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Emergent generalized symmetries in ordered phases
by Salvatore D. Pace
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Submission summary
| Authors (as registered SciPost users): | Salvatore Pace |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2308.05730v3 (pdf) |
| Date submitted: | 2023-09-29 20:50 |
| Submitted by: | Pace, Salvatore |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We explore the rich landscape of higher-form and non-invertible symmetries that emerge at low energies in generic ordered phases. Using that their charge is carried by homotopy defects (i.e., domain walls, vortices, hedgehogs, etc.), in the absence of domain walls we find that their symmetry defects in ${D}$-dimensional spacetime are described by ${(D-1)}$-representations of a ${(D-1)}$-group that depends only on the spontaneous symmetry-breaking (SSB) pattern of the ordered phase. These emergent symmetries are not spontaneously broken in the ordered phase. We show that spontaneously breaking them induces a phase transition into a nontrivial disordered phase that can have symmetry-enriched (non-)abelian topological orders, photons, and even more emergent symmetries. This SSB transition is between two distinct SSB phases$\unicode{x2013}$an ordinary and a generalized one$\unicode{x2013}$making it a possible generalized deconfined quantum critical point. We also investigate the 't Hooft anomalies of these emergent symmetries and conjecture that there is always a mixed anomaly between them and the microscopic symmetry spontaneously broken in the ordered phase. One way this anomaly can manifest is through the fractionalization of the microscopic symmetry's quantum numbers. Our results demonstrate that even the most exotic generalized symmetries emerge in ordinary phases and provide a valuable framework for characterizing them and their transitions.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024-3-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2308.05730v3, delivered 2024-03-19, doi: 10.21468/SciPost.Report.8734
Report
This article presents an interesting perspective on where generalized symmetries can appear in quantum systems. The author describes how ordered systems of conventional or more generally invertible/higher group symmetries could provide a pretty general avenue to find more exotic higher categorical symmetries. While this is an interesting and less explored work in the literature thus far, I think the work could benefit from having more examples formulated as conventional quantum field theories and Hamiltonian lattice models to convey the theoretical ideas.
Before recommending this work, I also have some slightly more specific questions:
1. Can the author describe the structure of 3Rep(G) symmetries concretely in some G symmetry broken phase. This category has infinitely many simple objects that presumably act identically on the charged operators. How are these simple objects represented within a concrete model?
2. In examples where the symmetry is 2-group with a non-trivial Postnikov class, how does the Postnikov class appear in the properties of the homotopy defects?
3. What are the forms of the condensation defects in 2Rep(S3) or 2Rep(D8) concretely realised within an ordered model? How do these defects act the homotopy defects?
4. Similarly is it possible to write the form of Q8 non-invertible 1-form symmetry generators, say wrapping a non-contractible cycle of space, in a concrete model displaying SO(3)—> Z2 x Z2 SSB? What determines the vacuum expectation value of such an operator?
Report #1 by Anonymous (Referee 1) on 2024-2-12 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2308.05730v3, delivered 2024-02-12, doi: 10.21468/SciPost.Report.8547
Report
The manuscript discuss emergent symmetries in ordered phase with spontaneously symmetry breaking. The emergent symmetries are organized in terms of homotopy groups of the sigma model.
Before I can recommend it for publication, here are a few questions to be addressed:
- The author does not discuss possible topological action such as theta term or Wess-Zumino term: even when the symmetry breaking pattern is the same, there are different sigma models distinguished by topological actions, and they can have different symmetries.
- The author discuss symmetry in terms of homotopy groups instead of cohomology. However, homotopy group does not always give the correct symmetry, see e.g. https://arxiv.org/pdf/1707.05448.pdf
https://arxiv.org/abs/2210.13780
- The author discuss whether homotopy defects are invertible. But fusing two homotopy defects can produce nontrivial non-topological defects with topological charge zero (e.g. most elementary excitations have zero topological charges). Can the author clarify how the fusion is defined?
- There is a discussion using Postnikov system. What is the physical meaning in terms of defects, e.g. does it imply some relations between correlation function? (as the defects are generally not topological, it is hard to imagine there is such universal relation just from homotopy groups)
Requested changes
See the questions in the report