SciPost logo

SciPost Submission Page

Crystal gravity

by Jan Zaanen, Floris Balm, Aron J. Beekman

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Aron Beekman · Jan Zaanen
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-05-04
Date submitted: 2022-04-22 14:28
Submitted by: Zaanen, Jan
Submitted to: SciPost Physics
Ontological classification
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".

Author comments upon resubmission

Dear Editor,

Being engaged in editorial activities, I am well aware of the tedium associated with processing long manuscripts with a content that falls in between the various subdisciplines. But we are grateful for the work and thought you invested in finding referee’s in your network responding with reports that turned out to be quite helpful.

Eventually, this revolves around the communication over the barriers between the various sub-disciplines. Frankly, when writing the first version the precise nature of the barrier was not sharply on our own radar. In the mean-time this has clarified. While waiting for the reports we tested this affair to the primary “customer base” -- the relativists. This clarified that what we call in the paper the “topologization of the gauge curvature” as primary motive of the Higgs phase is highly unfamiliar in this community: the affair of the Abrikosov flux lattices that became the dominant theme in the study of superconductivity in the condensed matter community. This never played any role in GR, but it is our experience that upon getting it across it is received as an intriguing eye opener.

This is also reflected in the referee reports. The substance of the paper is in the sections V-IX – the first 4 sections are setting the stage -- and you surely noticed that there is nothing found in any of the reports alluding to this. The reports of referee’s 2 and 3 reflect their struggle in trying to position it in the terrain that is familiar to them but the issue is that it is just something else.

Our formulation of the introduction is to be blamed, being in this regard not explicit enough. In the revised manuscript we have reformulated these passages putting the emphasis now much more on the “topologization” motive. We have rewritten Section I.A in this spirit, and added a new section I.B concisely summarizing the state of the art of the research on “solids in gravity” and how this relates to what we are looking at. This is intended to spell out what our paper is not about and here we are much helped by the reports. The literature on “solid cosmology” (referee 2), neutron star crusts etcetera and holographic applications (referee 3) is quite extensive. We are grateful for their listing of recent literature that we employ in this section.

Sections 1C-1G are mildly edited to adjust to this modified narrative and we did not touch section 1H, the executive summary of the results in our paper.

With regard to the substance, we were delighted by the observation of referee 3 that in the mean time the “working horse” Eq. (10) has entered the holography inspired literature: arXiv:1908.01175. We knew about it already some ten years ago and the original motivation of our work was holography related, aiming at correcting wrong tensor structure in the bulk as deduced from homogeneous ploys (e.g., Q-lattices). But in the process we found out that it is just way more fun to aim at the more general topologization affair. We have put arXiv:1908.01175 now in the limelight, both in the new general introduction, as well as in the introduction of the relevant section IIA. In one regard we actually disagree with this referee: \mathcal{W}__{ma} is not at all intuitive, it captures the bare essence of the “frame fixing” behind any Higgs mechanism but in this context it is directly obvious for the human visual system (the “intuition” of the referee). The mathematical formalisms are just not adding anything, these take this as input. We point this out in the modified introductory sentences of section IIA.

It appears that referee 1 invested quite some energy in studying in detail how it all works – we are very grateful! He arrives at excellent suggestions. Indeed, to keep track of the duality-mapping relations between vortices, dislocations and disclinations a “route map” table is most useful. Following his recommendation, we have added this in a new section 1.I. As also mentioned by referee 3 our “sloppy” handling of upper-\lower indices may be confusing for the relativists. But this is actually very easy: getting to business with “crystal geometry” we are invariably dealing with manifolds “flattened” by the crystal while in the compiutations we rely on Euclidean signature: the covariant/contravariant index structure is therefore redundant. At the instance that the Lorentzian signature of the background comes into play (e.g., gravitons) we pay tribute to the index structure. We spell this out in the new section 1J.

