are formulated in the spirit of Topological Holography.

The manuscript provides detailed, step-by-step derivations of the results, which have the potential to lead to important physical insights. Since the field of generalized symmetries is rapidly developing, shedding light on concrete examples is highly valuable at this stage. Nevertheless, I believe there is room for further improvement in the manuscript, and I list my requested changes below.

1-Is the Hamiltonian (2.8) originally written by the authors? Otherwise, I encourage the authors to cite proper references around (2.8).

2-The authors introduce a new SSB model (2.31) and claim that "$\alpha \in \mathbb{Z}_N$ characterizes a distinct SPT". Though I found that the authors apply a KT transformation to get an SPT model (2.36), it could be misleading in that readers may understand (2.31) as a new SPT model. I recommend the authors to add some sentences to clarify this.

3-I personally found that the authors' detailed calculation is helpful to understand the content of the manuscript, but some readers might be overwhelmed by hundreds of equations. It would be better if the authors could include a section right after Introduction that summarizes main results with selected equations.

4-As the authors point out in the last paragraph of Summary and Discussion, the different SSB phases proposed in the manuscript are only distinguished by their KT duals. To claim this, it is needed to be ensured that the KT transformation is injective. I wonder if it has been proven and encourage the authors to include remarks on this.

5-As the authors write in Summary and Discussion, it seems that the authors try to claim that the presence of noninvertible symmetry and a corresponding bulk topological theory is a special property of modulated symmetries. However, the claim might be inappropriate and should be revised because there are examples (e.g. the standard transverse-field Ising model) where non-modulated symmetries accompany KW self-duality, which can be interpreted as noninvertible symmetries. I recommend the authors to revise the paragraphs to avoid confusion.

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The authors study the symmetry protected topological (SPT) phases protected by non-invertible symmetries that include modulated symmetries. By way of examples, the authors construct SPT ground states of dipole and exponential modulated symmetries together with an appropriate non-anomalous non-invertible self-duality symmetry.

The paper is well-written with many details explicitly shown. I found the results interesting and significant enough to warrant a publication at Scipost Physics. However, I think the manuscript would benefit from a revision according to the following points.

1) The authors start with reviewing what they call "charge SPT" which is the generalization of the well-known cluster model to that with $\mathbb{Z}_N\times\mathbb{Z}_N$ symmetry. However, the symmetry here is a "charge" symmetry only if one combines the even and odd sites into larger unit cells supporting two copies of $\mathbb{Z}_N$ degrees of freedom. In particular, the symmetry operators $\prod X$ on even and odd sites are also modulated with the modulating function being $f_j = (j \text{ mod } 2)$. Therefore, I suggest reviewing this section to justify why these symmetries are "charge" and not modulated (perhaps by enlarging the unit cells so that both symmetry generators are translationally invariant).

2) Related to the above point, an important property of the modulated symmetries is that the translation symmetry (more generally spatial symmetries) has a non-trivial action on such internal symmetries. I think the lack of emphasis and discussion on this point weakens the point authors are trying to make. In particular, if we completely disregard spatial symmetries then all the examples in the manuscript can be recast into ordinary internal symmetries by appropriately enlarging the unit cells. This renders the adjactive "modulated" unnecessary. I suggest authors to incorporate translation (or reflection) symmetries into the discussion in all sections.

3) While the authors give examples of SPT states with non-invertible symmetries, the corresponding "holographic description" is only for the invertible part of the symmetries. This leads to two questions to be addressed. First, can the authors comment on what would be the bulk topological order/topological field theory (TFT) that would describe the non-invertible modulated SPTs fully? Such TFTs should be obtained by gauging in the bulk the automorphisms (such as layer swap) that correspond to the non-invertible symmetry in the boundary. Second, related to the comment 2), what is the role of the spatial symmetries in the symTFT description so that we know the boundary theory has indeed a modulated symmetry?

4) Do the authors know if their examples of noninvertible SPTs are in any sense complete? In other words, do the authors know the number of fiber functors of the fusion category smmetry protecting such non-invertible SPTs. It would be interesting to see if the families of Hamiltonians presented here exhaust all possible SPT phases with the appropriate fusion category symmetry.

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