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Crystal gravity

by Jan Zaanen, Floris Balm, Aron J. Beekman

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Submission summary

As Contributors: Aron Beekman · Jan Zaanen
Arxiv Link: (pdf)
Date submitted: 2021-09-27 11:25
Submitted by: Zaanen, Jan
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
  • Statistical and Soft Matter Physics
Approach: Theoretical


We address a subject that could have been analyzed century ago: how does the universe of general relativity look like when it would have been filled with solid matter? Solids break spontaneously the translations and rotations of space itself. Only rather recently it was realized in various context that the order parameter of the solid has a relation to Einsteins dynamical space time which is similar to the role of a Higgs field in a Yang-Mills gauge theory. Such a "crystal gravity" is therefore like the Higgs phase of gravity. The usual Higgs phases are characterized by a special phenomenology. A case in point is superconductivity exhibiting phenomena like the Type II phase, characterized by the emergence of an Abrikosov lattice of quantized magnetic fluxes absorbing the external magnetic field. What to expect in the gravitational setting? The theory of elasticity is the universal effective field theory associated with the breaking of space translations and rotations having a similar status as the phase action describing a neutral superfluid. A geometrical formulation appeared in its long history, similar in structure to general relativity, which greatly facilitates the marriage of both theories. With as main limitation that we focus entirely on stationary circumstances -- the dynamical theory is greatly complicated by the lack of Lorentz invariance -- we will present a first exploration of a remarkably rich and often simple physics of "Higgsed gravity".

Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission 2109.11325v2 on 22 April 2022

Reports on this Submission

Anonymous Report 3 on 2022-3-7 (Invited Report)


This paper studies the coupling between (dynamical) gravity and elastic matter. This is an old subject.
What is new to some extent in the context of this paper is that the coupling is done at the level of an action and, additionally, care is given to the possibility of defects in elastic mater.
This is an interesting addition to this literature. However, I have various comments that I believe should be addressed or given some consideration before this paper is considered for publication.

- The paper is very long with a very long introduction and in many ways akin to a review. This long introduction does not help significantly to understand the new contributions of this paper. This would have been a better paper if it had been condensed into 20 pages with the new contributions. Can the authors attempt at making clear a distinction between what is new in this paper and what is not?

There is a wide variety of literature missing which leads me to disagree with various statements.

- I am in disagreement with the comments in section IA and in the abstract suggesting that this is the first time that such study has led to a proper understanding of the coupling of elastic degrees of freedom and gravity.
Relativistic elasticity is typically used to study neutron star crusts, a literature that seems to have been completely ignored in this paper (e.g. arXiv:2003.05449 and referees therein). In this context, the coupling is usually done at the level of the equations of motion: elastic matter is coupled to Einstein’s equations via an appropriate elastic stress tensor.

- The authors motivate their work in IB, page 3, as “aimed at informing this community regarding the role of elasticity in this context”, referring to the holographic study of elasticity and in particular references [11,12] and progress “hindered by unfamiliarity with elasticity”. Additionally, when referring to relativistic elasticity the authors cite the work of Carter and Quintana. I think that the authors missed crucial references in the context of modern treatments of relativistic elasticity, also in relation to the holographic community. For instance, the work of Fukuma et al. (arXiv:1104.1416 and references therein) and in particular the work of Armas et al. (arXiv:1908.01175, in particular section 2 and appendix A), which provide a more rigorous geometric formulation of relativistic elasticity than any other literature the authors refer to.

- The authors do not mention the role elasticity has played in the realm of quantum matter and describing pinned crystals and charge density waves. For instance arXiv:cond-mat/0103392 and the more modern treatments such as arXiv:1702.05104 and arXiv:2001.07357.

- This paper is rather close in spirit to a recent paper trying to address more rigorously how matter couples to dynamical gravity. The paper arXiv:1907.04976 is a recent exploration that discusses coupling to fluid degrees of freedom, instead of elastic ones. The overlap between the two works, however, would be visible when discussing a more rigorous approach to this work in which dynamics is taken into account.

Could the authors perhaps revise some of their comments and take into account this relevant literature?

Technical details:

- The authors claim to address mostly the stationary case but fluctuations (gravitons/phonons) are considered. Could the authors make it more clear when the stationary assumption is being used and when it is not?
In relation to this, would it be possible to write the full action that is being considered somewhere, that is the elastic part with kinetic terms and the Einstein-Hilbert action?

- The “working horse of crystal gravity” as referred to equation (10) has already been written down, in particular in section 2.2 of arXiv:1908.01175. Eq. (10) corresponds to the linearised version in the Goldstones and background metric of (2.14) of arXiv:1908.01175. Perhaps the authors could refer to this? The difference is that the authors will supplement this action with the Einstein-Hilbert action.

