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Volumepreserving diffeomorphism as nonabelian higherrank gauge symmetry
by YiHsien Du, Umang Mehta, Dung Xuan Nguyen, Dam Thanh Son
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Submission summary
Authors (as registered SciPost users):  Dung Nguyen 
Submission information  

Preprint Link:  https://arxiv.org/abs/2103.09826v3 (pdf) 
Date submitted:  20211007 05:13 
Submitted by:  Nguyen, Dung 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We propose nonabelian higherrank gauge theories in 2+1D and 3+1D. The gauge group is constructed from the volumepreserving 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 GirvinMacDonaldPlatzman (GMP) algebra as well as the topological WenZee 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 nonlinear sigma models of ferromagnet in 2+1D and 3+1D exhibit the higherrank 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 higherrank gauge symmetry.
Author comments upon resubmission
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.
Sincerely
YiHsien 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 volumepreserving 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 higherrank 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 nonlinear 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 nonlinear transformation of $A_0$ in Eq. (24).
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2021112 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2103.09826v3, delivered 20211102, doi: 10.21468/SciPost.Report.3778
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.
Report #1 by Anonymous (Referee 2) on 2021111 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2103.09826v3, delivered 20211031, doi: 10.21468/SciPost.Report.3768
Strengths
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 fractonlike properties in conventional condensed matter systems.
Weaknesses
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 higherrank 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)
Report
The central observation of the paper is that the higherrank gauge symmetries (HRGS) can occur as a linearization of the symmetry under volumepreserving 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 volumepreserving 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).
Author: Dung Nguyen on 20211213 [id 2027]
(in reply to Report 1 on 20211101)
Comment: 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 exhibit this symmetry, if one couples a gauge potential of the higherrank 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)
Reply: We did not consider the physical gauge fields that skyrmions couple to. In fact, in the manuscript, the background fields were used as a formal device to provide an alternative interpretation of the emergent symmetries (from which the conserved quantities follow).
In principle, one can think of a physical situation where the higherrank gauge potential can be realized. The variation of metric $h_{ij}$ can be realized through the local deformation of the lattice (which induces the spatial dependent magnetic coupling). In addition, the source of skyrmion density (scalar potential $A_0$) is related to the applied spinpolarized current in a complicated way (One can see more detail in Chapter 7 of Ref. [54]). However, the topic is a little bit complicated, thus we do not discuss further the physical realization of the gauge potentials in our manuscript.
Comment: Multiple typos and grammar inconsistencies should be corrected.
Reply: We thank the referee for the comment. Indeed, in the revised manuscript, we have tried to carefully fix all the typos.
Comment: In the generally phrased citation "The connection between volumepreserving 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).
Reply: We cited more references that studied the connection between volumepreserving diﬀeomorphism and quantum Hall physics, including Refs. 33, 34 and the new Refs. 4750.
Author: Dung Nguyen on 20211213 [id 2026]
(in reply to Report 2 on 20211102)We would like to thank the referee for the recommendation for publication and previous feedback that helped us improve our manuscript significantly.