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Fractons, dipole symmetries and curved spacetime
by Leo Bidussi, Jelle Hartong, Emil Have, Jørgen Musaeus, Stefan Prohazka
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Submission summary
| Authors (as registered SciPost users): | Emil Have · Stefan Prohazka |
| Submission information | |
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| Preprint Link: | scipost_202201_00038v1 (pdf) |
| Date submitted: | 2022-01-27 12:35 |
| Submitted by: | Have, Emil |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study complex scalar theories with dipole symmetry and uncover a no-go theorem that governs the structure of such theories and which, in particular, reveals that a Gaussian theory with linearly realised dipole symmetry must be Carrollian. The gauging of the dipole symmetry via the Noether procedure gives rise to a scalar gauge field and a spatial symmetric tensor gauge field. We construct a worldline theory of mobile objects that couple gauge invariantly to these gauge fields. We systematically develop the canonical theory of a dynamical symmetric tensor gauge field and arrive at scalar charge gauge theories in both Hamiltonian and Lagrangian formalism. We compute the dispersion relation of the modes of this gauge theory, and we point out an analogy with partially massless gravitons. It is then shown that these fractonic theories couple to Aristotelian geometry, which is a non-Lorentzian geometry characterised by the absence of boost symmetries. We generalise previous results by coupling fracton theories to curved space and time. We demonstrate that complex scalar theories with dipole symmetry can be coupled to general Aristotelian geometries as long as the symmetric tensor gauge field remains a background field. The coupling of the scalar charge gauge theory requires a Lagrange multiplier that restricts the Aristotelian geometries.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2022-4-1 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202201_00038v1, delivered 2022-04-01, doi: 10.21468/SciPost.Report.4844
Strengths
Solid results that a clearly correct.
Weaknesses
The only weakness I find is that the paper is written as a chapter in a textbook. It is hard to look for main results, which could have easily been summarized on a single page, and it is hard to chase some of the definitions to make sense of equations. I would recommend to add a section where the main equations are summarized and all notations needed to understand those are introduced. However, if the authors feel strongly about keeping the paper the way it is I will not object to the publication ``as is''.
Report
The authors explain the following issues:
(1) Gaussian fracton theories have Carroll symmetry. This observation is extremely simple, but, to the best of my knowledge, has not been stated previously and counts as new.
(2) They explain how to couple dipole conserving theories to Aristotelian geometry. They carefully write the covariant version of the complex scalar theory with dipole conservation. The main result is Eq.(65).
(3) They also show how to couple the tensor gauge theory to curved space and recover previous result about compatibility of the gauge structure with Einstein geometry. Incidentally, this was first done in 1712.06600 in 2 spatial dimensions.
These results are very good and should certainly be published.
Report #1 by Anonymous (Referee 2) on 2022-3-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202201_00038v1, delivered 2022-03-03, doi: 10.21468/SciPost.Report.4613
Strengths
1) Work is generally quite thorough
2) Coupling fractons to curved spacetime is novel
Weaknesses
1) Writing style is sometimes confusing and notation-dense
Report
In this paper, the authors carefully couple the traceful and traceless scalar charge higher-rank gauge theories and dipole-conserving matter theories to curved spacetime, as well as coupling the matter theories to the gauge theories. In the process, they give some classification results for dipole-conserving matter theories. Some of the classification results were informally in the “lore” of the field, but this paper has formalized them sharply (although I have one important piece of confusion about this part of the work – see second requested change). As the authors point out, these gauge theories were coupled to curved space in Ref. 31; the present generalization to curved Aristotelian spacetime is nontrivial. Although I find the writing occasionally confusing and notation-dense, the results appear interesting and suggest that it is important and useful to consider curved spacetime even in theories where there is no Lorentz or Galilean boost symmetry. It would be helpful for the authors to address the following points and questions in the "requested changes" section, but once those points are addressed, I recommend this work for publication. This work prompts a number of interesting follow-up directions.
Requested changes
1) The statement “The restricted mobility of isolated fracton particles can be viewed as a consequence of conservation of their dipole moment” is not particularly generic and should be modified. While dipole conservation is the simplest way to obtain fractonic mobility restrictions, it is not the only way, particularly in more exotic quantum fracton models like Haah’s code.
