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Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains

by Ömer M. Aksoy, Christopher Mudry, Akira Furusaki, Apoorv Tiwari

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Submission summary

Authors (as registered SciPost users): Ömer M. Aksoy · Apoorv Tiwari
Submission information
Preprint Link: scipost_202308_00019v2  (pdf)
Date accepted: 2023-12-19
Date submitted: 2023-12-05 17:54
Submitted by: Aksoy, Ömer M.
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


Lieb-Schultz-Mattis (LSM) theorems impose non-perturbative constraints on the zero-temperature phase diagrams of quantum lattice Hamiltonians (always assumed to be local in this paper). LSM theorems have recently been interpreted as the lattice counterparts to mixed 't Hooft anomalies in quantum field theories that arise from a combination of crystalline and global internal symmetry groups. Accordingly, LSM theorems have been reinterpreted as LSM anomalies. In this work, we provide a systematic diagnostic for LSM anomalies in one spatial dimension. We show that gauging subgroups of the global internal symmetry group of a quantum lattice model obeying an LSM anomaly delivers a dual quantum lattice Hamiltonian such that its internal and crystalline symmetries mix non-trivially through a group extension. This mixing of crystalline and internal symmetries after gauging is a direct consequence of the LSM anomaly, i.e., it can be used as a diagnostic of an LSM anomaly. We exemplify this procedure for a quantum spin-1/2 chain obeying an LSM anomaly resulting from combining a global internal $\mathbb{Z}^{\,}_{2}\times\mathbb{Z}^{\,}_{2}$ symmetry with translation or reflection symmetry. We establish a triality of models by gauging a $\mathbb{Z}^{\,}_{2}\subset\mathbb{Z}^{\,}_{2}\times\mathbb{Z}^{\,}_{2}$ symmetry in two ways, one of which amounts to performing a Kramers-Wannier duality, while the other implements a Jordan-Wigner duality. We discuss the mapping of the phase diagram of the quantum spin-1/2 $XYZ$ chains under such a triality. We show that the deconfined quantum critical transitions between Neel and dimer orders are mapped to either topological or conventional Landau-Ginzburg transitions. Finally, we extend our results to $\mathbb{Z}^{\,}_{n}$ clock models with $\mathbb{Z}^{\,}_{n}\times\mathbb{Z}^{\,}_{n}$ global internal symmetry, and provide a reinterpretation of the dual internal symmetries in terms of $\mathbb{Z}^{\,}_{n}$ charge and dipole symmetries.

Author comments upon resubmission

We would like to thank all three referees for their reports.

Referee 1:

We thank Referee 1 for delivering very quickly their report. We followed their suggestion to ``... add a comparison of the main results with the existing statements in the literature.'' which we placed in Subsec. 1.3 of the introduction.

Referee 2:

We thank Referee 2 for assessing the validity, significance, originality, and grammar of our paper as high, good, high, and perfect, respectively.

We agree with Referee 2 that it is better to ``... present in the body of the paper only the important results.'' This is done with the addition of Subsecs. 1.2 and 1.3. We hope that these changes make our results more transparent and facilitate an easier read.

Our paper is comparable in length and is written in the style of Refs.

[39] M. Cheng and N. Seiberg, ``Lieb-Schultz-Mattis, Luttinger, and ’t Hooft - anomaly matching in lattice systems,'' SciPost Phys. 15, 051 (2023), doi:10.21468/SciPostPhys.15.2.051.

[41] N. Seiberg and S.-H. Shao, ``Majorana chain and Ising model –(non-invertible) trans- lations, anomalies, and emanant symmetries,'' arXiv e-prints arXiv:2307.02534 (2023), doi:10.48550/arXiv.2307.02534, 2307.02534 .

[53] S. Seifnashri, ``Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries,'' arXiv e-prints arXiv:2308.05151 (2023), doi:10.48550/arXiv.2308.05151, 2308.05151.

The first is already published, while the other two are submitted to SciPost Physics.

We appreciate the Referee's suggestion to reorganize the entire paper. However, we have decided to maintain the current structure,
as we believe it effectively communicates our ideas. Additionally, both Referees 1 and 3 share our perspective, emphasizing that the earlier sections are valuable in elucidating key concepts crucial for comprehending the results of our work.

Referee 3:

We thank Referee 3 for their report and praise of our Sec. 2. Sections 1.2 and 1.3 address the request of Referee 3 to ``... more clearly highlight their new contributions versus already known results.''

Regarding weakness 1:
Most LSM theorems that we know of mix crystalline and internal symmetries. We do not understand what is the meaning of gauging a crystalline symmetry on its own right and do not claim any breakthrough in this direction.

Regarding weakness 2:
We followed with Sec. 1.2 the suggestion to present ``..., a clearer summary of the new results in the introduction, for example in the form of bullet points (since the introduction is a large block of text).''

List of changes

1. The main changes are the partitioning of the introduction (Sec. 1) into four subsections. Subsections 1.1 and 1.4 are not new. Subsections 1.2 and 1.3 are new. In Subsec. 1.2, we present in a compact and itemized format our new results. In Subsec. 1.3, we explain the difference between our results and the literature.

2. In the bulk of the paper, we added new results with Eqs. (3.16), (3.17), (3.28), (3.29), and below Eq. (5.23). This is a reinterpretation of the absence of the LSM anomaly in the KW and JW dual theories as a manifestation of the presence of simultaneous charge, dipole, and crystalline symmetries. Correspondingly, we added a few lines to the abstract.

3. We made a correction around Eq. (4.14). Other changes are stylistic or typos.

Published as SciPost Phys. 16, 022 (2024)

Reports on this Submission

Anonymous Report 2 on 2023-12-13 (Invited Report)


I find the paper interesting and suitable for publications. The revisions made it clearer.

One could argue that the paper is too long. But as a matter of principle, it is better for a paper to be too long than to be too short and therefore incomprehensible. Personally, I do not think this paper is too long.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2023-12-9 (Invited Report)


I have read and reviewed the resubmitted version of this paper. I appreciate that the authors decided to include two new subsections in the Introduction, one summarizing the salient results and the other on the relation between this work and other published paper. I am disappointed that the authors did not appreciate my other recommendations as being difficult to implement. It is a pity as the paper is indeed much too long (as are the published papers cite as examples) and this will most likely result in a paper that is not as impactful as it could be. But this is their choice. I recommend that this paper be published.

  • validity: top
  • significance: high
  • originality: top
  • clarity: good
  • formatting: reasonable
  • grammar: perfect

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