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Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
by Sahand Seifnashri
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Submission summary
| Authors (as registered SciPost users): | Sahand Seifnashri |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2308.05151v2 (pdf) |
| Date submitted: | 2023-11-05 19:41 |
| Submitted by: | Seifnashri, Sahand |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study 't Hooft anomalies of global symmetries in 1+1d lattice Hamiltonian systems. We consider anomalies in internal and lattice translation symmetries. We derive a microscopic formula for the "anomaly cocycle" using topological defects implementing twisted boundary conditions. The anomaly takes value in the cohomology group $H^3(G,U(1)) \times H^2(G,U(1))$. The first factor captures the anomaly in the internal symmetry group $G$, and the second factor corresponds to a generalized Lieb-Schultz-Mattis anomaly involving $G$ and lattice translation. We present a systematic procedure to gauge internal symmetries (that may not act on-site) on the lattice. We show that the anomaly cocycle is the obstruction to gauging the internal symmetry while preserving the lattice translation symmetry. As an application, we construct anomaly-free chiral lattice gauge theories. We demonstrate a one-to-one correspondence between (locality-preserving) symmetry operators and topological defects, which is essential for the results we prove. We also discuss the generalization to fermionic theories. Finally, we construct non-invertible lattice translation symmetries by gauging internal symmetries with a Lieb-Schultz-Mattis anomaly.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2024-1-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2308.05151v2, delivered 2024-01-30, doi: 10.21468/SciPost.Report.8475
Strengths
1. A very detailed discussion on various properties of the topological defects for internal and translation symmetry on the lattice such as moving and fusing of the defects.
2. A concrete procedure to construct topological defects on the lattice.
3. Discussion on how to obtain the symmetry operators from the defects and vice versa.
4. Provide a concrete formula to compute the anomalies for internal and spatial symmetries on the lattice.
5. A very clear discussion on the general gauging procedure and on the anomaly as obstruction of gauging.
Report
This paper provides a very detailed discussion on the topological defects for both internal and translation symmetry and how to compute the anomalies on the lattice. In addition to the strengths mentioned above, it contains a very nice discussion on the topological defects for the lattice translation symmetry and antiunitary symmetry. I believe the framework developed in this work makes an important contribution to our understanding of the topological defect on the lattice. This paper is very well-written and easy to follow. I recommend this paper to be published in SciPost Physics.
Requested changes
1. The author claims that the method introduced in this work can also be applied to reflection symmetry but doesn't provide explicit discussion. It would be nice to have an example for the reflection symmetry defect.
2. What modifications for the lattice calculation of the anomaly are needed if the anomaly involves reflection symmetry.
3. It would be nice to have an example on gauging translation symmetry explicitly on the lattice.
Report #1 by Anonymous (Referee 2) on 2023-12-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2308.05151v2, delivered 2023-12-25, doi: 10.21468/SciPost.Report.8342
Strengths
1. The authors develop a solid theory of LSM anomalies from the perspective of gauging. They clarify some subtleties left in the previous literature. Discussions on symmetry operators, topological defects, and their relations are quite clear, and easy to follow.
2.The systematic constructions for truncating a non-onsite symmetry, e.g. translations, provide stable foundations for some intuitive ideas.
3.The gauging procedure provides perspectives for other areas, such as lattice gauge theory and non-invertible symmetry.
Report
The paper presents a systematic study of anomaly for both onsite and non-onsite symmetries, from the perspective of gauging obstructions. Some mathematical formalisms are developed to make this gauging procedure clear. Cocycle data can be extracted, providing direct evidence for ungaugability.
The systematic constructions for gauging a non-site symmetry make this appliable to many known models.
I would recommend acceptance.
Requested changes
1. The F-symbol satisfies a modified pentagon equation, right hand side of Eqn. (3.5). Is this specialized to this context or universal on the lattice? Maybe a footnote for clarification is needed.
2. In Sec. 4.1 and 4.2, examples for gauging internal symmetry are given. A short parallel discussion on gauging translation symmetry would confirm the mixed ‘t Hooft anomaly between them.
3. In the third paragraph of page 41, I think there is a logical gap for people who are not familiar with splitable states. “The fact that each region has d > 1 ground states is in contradiction with …” This sentence should be sharpened to make this contradiction clear.