SciPost Submission Page
Gauging non-invertible symmetries on the lattice
by Sahand Seifnashri, Shu-Heng Shao, Xinping Yang
Submission summary
| Authors (as registered SciPost users): | Sahand Seifnashri |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2503.02925v1 (pdf) |
| Date accepted: | July 10, 2025 |
| Date submitted: | April 15, 2025, 5:40 p.m. |
| Submitted by: | Sahand Seifnashri |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We provide a general prescription for gauging finite non-invertible symmetries in 1+1d lattice Hamiltonian systems. Our primary example is the Rep(D$_8$) fusion category generated by the Kennedy-Tasaki transformation, which is the simplest anomaly-free non-invertible symmetry on a spin chain of qubits. We explicitly compute its lattice F-symbols and illustrate our prescription for a particular (non-maximal) gauging of this symmetry. In our gauging procedure, we introduce two qubits around each link, playing the role of "gauge fields" for the non-invertible symmetry, and impose novel Gauss's laws. Similar to the Kramers-Wannier transformation for gauging an ordinary $\mathbb{Z}_2$, our gauging can be summarized by a gauging map, which is part of a larger, continuous non-invertible cosine symmetry.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Published as SciPost Phys. 19, 063 (2025)
Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2025-6-29 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.02925v1, delivered 2025-06-28, doi: 10.21468/SciPost.Report.11475
Strengths
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Giving a general framework for gauging noninvertible symmetry on lattice.
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Illustrating this framework with the Rep$(D_8)$ example.
Weaknesses
- The calculation for the Rep$(D_8)$ example is too heavy and reader might be lost.
Report
Requested changes
Suggestions: 1. The calculations for Rep$(D_8)$ is too heavy and the reader might be lost in the details. It is better to integrate Appendix A to the main text to help readers to grasp the essence of gauging (non)invertible symmetries on the lattice in the language of algebraic objects. I think even reviewing gauging invertible symmetry in the language of algebraic objects is useful.
- Could author comment on the connections between lattice and continuum? How do the two gauge fields become in the continuum? Can we turn on these background gauge fields in the partition functions?
Questions: 1. Is gauging a noninvertible symmetry a duality transformation?
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For ordinary symmetries, we can impose the Gauss law energetically. Can we do the same trick here for gauging noninvertible symmetries?
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The authors claim gauging in Rep$(D_8)$ is the simplest example. But since you have realized lattice models with cosine symmetry, I guess non-maximal gauging in Rep$(S_3)$ should be simpler?
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It might be different realization of noninvertible symmetry on the lattice. For example, the noninvertible symmetry of Rep($D_8$) can be realized as gauging $\mathbb Z_2\times \mathbb Z_2$ (S) or KT (TST). Does it matter in the non-maximal gauging?
Recommendation
Ask for minor revision
Report #2 by Anonymous (Referee 2) on 2025-6-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.02925v1, delivered 2025-06-23, doi: 10.21468/SciPost.Report.11448
Strengths
- Concrete realization of abstract algebraic structures
- General gauging framework
- Explicit, fully worked-out example
Report
I only have a small question: the paper provides a clear and detailed derivation of gauging Rep(D₈) symmetries on the lattice in Section 3, but is there any physical intuition behind some of the key results, such as the form of the Gauss law or the choice of minimal coupling to the gauge field?
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
We thank the referee for their positive feedback. Our gauging procedure generalizes the gauging of invertible symmetries, for which the physical intuition is more transparent, as reviewed in Appendix A.
Report #1 by Anonymous (Referee 1) on 2025-6-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.02925v1, delivered 2025-06-23, doi: 10.21468/SciPost.Report.11447
Strengths
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Detailed description of the gauging procedure of $\mathrm{Rep}(D_8)$ symmetry realized in 1+1d spin chains of qubits
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A new prescription for gauging general finite non-invertible symmetries in 1+1d lattice systems
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Detailed computations and helpful background information are provided in the appendices, which make the paper self-contained.
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The manuscript is written clearly.
Report
This paper provides new insights into the gauging of finite non-invertible symmetries. In particular, it clarifies the notion of gauge fields for non-invertible symmetries on the lattice and their relation to topological defects. The definition of the gauging in this paper is complementary to the existing definition of the gauging in the anyon chain model, whose state space typically does not admit a tensor product decomposition. The conceptual and technical developments in this paper would be a valuable addition to the existing literature. I would thus recommend this paper for publication.
