SciPost Submission Page
(SPT-)LSM theorems from projective non-invertible symmetries
by Salvatore D. Pace, Ho Tat Lam, and \"{O}mer M. Aksoy
Submission summary
| Authors (as registered SciPost users): | Salvatore Pace |
| Submission information | |
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| Preprint Link: | scipost_202410_00012v1 (pdf) |
| Date submitted: | 2024-10-09 00:33 |
| Submitted by: | Pace, Salvatore |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Projective symmetries are ubiquitous in quantum lattice models and can be leveraged to constrain their phase diagram and entanglement structure. In this paper, we investigate the consequences of projective algebras formed by non-invertible symmetries and lattice translations in a generalized $1+1$D quantum XY model based on group-valued qudits. This model is specified by a finite group $G$ and enjoys a projective $\mathsf{Rep}(G)\times Z(G)$ and translation symmetry, where symmetry operators obey a projective algebra in the presence of symmetry defects. For invertible symmetries, such projective algebras imply Lieb-Schultz-Mattis (LSM) anomalies. However, this is not generally true for non-invertible symmetries, and we derive a condition on $G$ for the existence of an LSM anomaly. When this condition is not met, we prove an SPT-LSM theorem: any unique and gapped ground state is necessarily a non-invertible weak symmetry protected topological (SPT) state with non-trivial entanglement, for which we construct an example fixed-point Hamiltonian. The projectivity also affects the dual symmetries after gauging $\mathsf{Rep}(G)\times Z(G)$ sub-symmetries, giving rise to non-Abelian and non-invertible dipole symmetries, as well as non-invertible translations. We complement our analysis with the SymTFT, where the projectivity causes it to be a topological order non-trivially enriched by translations. Throughout the paper, we develop techniques for gauging $\mathsf{Rep}(G)$ symmetry and inserting its symmetry defects on the lattice, which are applicable to other non-invertible 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
Strengths
1. The paper studies a very interesting generalization of traditional LSM theorems to systems with non-invertible symmetries.
2. The generalized XY model studied in the paper encompasses numerous intriguing details and a wide range of interesting phenomena is demonstrated by this model.
Weaknesses
1. The paper focuses solely on non-invertible symmetry $Rep(G)\times Z(G)$, and potential generalizations to other non-invertible symmetries are not straightforward.
Report
The paper investigates the consequences of projective algebras formed by non-invertible symmetries and lattice translations in a generalized 1+1D quantum XY model based on group-valued qudits. This model is written in terms of G-qudits and enjoys a projective $Rep(G)\times Z(G)$ symmetry and translation symmetry. Depending on the detailed properties of $G$, this paper uncovers that the model either has some LSM anomaly or satisfies the condition of some SPT-LSM theorem.
As stated in the Strengths, this paper is very well-written and serves as a very timely addition to the subject. I recommend it to be published on Scipost.
Here are several questions that I have about the paper. The authors may consider integrating them in the paper if these questions are relevant.
1. A natural question is how to generalize these results to any fusion-category symmetry in 1D lattice systems. Hence there are some natural questions regarding this. For example, for invertible symmetry $G$, projective representations of $G$ are classified by $H^2(G, U(1))$. Is there a similar classification for non-invertible symmetry? It is known that LSM anomaly is characterized by an element $H^3(G, U(1))$ (for example, see https://arxiv.org/abs/2401.02533 for the most recent account on this), where G contains both internal as well as translation symmetries, can I obtain a similar classification in the context of non-invertible symmetries? What about SPT-LSM?
2. I am confused about the details of the SymTFT construction, and some clarification may be helpful. For example, why can we not also gauge the translation symmetry (by adding an extra $Z(\mathbb{Z}_L)$ quantum double in the symmetry-topological order, instead of SET)? This paper demonstrates that we can consider the nontrivial SPT state by calculating the Lagrangian algebra for $G=D_8$. Can we identify the corresponding Lagrangian algebra for general $G$ as well, whenever $Z(G)\subset [G, G]$?
What about just LSM anomalies from the point of view of SymTFT? Do they correspond to condensable algebras, or something else? Also, according to https://journals.aps.org/prb/abstract/10.1103/PhysRevB.100.115147, I think that a consistent set of U-symbols has to be chosen (which are possibly trivial, but I think that it is still worth mentioning) when specifying the action of translation on the topological order. And do we need to choose a consistent set of eta-symbols as well?
3. A smaller question, in Appendix C, can we just classify the Lagrangian algebra for $Z(Rep(D_8)\times \mathbb{Z}_2 \times \mathbb{Z}_2)$?
Requested changes
Just consider questions in the report and integrate them in the paper if relevant.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)