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Towards Non-Invertible Anomalies from Generalized Ising Models

by Shang Liu, Wenjie Ji

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Submission summary

Authors (as registered SciPost users): Shang Liu
Submission information
Preprint Link: scipost_202211_00017v2  (pdf)
Date accepted: 2023-08-29
Date submitted: 2023-08-09 06:06
Submitted by: Liu, Shang
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


We present a general approach to the bulk-boundary correspondence of noninvertible topological phases, including both topological and fracton orders. This is achieved by a novel bulk construction protocol where solvable $(d+1)$-dimensional bulk models with noninvertible topology are constructed from the so-called generalized Ising (GI) models in $d$ dimensions. The GI models can then terminate on the boundaries of the bulk models. The construction generates abundant examples, including not only prototype ones such as $\mathbb{Z}_2$ toric code models in any dimensions no less than two, and the X-cube fracton model, but also more diverse ones such as the $\mathbb{Z}_2\times \mathbb{Z}_2$ topological order, the 4d $\mathbb{Z}_2$ topological order with pure-loop excitations, etc. The boundary of the solvable model is potentially anomalous and corresponds to precisely only sectors of the GI model that host certain total symmetry charges and/or satisfy certain boundary conditions. We derive a concrete condition for such bulk-boundary correspondence. The condition is violated only when the bulk model is either trivial or fracton ordered. A generalized notion of Kramers-Wannier duality plays an important role in the construction. Also, utilizing the duality, we find an example where a single anomalous theory can be realized on the boundaries of two distinct bulk fracton models, a phenomenon not expected in the case of topological orders. More generally, topological orders may also be generated starting with lattice models beyond the GI models, such as those with SPT orders, through a variant bulk construction, which we provide in an appendix.

Published as SciPost Phys. 15, 150 (2023)

Author comments upon resubmission

Dear Editor,

Thank you for sending us the referee reports, and sorry for the delayed reply.

We have addressed all questions from both referees, and made necessary changes to the manuscript accordingly. We believe our manuscript is now ready for publication.

Best regards,
The Authors

List of changes

1. We added a more precise discussion about the definition of $\Omega^Z$ after it is first mentioned in Section V.

2. We have added a clarification about the meaning of "GI model terminates on the boundary" at the first paragraph of Section V.

3. There are many different symbols in this paper. To assist the readers, we have added a table at the beginning of Section IV which summarizes our notations. A reference to this table has also been added right above the expression of the bulk Hamiltonian (previous Eq.6, current Eq.7).

4. In the previous version of our manuscript, we did not explicitly write down the general Hamiltonian of the dual model, and this has now been fixed.

Reports on this Submission

Anonymous Report 1 on 2023-8-24 (Invited Report)


I'm pleased to observe that the authors have effectively addressed my earlier concerns. The added explanations have enhanced the paper's clarity. Furthermore, the inclusion of the general Hamiltonian of the dual model addresses a previous omission, enhancing the manuscript's comprehensiveness. Considering these changes, I recommend the manuscript for publication.

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