- Related to the above comment, the authors could have used a more geometric language to describe the crystal. In particular \mathcal{W}_{ma} introduced in equation (10) is just the strain tensor expanded linearly in the Goldstones and background metric. This strain coincides with the linear version of (2.4) of arXiv:1908.01175. It would be useful to comment on what the meaning of \mathcal{W}_{ma} is, as at the moment it appears as if it was an educated guess.

- The language employed throughout the paper is rather non-covariant, Eq. (10) being an example with multiple non-contracted indices, which is a bit at odds with the relativistic nature of Einstein general relativity. More consideration could have been given to carefully describe the crystal space and the background spacetime, and in particular mention what is the geometric structures associated with both. How are the indices m and a raised/lowered in Eq.(10)?

- The Einstein equations coupled to elastic matter are never written down explicitly, except in very special circumstances in which the action is taken on-shell for specific configurations. This is a bit strange for any relativist. Could the authors write down the Einstein equations and equations of motion for the crystal in generality with the stress tensor for elastic matter that they are considering? This would surely make the connection with earlier literature more clear.

- Extending this work to dynamical settings also requires a non-linear approach to elasticity. I believe that the language in this paper would have to substantially change in order to describe such situations. Could the authors comment on this and what previous works they expect to be useful in order to do so?

I would recommend the paper for publication once the authors review these comments.

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

Anonymous Report 2 on 2022-1-25 (Invited Report)


This paper is an interesting read: it is well written, comprehensive, and overall worth publishing. A significant amount of it brings together and reviews results that are not necessarily new, but fit nicely together and are meant to provide the reader with the necessary background. However, despite the fairly extensive bibliography, the authors miss several opportunities to acknowledge prior work and place their own in the appropriate context. In light of sentences such as "We address a subject that could have been analyzed a century ago", the reader might be left with the mistaken impression that this paper introduces an entirely novel viewpoint on the subject. I recommend that the authors consider the following points and comment on the prior literature when appropriate:

1. First, the Higgsing gf gravity has been considered extensively in the high energy physics literature. See the following paper and all the references therein for a sample:

- Bonifacio, James, Kurt Hinterbichler, and Rachel A. Rosen. "Constraints on a gravitational Higgs mechanism." Physical Review D 100.8 (2019): 084017.

2. The Higgsing gf gravity is also tightly related to massive gravity, a topic that has been studied very extensively. Of particular relevance to this paper are Lorentz-violating theories of massive gravity, which have been first studied systematically in:

- Dubovsky, Sergei L. "Phases of massive gravity." Journal of High Energy Physics 2004.10 (2004): 076.

3. The author's skepticism that this topic could be "of relevance to the physics of our universe" is perhaps a bit premature. For instance, interesting models of inflation drive by solids have been proposed, see:

- Gruzinov, Andrei. "Elastic inflation." Physical Review D 70.6 (2004): 063518.
- Endlich, Solomon, Alberto Nicolis, and Junpu Wang. "Solid inflation." Journal of Cosmology and Astroparticle Physics 2013.10 (2013): 011.
- Kang, Jonghee, and Alberto Nicolis. "Platonic solids back in the sky: Icosahedral inflation." Journal of Cosmology and Astroparticle Physics 2016.03 (2016): 050.
- Nicolis, Alberto, and Guanhao Sun. "Scalar-tensor mixing from icosahedral inflation." Journal of Cosmology and Astroparticle Physics 2021.04 (2021): 074.

The last two paper seem also relevant for the discussion of platonic solids in Sec. VIII.

4. Although holography is not the main focus of this paper, it is mentioned in the introduction as one of the motivations for this study. It is worth pointing out that holographic solid states have been found to be related to monopoles in AdS, see e.g.

- Bolognesi, Stefano, and David Tong. "Monopoles and holography." Journal of High Energy Physics 2011.1 (2011): 1-28.
- Esposito, Angelo, et al. "Conformal solids and holography." Journal of High Energy Physics 2017.12 (2017): 1-29.

5. I find the discussion in the first column of p.33 about the symmetries broken by a solid confusing. The statement that "finite translations are the same as rotations" is at best very misleading, at worst plain wrong. Furthermore, the reason why solids don't feature Goldstone modes associated with the breaking of rotation (or boosts, for that matter) is well understood as is a manifestation of the inverse Higgs mechanism:

- Ivanov, Evgeny Alexeevich, and Victor Isaakovich Ogievetskii. "Inverse Higgs effect in nonlinear realizations." Theoretical and Mathematical Physics 25.2 (1975): 1050-1059.
- Low, Ian, and Aneesh V. Manohar. "Spontaneously broken spacetime symmetries and Goldstone’s theorem." Physical review letters 88.10 (2002): 101602.
- Nicolis, Alberto, Riccardo Penco, and Rachel A. Rosen. "Relativistic fluids, superfluids, solids, and supersolids from a coset construction." Physical Review D 89.4 (2014): 045002.

