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Generalized symmetry enriched criticality in (3+1)d
by Benjamin Moy
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Benjamin Moy |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202507_00088v1 (pdf) |
| Date submitted: | July 30, 2025, 8:51 p.m. |
| Submitted by: | Benjamin Moy |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We construct two classes of continuous phase transitions in 3+1 dimensions between phases that break distinct generalized global symmetries. Our analysis focuses on $SU(N)/\mathbb{Z}_N$ gauge theory coupled to $N_f$ flavors of Majorana fermions in the adjoint representation. For $N$ even and sufficiently large odd $N_f$, upon imposing time-reversal symmetry and an $SO(N_f)$ flavor symmetry, the massless theory realizes a quantum critical point between two gapped phases: one in which a $\mathbb{Z}_N$ one-form symmetry is completely broken and another where it is broken to $\mathbb{Z}_2$, leading to $\mathbb{Z}_{N/2}$ topological order. We provide an explicit lattice model that exhibits this transition. The critical point has an enhanced symmetry, which includes non-invertible analogues of time-reversal symmetry. Enforcing a non-invertible time-reversal symmetry and the $SO(N_f)$ flavor symmetry, for $N$ and $N_f$ both odd, we demonstrate that this critical point can appear between a topologically ordered phase and a phase that spontaneously breaks the non-invertible time-reversal symmetry, furnishing an analogue of deconfined quantum criticality for generalized symmetries.
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
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-10-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202507_00088v1, delivered 2025-10-09, doi: 10.21468/SciPost.Report.12096
Report
The topic is timely. The interplay of generalized (higher-form / non-invertible) symmetries, topological order, and critical phenomena is at the frontier of condensed matter / high-energy duality lines of research. The paper finds interesting new examples of critical points enriched by generalized symmetries.
I believe this is a compelling and high-quality contribution to the study of generalized symmetries and critical phenomena. I recommend acceptance after minor revisions. Maybe it would be helpful that the author more more explicitly situate the results relative to prior work on deconfined quantum criticality, generalized symmetries in critical phases, and non-invertible symmetry studies. Perhaps a table or comparative discussion would help.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report #1 by Anonymous (Referee 1) on 2025-10-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202507_00088v1, delivered 2025-10-08, doi: 10.21468/SciPost.Report.12080
Strengths
Weaknesses
1) The point of the lattice model section 4.5 is unclear. If the lattice analysis provides additional insight into the phase structure, the authors should make it explicit. Otherwise, I suggest moving it to the appendix.
2) The discussion on page 23 about the symmetry fractional class is rough and incomplete. The author seems to aim at emphasizing the role of global symmetry in the topological phases, and I believe it would be worthwhile to expand this part of the discussion. I would suggest the author specify the group cohomology and provide more details if possible.
Report
On page 5: ''we emphasize that these quasiparticles need not be magnetic monopoles of the electromagnetic field'' This phrasing is somewhat misleading, as the term ''electromagnetic field'' almost exclusively refers to U(1) gauge fields.
Equation (2.3): I suggest the author provide a more detailed definition in terms of the singular boundary condition.
Equations (2.8) and (2.11): the author has used two different constants $\rho$ and $\tilde{\rho}$ without an explanation. I believe it is beneficial to note that the constants are generally schema-dependent.
Equation (4.19) and (5.25): I assume this is the leading result at the limit $g\to 0$. Can the author provide what parameter controls this expansion? Is the next-order correction at the order $O(g^2)$, $O(m/\Lambda)$, or something else?
On page 34 above equation (5.6): As far as I understand, $T$ and $D_n$ by themselves are not symmetries for theoreis with $\theta=\pi n$. Rather, they are topological interfaces between different theories. The author's statement "T is the standard invertible time-reversal operator" is slightly misleading. I suggest an improvement.
Recommendation
Ask for major revision
Thank you for your detailed and thoughtful comments. Below are replies to specific comments:
Referee's comment:
The point of the lattice model section 4.5 is unclear. If the lattice analysis provides additional insight into the phase structure, the authors should make it explicit. Otherwise, I suggest moving it to the appendix.
Reply:
I was independently considering whether to move the lattice model subsection to an appendix, so thank you for this constructive feedback. The primary utility of the lattice model is that it provides a natural way to study the phase structure upon explicitly breaking the $\mathbb{Z}_N$ magnetic one-form symmetry, which is more typical in condensed matter settings. Since this analysis was left to future work, I agree that it is better to move this subsection to the appendix and have done so in the updated version.
Referee's comment:
The discussion on page 23 about the symmetry fractional class is rough and incomplete. The author seems to aim at emphasizing the role of global symmetry in the topological phases, and I believe it would be worthwhile to expand this part of the discussion. I would suggest the author specify the group cohomology and provide more details if possible.
Reply:
Thanks for this suggestion. I have added more details to the discussion of symmetry fractionalization in Section 4.2 including the group cohomology. Correspondingly, I have added two paragraphs in Section 4.3 (on pp. 27-28) commenting on the role of the symmetry fractionalization in the other topologically ordered phase.
Referee's comment:
On page 5: ''we emphasize that these quasiparticles need not be magnetic monopoles of the electromagnetic field'' This phrasing is somewhat misleading, as the term ''electromagnetic field'' almost exclusively refers to U(1) gauge fields.
Reply:
I meant that the monopoles of the PSU(N) gauge theory should not be confused with monopoles of the U(1) electromagnetic gauge field. The paper has been edited to reflect that.
Referee's comment:
Equation (2.3): I suggest the author provide a more detailed definition in terms of the singular boundary condition.
Reply:
I have added a more explicit definition of the 't Hooft loop.
Referee's comment:
Equations (2.8) and (2.11): the author has used two different constants $\rho$ and $\tilde{\rho}$ without an explanation. I believe it is beneficial to note that the constants are generally schema-dependent.
Reply:
I have added a note that the constants depend on scheme.
Referee's comment:
Equation (4.19) and (5.25): I assume this is the leading result at the limit $g\to 0$. Can the author provide what parameter controls this expansion? Is the next-order correction at the order $O(g^2)$, $O(m/\Lambda)$ , or something else?
Reply:
The expansion parameter is $|m|/\Lambda_{\mathrm{UV}}$. To obtain higher order corrections, one should solve the coupled RG equations for $g$ and $m$ to lowest order in both parameters. This was done numerically in References 128 and 129, and their results agree with Eq. 4.19 for $|m|/\Lambda_\mathrm{UV}\ll 1$. Because $g$ is marginally irrelevant at the critical point, I expect that the next order correction to Eq. 4.19 should introduce logarithmic corrections to scaling, like for the Ising critical point in (3+1)d.
Referee's comment:
On page 34 above equation (5.6): As far as I understand, $T$ and $D_n$ by themselves are not symmetries for theoreis with $\theta=\pi n$. Rather, they are topological interfaces between different theories. The author's statement "T is the standard invertible time-reversal operator" is slightly misleading. I suggest an improvement.
Reply:
Thanks for catching this. I revised the statement, now referring to $\mathsf{T}$ as an interface that reverses orientation.
Author: Benjamin Moy on 2025-10-20 [id 5944]
(in reply to Report 2 on 2025-10-09)Thank you for your positive feedback and your suggestion. I have added a paragraph to the introduction at the bottom of p. 6 discussing previous work on critical phases and critical points for which non-invertible symmetry plays an important role.