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Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall Conductance
by Po-Shen Hsin, Ryohei Kobayashi, Carolyn Zhang
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Submission summary
| Authors (as registered SciPost users): | Po-Shen Hsin · Ryohei Kobayashi |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2405.20401v2 (pdf) |
| Date submitted: | 2024-06-15 01:13 |
| Submitted by: | Kobayashi, Ryohei |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible $0$-form symmetries. These symmetries can arise as global symmetries in quantum spin liquids, given by the quotient of the projective symmetry group by a non-normal subgroup as invariant gauge group. We point out that such coset non-invertible symmetries in topological orders can exhibit symmetry fractionalization: each anyon can carry a "fractional charge" under the coset non-invertible symmetry given by a gauge invariant superposition of fractional quantum numbers. We present various examples using field theories and quantum double lattice models, such as fractional quantum Hall systems with charge conjugation symmetry gauged and finite group gauge theory from gauging a non-normal subgroup. They include symmetry enriched $S_3$ and $O(2)$ gauge theories. We show that such systems have a fractionalized continuous non-invertible coset symmetry and a well-defined electric Hall conductance. The coset symmetry enforces a gapless edge state if the boundary preserves the continuous non-invertible symmetry. We propose a general approach for constructing coset symmetry defects using a "sandwich" construction: non-invertible symmetry defects can generally be constructed from an invertible defect sandwiched by condensation defects. The anomaly free condition for finite coset symmetry is also identified.
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 2024-8-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2405.20401v2, delivered 2024-08-16, doi: 10.21468/SciPost.Report.9603
Report
The authors discuss non-invertible symmetries in 2+1d dimension from gauging non-normal discrete subgroups of ordinary invertible symmetries. In particular, they focus on continuous non-invertible symmetries and the analog of Hall conductance and fictionalization for such symmetries. The manuscript has a valuable general discussion in Section 3 (and Section 6) that can be of independent interest. Section 4 discusses lattice models realizing such phenomena.
I recommend the draft for publication and have the following minor comments and questions below:
Requested changes
1. Around equation (3.2), the authors discuss the general case of gauging a discrete subgroup $K$ of a $G \rtimes_\rho K$ symmetry. However, the example in equation (3.5) is not of this type. It is worthwhile either expanding the general case or explaining why the subgroup $K$ is not normal in this case. In addition, it is worthwhile to describe why gauging a non-normal subgroup leads to a non-invertible symmetry or add appropriate references.
2. Related to the previous point, I recommend citing the references [arXiv:1704.02330] and [arXiv:1712.09542], which discuss gauging discrete subgroups and non-invertible symmetries in 1+1d.
3. Does Figure 1 apply to discrete symmetries?
Recommendation
Publish (meets expectations and criteria for this Journal)
Report #1 by Anonymous (Referee 1) on 2024-8-13 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2405.20401v2, delivered 2024-08-13, doi: 10.21468/SciPost.Report.9590
Report
The manuscript explores symmetry fractionalization for non-invertible 0-form symmetry. The non-invertible symmetry is for the coset symmetry which can occur in quantum spin liquids. The manuscript includes both concrete examples and general abstract treatment, which would satisfy broad audience. In addition the manuscript is well organized and composed. Hence I recommend publishing the manuscript in SciPost, up to minor points listed below.
Requested changes
1. Eq. (3.1): This is correct when the stabilizer of $g$ in $K$ is trivial. It would be good either modify the equation so that it is always correct, or be explicit about that the trivial stabilizer is assumed.
2. Page 12: The non-invertible surface operator in $U(1)_8$ was found in https://arxiv.org/abs/1012.0911 .
3. Page 31: When a subgroup $K$ acts trivially on a low-energy effective theory, the authors considered $K$ to be gauged. While this notion seems standard in the context of spin liquid, in a general context $K$ is treated as a mere decoupled factor and not gauged. Although this is supposed to be explained in the cited reference [44], it would be nice to include a concise explanation to be self-contained for broader audience.
4. In section 6, the concrete example of application is for $S_3$ case. I wonder how the $O(2)$ exotic FQH example can occur in a microscopical model (or more desirably in an experiment). It would be nice to highlight it in this section if authors know the answer.
Recommendation
Ask for minor revision