Requested changes
It might be good to address the following points.
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p.27, around (3.33): could you clarify why the local terms in the Hamiltonian $\tilde{H}$ commute with the projection $P_j$ by construction?
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p.30, Section 4: the first paragraph says that the gauging procedure in this paper can be applied to any finite non-invertible symmetry. Is it implicitly assumed here that the finite non-invertible symmetry under consideration is realized on a tensor product Hilbert space without mixing with the lattice translation?
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p.30, the last sentence: could you elaborate on why "the gauging map has an ambiguity by composing it with a local unitary transformation from the left"?
Please see also the following typos.
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p.8, Figure 1: "For $\lambda = - h_0/h_1 \in [-1, 1)$" would be a typo of "For $\lambda = - h_0/h_1 \in (-1, 1]$"
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p.12, above (2.23): "$H_{\eta^o}^{(2, 3)} = H_{\eta^o}^{(1, 2)}$, and $H_{\eta^e}^{(0, 1)} = H_{\eta^e}^{(1, 2)}$" would be a typo of "$H_{\eta^e}^{(2, 3)} = H_{\eta^o}^{(1, 2)}$, and $H_{\eta^o}^{(0, 1)} = H_{\eta^e}^{(1, 2)}$"
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p.13, footnote 7: "$\mathcal{H}_{\mathcal{L}; \mathcal{M}}^{(j-1, j); (j, j+2)}$" on the first and second lines would be a typo of "$\mathcal{H}_{\mathcal{L}; \mathcal{M}}^{(j-1, j); (j, j+1)}$"
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p.14, below (2.27): "a $\mathcal{D}$ defects" would be a typo of "a $\mathcal{D}$ defect"
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p.18, the first line of Section 3: ref [82] is the same as ref [56].
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p.28, below (3.37): "gauged invariant" would be a typo of "gauge invariant"
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p.33, above (4.12): "gauging fixing" would be a typo of "gauge fixing"
8.p.37, above (A.12): "This fusion operator be" would be a typo of "This fusion operator can be"
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p.47, footnote 16: "if $O_j$ is commutes with" would be a typo of "if $O_j$ commutes with"
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p.51, the first line of Section D.1: "Here, compute" would be a typo of "Here, we compute"
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p.55, (E.13): "$(m_{\mathcal{A}_2 \otimes \mathcal{A}_3}^{\mathcal{A}_2})^{\dagger}$" would be a typo of "$(m_{\mathcal{A}_2 \otimes \mathcal{A}_3}^{\mathcal{A}_3})^{\dagger}$"
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
We thank the referee for their thoughtful comments and valuable feedback.
1. This point follows from our procedure for coupling to the gauge field. While it may not be immediately evident, the result emerges from a diagrammatic representation of defects and $P_j$, as discussed in Section 4 and Appendix A.
2. A finite non-invertible symmetry cannot mix with lattice translation, as such mixing would generate an infinite number of simple objects. Therefore, no additional assumptions are required.
3. This refers to the invariance of the left-hand side of equation (4.3) under the transformation $\mathsf{G} \mapsto U \mathsf{G}$.
We also thank the referee for identifying the typographical errors. All have been corrected except for item 2, which is not a typo and is correct as written.
Author: Sahand Seifnashri on 2025-07-19 [id 5653]
(in reply to Report 3 on 2025-06-29)We thank the referee for their thoughtful comments and feedback.
1 - This is an excellent question. Unfortunately, we do not currently have an answer, and we believe it remains an open question worth exploring in future work.
Absolutely — one can impose Gauss’s law energetically, and in this respect, there is no difference from the standard case of gauging an invertible symmetry.
Indeed, Rep(S₃) is simpler. The advantage of using Rep(D₈) is that the cosine symmetry provides an alternative route for computing the gauging map, which we have used to independently verify our result. That said, applying our method to the non-maximal gauging of Rep(S₃), which is comparatively simple, would be worthwhile.
Indeed, it does not matter, as the choices are related by a local unitary transformation, i.e., a local change of basis.