Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections

We thank both referees for carefully reading our manuscript, encouraging comments, and valuable suggestions. We detail below the changes we made and address the referees’ comments:

Referee 1

Referee Report: In this paper the authors give a treatment of the gauging of spatially modulated symmetries in translation-invariant spin chains. The authors apply this to a class of generalized Ising models with such modulated symmetries, and identify non-invertible Kramers-Wannier type dualities (sometimes involving a spatial reflection). A number of natural follow up questions are suggested. The material on gauging spatial symmetries consists of a mix of examples and general results. The examples are certainly helpful, but it is not always clear what the most important material is. The authors give a summary in the introduction, nevertheless it would be helpful to streamline these two sections, perhaps by moving some of the examples to an appendix.

Authors’ Response: We appreciate the referee’s point and the opportunity to clarify our strategy for organizing the paper. As the referee pointed out, we did include a summary of our results in the introduction section. Given the paper’s length, we wrote this summary to allow the interested reader to take away the most important results of our study quickly. The summary section also states the most important material of the paper. The main text of the paper was written to be a pedagogical and thorough investigation. Indeed, there are numerous examples throughout the paper, each self-contained and containing unique features to emphasize different aspects of our general results. Some readers are interested only in examples, while others are more interested in the general and formal results. With the aid of the summary section and the detailed table of contents, we feel that the reader can pick and choose what they want to read to get the most out of our paper with its current organization.

Referee Report: The treatment of the Kramers-Wannier dualities is interesting. It would be useful to comment on whether the generalized Ising models have previously appeared in the literature, and if not, more could be said about their physics. For example, the duality becomes a symmetry at the point J=h - is this a critical point of the model?

Authors’ Response: We thank the referee for their interesting questions. To our knowledge, not much is known about these generalized Ising models for general $ \Delta$. They have appeared, however, for particular choices of $\Delta$. In these known cases, the models’ symmetries are $\mathbb{Z}_N$ dipole symmetry and $\mathbb{Z}_N$ exponential symmetry, and we have added references to these previous studies. Depending on the details of $\Delta$, the Kramers-Wannier duality symmetry may or may not be anomalous. For instance, it is known that for $\mathbb{Z}_N$ dipole symmetry, the corresponding Kramers-Wannier duality symmetry is anomalous for $N=3$ and anomaly-free when $N=2$. This means that the point J=h cannot be a non-degenerate and gapped when $N=3$, i.e., it should be gapless or degenerate, while there is no such condition when $N=2$. We have expanded on the outlook section to include this important discussion.

Referee Report: Requested changes: Some additional comments on the generalized Ising models.

Consider streamlining the main text to improve readability of the paper.

I found the following typos that should be corrected: 1) Footnote 5 "refereed to” 2) (2.47) right hand equation j→j+1. [Also consider writing p=2 in (2.44) and replacing Z† with Z in (2.48).]

Authors’ Response: We thank the referee for their careful reading that spotted these typos! We have fixed these in the resubmitted version.

Referee 2

Referee Report: The authors discuss the construction of generalized reflection symmetries in 1+1d lattice Hamiltonian systems. They study generalized Ising models with internal symmetries that do not necessarily commute with lattice translations. Furthermore, they gauge these internal symmetries and construct non-invertible reflection symmetries when the system is self-dual under gauging. I recommend the draft for publication and have the following questions: The authors implement gauging using Gauss's law in equation (1.15). They should comment on how this equation is derived and whether the expression is unique. In particular, there might be multiple (untwisted/twisted) ways to gauge the symmetry, corresponding to different forms of Gauss's law.

Authors’ Response: We appreciate the referee’s comment and thank them for the opportunity to improve our paper. In our paper, we explained derivations and the validity of the general Gauss law when introducing it in the main text (e.g., see the discussion before Eq 2.84). However, we did not provide a similar justification in the summary and have expanded the summary to include one. The referee is correct that the Gauss law used for gauging the symmetries is not unique. In fact, as referenced in footnote 8, we present an alternative Gauss law in Appendix C that also gauges the entire modulated symmetry. Furthermore, exploring twisted gauging was mentioned in the outlook section as an interesting follow-up direction. We have updated footnote 8 to emphasize that this Gauss law is not unique.

Referee Report: What are the possible gapped or gapless phases that remain invariant under non-invertible reflection symmetries? Specifically, what is the low-energy behavior of these systems with non-invertible reflection symmetries at J=h?

Authors’ Response: We thank the referee for this very interesting question! We listed this as an important follow-up in the outlook section. We are pursuing this direction now and think it is more suited for a stand-alone paper as it falls outside the scope of this paper. However, we have expanded our current discussion to emphasize how the non-invertible symmetry can generally be anomalous or anomaly-free and referenced known cases of anomalous KW symmetries related to $\mathbb{Z}_N$ dipole symmetry.

Added equation 1.16

Updated footnote 8

Modified the now second and third paragraphs of the outlook section