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Gapless Phases in (2+1)d with Non-Invertible Symmetries
by Lakshya Bhardwaj, Yuhan Gai, Sheng-Jie Huang, Kansei Inamura, Sakura Schafer-Nameki, Apoorv Tiwari, Alison Warman
Submission summary
| Authors (as registered SciPost users): | Apoorv Tiwari · Alison Warman |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2503.12699v1 (pdf) |
| Date submitted: | June 7, 2025, 12:27 a.m. |
| Submitted by: | Apoorv Tiwari |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The study of gapless phases with categorical (or so-called non-invertible) symmetries is a formidable task, in particular in higher than two space-time dimensions. In this paper we build on previous works arXiv:2408.05266 and arXiv:2502.20440 on gapped phases in (2+1)d and provide a systematic framework to study phase transitions with categorical symmetries. The Symmetry Topological Field Theory (SymTFT) is, as often in these matters, the central tool. Applied to gapless theories, we need to consider the extension of the SymTFT to interfaces between topological orders, so-called ``club sandwiches", which realize generalizations of so-called Kennedy-Tasaki (KT) transformations. This requires an input phase transition for a smaller symmetry, such as the Ising transition for $\mathbb{Z}_2$, and the SymTFT constructs a transformation to a gapless phase with a larger categorical symmetry. We carry this out for categorical symmetries whose SymTFT is a (3+1)d Dijkgraaf-Witten (DW) theory for a finite group $G$ with twist -- so-called all bosonic fusion 2-categories. We classify such interfaces using a physically motivated picture of generalized gauging, as well as with a complementary analysis using (bi-)module 2-categories.This is exemplified in numerous abelian and non-abelian DW theories, giving rise to interesting gapless phases such as intrinsically gapless symmetry protected phases (igSPTs) and spontaneous symmetry breaking phases (igSSBs) from abelian, $S_3$, and $D_8$ DW theories.
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
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Requested changes
- A brief discussion (even a paragraph) of potential physical realisations (condensed‐matter systems, lattice models) of the predicted gapless phases would be better (even if speculative).
Recommendation
Publish (meets expectations and criteria for this Journal)
Report
The paper generalizes the KT transformation and apply it to the study of gapless phases in (2+1) using SymTFT, and provide a way to construct 2nd order phase transition with a larger categorical symmetry from a given one with a smaller symmetry. The paper has a good balance between abstract approach and concrete examples. I believe it is beneficial for readers from both Math and Physics backgrounds. However, before I can recommend this paper to be published, I hope the authors could address the following questions:
- On page 14, given that $\mathcal{Z}(2\text{Vec}_{H/N}^\pi)$ will appear with a non-trivial $\pi \in H^4$. It is helpful if the authors to comment on the effect of non-trivial $\pi$ here.
- Around page 22, when discussing the interfaces $\mathcal{I}(H,N,\pi,\omega)$, for completeness, it will be nice if the authors could describe what happens to lines $\mathbf{Q}_1^R$ where $R \notin ker(\kappa)$ and surfaces $\mathbf{Q}_2^{[g]}$ where $g \notin H$.
- On page 31, below (3.1), can the authors explain what is physically meaning of assumption where $\mathcal{A}_e$ is fusion?
- At the end of page 45, for the case $H = G$ and $N = 1$, there could be non-trivial choice of $\omega \in H^3(H,U(1))$. Are the interfaces trivial for all choices of $\omega$ or it is only trivial when $\omega = 0$?
- In Section 4.2, it would be nice if the authors could describe the KT transformations in the case where $\omega$ in the $\mathcal{I}(H,N,\omega)$ is non-trivial.
- In Eq (5.17), will the RHS (currently being written as $2\text{Vec}(\mathbb{Z}_3^{(1)}\rtimes \mathbb{Z}_2^{(0)})$) depend the label for the twist $\omega \in H^3(\mathbb{Z}_3,U(1))$?
- In Eq (5.65), it would be helpful if the authors could move the definition of $\text{Mat}_n$ of some 2-categories from below (B.30) to here.
Besides the above questions, there are a few potential typos that the authors should check:
- Page 45, at the end of the first paragraph of Section 4, "." is missing.
- Page 49, "The $m^p$ surfaces becomes ..." should be "The $m^p$ surfaces become ..."; "Non-trival $\omega$ ..." should be "Non-trivial $\omega$".
- Page 51, Eq (4.17) and below, $\text{DW}(\mathbb{Z}{Z_4})\omega$ looks like a typo.
- Page 52, at the end of the first paragraph of Section 4.4, $D_1^{e20}$ looks like a typo.
- Page 57, Eq (4.35) $Z_2^{(1)}$ should be $\mathbb{Z}_2^{(1)}$.
- Page 74, below Eq (5.52), $\mathbb{Z}_3 \times \mathbb{Z}_3 \cong \in H^3(S_3,U(1))$ should be $\mathbb{Z}_3 \times \mathbb{Z}_2 \cong H^3(S_3,U(1))$.
- Page 77, below Eq (5.72), $\mathcal{Z}(A^{\text{id}})$ should be $\mathcal{Z}(\mathcal{A}^{id})$.
- Page 79, Eq (5.81), there shouldn't be a $j$ on the exponent of $\phi$? Below, $\mathcal{Z}(A^{\text{id}})$ should be $\mathcal{Z}(\mathcal{A}^{id})$.
- Page 84, below Eq (5.108), $\mathcal{Z}(\mathcal{A}^{\text{id}23}$ looks like a typo.
- Page 87, below Eq (5.132), $p' \in H^3(\mathbb{Z}_2,U(1)$ should be $p' \in H^3(\mathbb{Z}_3,U(1))$.
Recommendation
Ask for major revision