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Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems

by Meng Cheng, Nathan Seiberg

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Submission summary

Authors (as registered SciPost users): Meng Cheng · Nathan Seiberg
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-05-22
Date submitted: 2023-04-09 17:04
Submitted by: Cheng, Meng
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


We analyze lattice Hamiltonian systems whose global symmetries have 't Hooft anomalies. As is common in the study of anomalies, they are probed by coupling the system to classical background gauge fields. For flat fields (vanishing field strength), the nonzero spatial components of the gauge fields can be thought of as twisted boundary conditions, or equivalently, as topological defects. The symmetries of the twisted Hilbert space and their representations capture the anomalies. We demonstrate this approach with a number of examples. In some of them, the anomalous symmetries are internal symmetries of the lattice system, but they do not act on-site. (We clarify the notion of "on-site action.") In other cases, the anomalous symmetries involve lattice translations. Using this approach we frame many known and new results in a unified fashion. In this work, we limit ourselves to 1+1d systems with a spatial lattice. In particular, we present a lattice system that flows to the $c=1$ compact boson system with any radius (no BKT transition) with the full internal symmetry of the continuum theory, with its anomalies and its T-duality. As another application, we analyze various spin chain models and phrase their Lieb-Shultz-Mattis theorem as an 't Hooft anomaly matching condition. We also show in what sense filling constraints like Luttinger theorem can and cannot be viewed as reflecting an anomaly. As a by-product, our understanding allows us to use information from the continuum theory to derive some exact results in lattice model of interest, such as the lattice momenta of the low-energy states.

List of changes

We have added a few clarifications and comments to address questions from the referee reports and have corrected typos. An additional reference ([67]) was added.

Published as SciPost Phys. 15, 051 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-4-17 (Invited Report)


The authors have addressed my comments from the previous report and the paper is now ready for publication.

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

Anonymous Report 1 on 2023-4-12 (Invited Report)


Now that the authors have responded to the points in the Report, I recommend publication in SciPost Physics.

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

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