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Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
by Salvatore D. Pace, Guilherme Delfino, Ho Tat Lam, Ömer M. Aksoy
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Submission summary
| Authors (as registered SciPost users): | Ömer M. Aksoy · Salvatore Pace |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2406.12962v2 (pdf) |
| Date submitted: | 2024-07-30 22:01 |
| Submitted by: | Pace, Salvatore |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Modulated symmetries are internal symmetries that act in a non-uniform, spatially modulated way and are generalizations of, for example, dipole symmetries. In this paper, we systematically study the gauging of finite Abelian modulated symmetries in ${1+1}$ dimensions. Working with local Hamiltonians of spin chains, we explore the dual symmetries after gauging and their potential new spatial modulations. We establish sufficient conditions for the existence of an isomorphism between the modulated symmetries and their dual, naturally implemented by lattice reflections. For instance, in systems of prime qudits, translation invariance guarantees this isomorphism. For non-prime qudits, we show using techniques from ring theory that this isomorphism can also exist, although it is not guaranteed by lattice translation symmetry alone. From this isomorphism, we identify new Kramers-Wannier dualities and construct related non-invertible reflection symmetry operators using sequential quantum circuits. Notably, this non-invertible reflection symmetry exists even when the system lacks ordinary reflection symmetry. Throughout the paper, we illustrate these results using various simple toy models.
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
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The authors discuss the construction of generalized reflection symmetries in 1+1d lattice Hamiltonian systems. They study generalized Ising models with internal symmetries that do not necessarily commute with lattice translations. Furthermore, they gauge these internal symmetries and construct non-invertible reflection symmetries when the system is self-dual under gauging. I recommend the draft for publication and have the following questions:
The authors implement gauging using Gauss's law in equation (1.15). They should comment on how this equation is derived and whether the expression is unique. In particular, there might be multiple (untwisted/twisted) ways to gauge the symmetry, corresponding to different forms of Gauss's law.
What are the possible gapped or gapless phases that remain invariant under non-invertible reflection symmetries? Specifically, what is the low-energy behavior of these systems with non-invertible reflection symmetries at J=h?
Requested changes
1. Provide more detail on Guass' law and its uniqueness.
2. Comment on the low-energy phase of the generalized Ising models at the self-dual point.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report
In this paper the authors give a treatment of the gauging of spatially modulated symmetries in translation-invariant spin chains. The authors apply this to a class of generalized Ising models with such modulated symmetries, and identify non-invertible Kramers-Wannier type dualities (sometimes involving a spatial reflection). A number of natural follow up questions are suggested.
The material on gauging spatial symmetries consists of a mix of examples and general results. The examples are certainly helpful, but it is not always clear what the most important material is. The authors give a summary in the introduction, nevertheless it would be helpful to streamline these two sections, perhaps by moving some of the examples to an appendix.
The treatment of the Kramers-Wannier dualities is interesting. It would be useful to comment on whether the generalized Ising models have previously appeared in the literature, and if not, more could be said about their physics. For example, the duality becomes a symmetry at the point $J=h$ - is this a critical point of the model?
Requested changes
Some additional comments on the generalized Ising models.
Consider streamlining the main text to improve readability of the paper.
I found the following typos that should be corrected:
1) Footnote 5 "refereed to"
2) (2.47) right hand equation $j\rightarrow j+1$.
[Also consider writing $p=2$ in (2.44) and replacing $Z^\dagger$ with $Z$ in (2.48).]
Recommendation
Publish (meets expectations and criteria for this Journal)