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Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry

by Yi-Hsien Du, Umang Mehta, Dung Xuan Nguyen, Dam Thanh Son

Submission summary

As Contributors: Dung Nguyen
Arxiv Link: (pdf)
Date submitted: 2021-10-07 05:13
Submitted by: Nguyen, Dung
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


We propose nonabelian higher-rank gauge theories in 2+1D and 3+1D. The gauge group is constructed from the volume-preserving diffeomorphisms of space. We show that the intriguing physics of the lowest Landau level (LLL) limit can be interpreted as the consequences of the symmetry. We derive the renowned Girvin-MacDonald-Platzman (GMP) algebra as well as the topological Wen-Zee term within our formalism. Using the gauge symmetry in 2+1D, we derive the LLL effective action of vortex crystal in rotating Bose gas as well as Wigner crystal of electron in an applied magnetic field. We show that the non-linear sigma models of ferromagnet in 2+1D and 3+1D exhibit the higher-rank gauge symmetries that we introduce in this paper. We interpret the fractonic behavior of the excitations on the lowest Landau level and of skyrmions in ferromagnets as the consequence of the higher-rank gauge symmetry.

Current status:
Editor-in-charge assigned

Author comments upon resubmission

To the editors of SciPost Physics:

We thank the referee for the careful reading of our manuscript and comments, which have helped us improve our work.

We have edited the SciPost Physics manuscript in line with these comments. We believe that after these revisions, we have answered all of these referees’ queries, and our work is now appropriate for publication in SciPost Physics.

Yi-Hsien Du, Umang Mehta, Dung X. Nguyen and Dam Thanh Son

List of changes

-We changed the format of the manuscript to SciPost Physics format.

-We fixed typos.

- We added comments at the end of section 4.1 to discuss the two independent conservation laws. One is the consequence of the volume-preserving diffeomorphism (combining with a corresponding $U(1)$ gauge transformation). One is the charge conservation as the consequence of $U(1)$ gauge transformation only.

-We added some comments at the end of section 5 in the revised version to distinguish the differences between 3+1D higher-rank symmetry in this paper with the vector charge theory proposed in Ref[2].

- We added the definition of curly brackets under equation 74 and footnote [2] to define the explicit definition in the case of the curly brackets with 2 and 3 indices.

- Explained the acronym NLS in Eq. 88 (equation 95 in the new version) and add the reference for more details of the non-linear sigma model.

- Added the definition of the emergent gauge field $a_i$ in section VI.D (Section 6.4 in the new version)
- We split section VI.D to 6.4.1 and 6.4.2 to discuss Ferromagnets in 2+1D and 3+1D separately.
- Moved the discussion of the $\mathbb{CP}^1$ parametrization to subsection 6.4.1 (Ferromagnets in 2+1D)
- We added Appendix A to discuss the uniqueness of the non-linear transformation of $A_0$ in Eq. (24).

Reports on this Submission

Anonymous Report 2 on 2021-11-2 (Invited Report)


The authors have satisfactorily addressed the concerns and comments of my previous report. The only point I still find weak, is the argument of treating the gauge field $A_i$ and the metric $h_{ij}$ as independent fields. Actually, in the quantum Hall example (LLL) the momentum conservation is more a constraint than a conservation equation, this fact somehow seems to relate the fields $A_i, h_{ij}$. However, this is a minor aspect and I think the content of the paper is innovative and scientifically sound, therefore, I believe it satisfies the publication criteria of SciPost, and I recommend it for publication.

  • validity: -
  • significance: high
  • originality: high
  • clarity: ok
  • formatting: good
  • grammar: reasonable

Anonymous Report 1 on 2021-11-1 (Invited Report)


1. The paper is clearly written, and derivations are easy to follow.

2. Many examples are given where VPD can be found and lead to higher rank gauge symmetry upon linearization.

3. Provides a unified way of understanding some known fracton-like properties in conventional condensed matter systems.


1. Many annoying typos are not corrected during the resubmission. Some of them can be easily fixed using a spellchecker or simply by carefully reading the manuscript.

2. Some examples are very brief and do not provide a physical context. For example, in the conclusion section, there is a sentence, "We also show that the nonlinear sigma models of ferromagnetism also exhibits this symmetry, if one couples a gauge potential of the higher-rank symmetry with the topological charge density." While this is indeed shown in section 6, it remains mysterious what physical situation requires the presence of the gauge potential of higher rank. (also exhibits should be changed to exhibit)


The central observation of the paper is that the higher-rank gauge symmetries (HRGS) can occur as a linearization of the symmetry under volume-preserving diffeomorphisms (VPD). Therefore, symmetry under VPD can be thought of as a nonlinear version of HRGS.

The authors give a few examples of known systems where symmetry under VPD can be realized by appropriate coupling to background fields. As the linearization of VPD produces HRGS, one can derive from the latter some of the fractonic properties in those examples. As a result, many known fractonic properties can be understood as emerging from the underlying symmetry under VPD. However, it is not clear under what conditions the HRGS can be promoted to VPD. The authors provide an example of the abelian HRGS (see eq. 85), which they could not lift to VPD.

All examples considered in the paper are essentially nonlinear sigma models with the metric background on a target space.

I find the paper interesting and worth publishing in SciPost Physics. The manuscript is clearly written except for a few points listed below.

Requested changes

1. Multiple typos and grammar inconsistencies should be corrected. Examples of trivial typos are: "notice noticed", "that manifests the gauge invariant." (that is manifestly gauge invariant), "Up to quadratic order and ignore the total" (ignoring), and many others.

2. In the generally phrased citation "The connection between volume-preserving diffeomorphism and quantum Hall physics was also noticed in Refs. [14] and [26]." it would be very relevant to refer to much earlier works (current Refs 33, 34, of the manuscript).

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

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