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

by Omer 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_00019v1  (pdf)
Date submitted: 2023-08-14 23:02
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.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 3 on 2023-11-13 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202308_00019v1, delivered 2023-11-13, doi: 10.21468/SciPost.Report.8107


1. Section 2 was a clear introduction into the triality concept, using the well-known example of $\mathbb{Z}_2$ XYZ Heisenberg chain. I found it illuminating/a good refresher.

2. The subject of LSM and crystalline symmetry anomalies is of high value to the quantum matter community. Since the concept of gauging crystalline symmetries is complicated, it is nice to see the authors use a more well-known approach (i.e. gauging internal symmetries) to identify LSM type systems.

3. Results seem correct and are new, although potentially straightforward generalization of previous work. It would help if the authors could more clearly highlight their new contributions versus already known results.


1. Although this approach is demonstrative of some of the interesting properties of LSM, I feel it is not generalizable or illuminating to the more general non-onsite significance of translation/crystalline symmetries (for example, LSM-like theories when there is only translation symmetry). However, I understand that this is beyond the scope of this paper - perhaps the authors could describe the limitations of their methods in the conclusion/introduction.

2. I feel that the presentation could more clearly emphasize the new results versus what is already known in literature. Section 2 (although very nice) is very long, so it distract the readers from the actual flesh of the paper. Perhaps 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).


The manuscript "Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains" illuminates the procedure of gauging non-anomalous symmetry subgroups to detect existence of an LSM obstruction. The authors present their work clearly with the general subject being of great interest.

My main reservation is the limitation/novelty of this method to study general LSM type systems and illuminate the significance of non-onsite symmetries such as translation. However, with the appropriate changes, this manuscript certainly deserves publication in SciPost. I thank the author for their thought-provoking and interesting work!

  • validity: high
  • significance: good
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: excellent

Anonymous Report 2 on 2023-10-22 (Invited Report)


I have read this paper and I have a few comments. In my opinion this is an important paper which, in some form, deserved to be published. I say "in some form" because the paper is really hard to read. It is much (and unnecessarily) too long. There are too many formal results proven (or argued) one after another without a sense of importance or significance. I am not sure if the authors need all these results. Perhaps a fraction of them (the less important ones) can be presented in an appendix. The applications are very interesting but they get lost after the myriad of (claimed) rigorous results presented in the very long first part of the paper. I would recommend either to split the paper in two, or to present in the body of the paper only the important results. If the authors cannot decide which are the more important results they cannot possibly expect the readers to do it for them. The result is that their important results will be lost to the readership.

Thus, my recommendation is to publish after the authors submit a more readable manuscript.

Requested changes

see above

  • validity: high
  • significance: good
  • originality: high
  • clarity: low
  • formatting: below threshold
  • grammar: perfect

Anonymous Report 1 on 2023-8-22 (Invited Report)


Detailed analysis of several lattice models and the relations between them using duality or gauging.


No clear comparison of the main results with the existing statements in the literature.


I suggest to add a comparison of the main results with the existing statements in the literature. This can be added in the Conclusions.

After such an addition, the paper is suitable for publication.

Requested changes

See above.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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