6. The dual description of superfluids discussed in Sec. III has been studied extensively. For instance, see:

- F. Lund and T. Regge, Unified Approach to Strings and Vortices with Soliton Solutions, Phys. Rev. D 14 (1976) 1524
- A. Zee, Vortex strings and the antisymmetric gauge potential, Nucl. Phys. B 421 (1994) 111
- Horn, Bart, Alberto Nicolis, and Riccardo Penco. "Effective string theory for vortex lines in fluids and superfluids." Journal of High Energy Physics 2015.10 (2015): 1-58.

7. A good portion of this paper is about defects in gravity, which was a topic discussed for instance in

- M.O Katanaev, I.V Volovich, Theory of defects in solids and three-dimensional gravity, Annals of Physics, Volume 216, Issue 1, 1992, Pages 1-28
- Bennett, D. L., et al. "The relation between the model of a crystal with defects and Plebanski's theory of gravity." International Journal of Modern Physics A 28.13 (2013): 1350044.

It would be important the the authors comment on previous work on defects in gravity and how it relates to their own work.

Finally, I should emphasize that I am listing specific papers only to provide the authors with an entry point to the relevant literature. This is emphatically not a request for specific citations, although the authors may choose to cite some of the papers above if deemed relevant. Instead, it is a broad encouragement to strengthen their connections with the already existing literature.

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

Anonymous Report 1 on 2022-1-24 (Invited Report)


This is an extremely interesting piece of work that deals with a question of (well, to me) tremendous physical interest. We are very familiar with the fact that when a theory with (say) a U(1) spontaneously broken global symmetry is gauged, the putative Goldstone modes are eaten by the gauge field, and the resulting spectrum is massive, with the phenomenology associated superconductors (expulsion of magnetic flux, quantized vortices, etc.). What then happens if one considers a solid (which spontaneously breaks translational symmetry) in the real world (where — loosely speaking — translational symmetry is gauged by gravity)? Is the graviton gapped? What is a gravitational superconductor? Etc.

This work aims to answer that question in detail: for the most part it does so, explaining in detail the phenomenology of a gravitational theory coupled to a giant space-filling crystal. It does so from a rather sophisticated and effective field-theory point of view. The topic is rich and rather technical due to the proliferation of polarizations; analogies to the U(1) case help a lot here, but there is a lot going on. There are many great nuggets of wisdom that link ideas in “real-life” physics (e.g. associated with things like principles of Weber detector design) to formal considerations of symmetry breaking and effective theory, and there are quite a few surprises (e.g. that it is torsion that is expelled, and that the phonons remain massless though the graviton acquires a mass).

Overall, after a few readings I think I have been able to appreciate the key points and I believe that the paper is definitely worthy of publication; it addresses a very natural physical question, and it does so in a very creative and interesting way.

However, I did find the work rather tough going, and I would like to make a few suggestions to make the paper easier to read, and then another comment on a point that the authors did not (I believe) address.
1. The authors frequently compare the gravitational case to the U(1) superconductor; however this analogy is stretched out over several chapters and it can become difficult to maintain in memory how it works simply due to all the moving parts. I suggest that the authors create a table or dictionary relating all the key ingredients (e.g. a_{\mu} maps to h_{\mu\nu}, the torsion maps to the magnetic flux, etc.), together perhaps with a page number for each entry showing where the explanation is given.
2. The paper is not consistent about raising and lowering indices; there are also many different kinds of indices (m, a, i, j, \mu \nu) and I don’t think it is clear how they relate to each other, and which of them should be raised and lowered and which not. (I am not sure this was every fully explained in the paper). For a paper dealing with gravity in a fundamental way this is not good. I believe it’s mostly okay because gravity is often (but not always) linearized, but I think this should be made explicit, perhaps by consistently using the flat-space metric to raise and lower them. (This will also make the paper more accessible to a gravitational audience).
3. Finally, a physics question that should be addressed: the authors frequently state that it is unlikely that this will be phenomenologically relevant simply because space is not full of a giant crystal (though they do make interesting statements about dark matter etc). However could it be relevant even in principle? I am a bit confused because if I fill space with a giant crystal then once the spatial extent R of the crystal is bigger than its own Schwarzschild radius it will collapse into a black hole (unless the situation is intrinsically time-dependent like in cosmology). If it has energy density \rho then this will happen when G R^{3} \rho = R, i.e. when R = \sqrt{1/(G \rho)}, i.e. parametrically the same as the gravitational screening length (1) (I imagine the shear modulus is controlled by the same physics as the energy density). Thus there doesn’t seem to be a way to separate these scales, and there is a worry that the physics here can *never* be accessed, even in principle. In my eyes this does not invalidate the results, but I think it should be addressed cleanly, and I would be extremely interested in knowing if the above conclusion could be evaded. (In practice there is probably a time-dependent version of this theory that could be relevant for cosmology towards which this construction is a first step).

Overall, I am happy to recommend this for publication; however I think the authors should consider the points above to strengthen the paper.

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

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