2) I find the introduction of Eq. (2.27) very confusing. Why have we restricted to this particular expression? Is this the most general expression that obeys the symmetries (2.1-2.2)? This is very important given that the rest of the classification results of section 2 come from this expression.
3) Could the authors clarify the consequences of the constraint Eq. (6.17) for dipolar symmetries? On a curved background, what does the $\Lambda$ corresponding to a dipolar symmetry look like? Is it clear when Eq. (6.17) is satisfied for a dipolar symmetry?
4) The results of Sec. 7 appear to contradict the results of Ref. 31. Ref. 31 states that, when considering only curved space, the traceful scalar charge theory in any dimension can only be gauge-invariant on flat space. The present paper’s Eq. 7.17 appears to allow Einstein manifolds. Could the authors explain the relationship between these results?
5) Is there any generalization of this work’s approach to theories where, for example, continuous rotational symmetry is relaxed to discrete rotation symmetry? Or theories with subsystem symmetries (e.g. the off-diagonal or “hollow” scalar charge theories where $A_{ij}$ has only off-diagonal components)?
6) Two typos: Top of page 21 – “..are described by $B_i = \partial_i \Lambda$” $B_i$ should read $\Sigma_i$ . Also, broken reference just after Eq. 4.76c.
We are grateful to the referee for their time and their useful comments. Below we respond to the comments and suggestions of the referee report point by point. We hope that the improvements will satisfy the referee and we believe that they increase the quality of the paper.
We agree with the referee that a conserved dipole moment is just one way of obtaining mobility restrictions. We have modified the sentence: "The restricted mobility of isolated fracton particles can be viewed as a consequence of conservation of their dipole moment:" to "For some theories the restricted mobility of isolated fracton particles can be viewed as a consequence of conservation of their dipole moment:"
The Lagrangian in (2.27) is an example of a dipole-invariant term that appears at fourth order in derivatives and not the most general expression at that order. We took this to be the archetypical Lagrangian since similar expressions have appeared before in the literature. None of the subsequent results rely in a crucial way on the specific form of this Lagrangian: for the no-go theorem, we only use a specific form of the Lagrangian to show that it is possible to build a Gaussian theory that contains spatial derivatives when we allow for a non-linearly realised dipole symmetry. To clarify this point, we have added the following below Eq. (2.37): "The above shows that any theory of a complex scalar with a linearly realised dipole symmetry cannot simultaneously be Gaussian and contain spatial derivatives (i.e., gradient terms). If we allow for a non-linearly realised dipole symmetry, it is possible to build theories that are both Gaussian and contain spatial derivatives, as we illustrate by the following example." We also use this explicit Lagrangian when computing the symmetry algebra since it also has scale invariance, but any other choice of dipole invariant Lagrangian will lead to the same symmetry algebra as can be seen from the general expressions in (2.46) and (2.47), which are true for generic dipole-invariant Lagrangians. To emphasise this point, we have added the following sentence under (2.52): "At this stage, we emphasise that (2.52a) and (2.52b) are independent of the specific choice of dipole-invariant Lagrangian. Only scale invariance is a special property of the Lagrangian (2.43)."
Eq. (6.17) is the general condition for there to be an analogue of global dipole symmetry on curved space. On a generic background, this will have no (non-trivial) solutions. Thus, whether there is a notion of global dipole symmetry on a given background has to be checked on a case-by-case basis and is a property inherent to the background itself.
The interesting result of Reference [31] can be generalised by showing that the Lagrangian (7.17) is indeed gauge invariant and dependent on the trace of :math:` A_{ij}`. It would be nice to see if the argument in [31] can be generalised to include this result.
Generalising our results to cases with discrete rotational symmetry remains an interesting open problem beyond the scope of the present work, but we expect similar techniques to work. To our knowledge, the corresponding geometry has, in contrast to Aristotelian geometry, not previously appeared in the literature.
We thank the referee for pointing out these typos. The broken reference has been changed to a reference to Eq. (4.34).
Author: Emil Have on 2022-06-02 [id 2548]
(in reply to Report 2 on 2022-04-01)We are grateful to the referee for their time and their useful comments. Below we respond to the comments and suggestions of the referee report point by point. We hope that the improvements will satisfy the referee and we believe that they increase the quality of the paper.