These are the changes we have made in the revised paper. Let us now respond to the remaining issues raised by the referee’s:

Referee 1:
Again, we thank the referee for his efforts to study our paper thoroughly and we of course like his appreciation of our effort! We alluded in the above already to his first two points. With regard to his third point: we fear that the referee is cutting here a corner. The gravitational penetration depth is actually determined by the shear modulus, a material property, while the Schwarzschild radius only requires a mass density – there is no universality in this dimensional analysis. More pressing, such questions belong to the realm of “solid cosmologies”. As we emphasize, we stay away from this substance matter. Our impression is that this community is on the right track, wiring in the ‘Higgsing’ through the breaking of spatial shift symmetries. However, when we approach it from the elasticity side we do hit a brick wall in the form of the “messing up” of the (GR) tensor structure due to the bad breaking of Lorentz invariance that by back reaction should also imprint on the dynamical evolution of the background geometry. We find this confusing and it begs for a further analysis. In summary, for these reasons we prefer to ignore this issue raised by the referee in the paper.

Referee 2:
As already announced, the referee is of course right with his recommendation for references to state of the art papers in closely related subjects. We now discuss and cite the papers mentioned in his first three points. We actually ignore the last two papers mentioned under his point three since these appear to be completely unrelated to what we find – as pointed out in Section VII departing from real solids this will not happen (the “dislocation gas” affair).
With regard to his comment in point 3 that we are too sceptical: this is a misunderstanding, we refer in this context just to effects of the “topologization”, given the revised introduction it should now be clear that we having nothing to say regarding elastic inflation and so forth. Point 4: we are quite familiar with this holographic work but the only relation between these monopoles and our disclinations is merely in the word “topology”. Point 5: this is easy, in so far the Goldstone bosons are at stake it is just the same affair discussed in a different language. In the “algebraic” approaches he refers to the absence of rotational modes is eventually rooted in the fact that [L^a, P^b] = i epsilon_{abc} P^c (L and P angular and linear momentum) which is just expressing the semidirect relations between translations and rotations. Notice that the Golstobe counting is in our text used merely as a transition towards the introduction of the topological incarnation, in the form of the well understood “lego-topology” discussed at length in VI-B and Appendix C. Point 6: in condensed matter the “vortex-boson” duality reviewed in Section III has turned into a main stream affair with an associated huge literature in condensed matter physics. E.g., Matthew Fisher received not long ago a Buckly prize for introducing it in the context of the superconductor-insulator transition in the late 1980’s. The papers he mentions have no special standing in the particular context where we discuss this affair. Our quantum liquid crystal review is just very convenient for turorial purposes. Point 7: this is similar, these papers just fit in the “Kleinert” tradition that also summarizes a huge literature and they do not anything useful for our particular purposes.

Referee 3:
We do have the impression that this referee did not manage to get a grip on our material. As acknowledged by the other two referee’s our paper is long because there is much novelty to be found, while it is just beneficial for the readership to take the time to explain the unfamiliar background material. We can assure the referee that the substances we have in the offering (Sections IV-IX) are concisely written and impossible to compress significantly. Reading his report it is obvious that he/she did not even get the elementary points. His first point: as we discussed in the beginning, we fully agree with his statement that a sophisticated literature emerged dealing with neutron star crusts and we refer to that in the new section I-B. Ironically, he seems to have missed our observation in Section IV-E that neutron stars are still small compared to the gravitational penetration depth, being therefore unrelated. His second and technical details (TD) point 2 : attracting the attention to Armas et al., arXiv:1908.01175 is indeed most helpful and we discussed in the beginning this is now put in the limelight. His third and fourth point: this reveals his confusion. Dealing with pinned charge density and especially with fluids (visco-elastic or not) the topologization which is the subject of our paper is just not in existence! This is just a different chapter in the book of physics. With regard to the holography inspired tradition in this regard, it is ironic that since the work by Frenkel in the 1930’s it is generally accepted in the professional fluid community that viscoelastic behaviour is best understood in terms of a solid littered with a low density of free dislocations. Dislocations are not quite part of the canon of this holographic community and it could well be beneficial for these developments to study our material to familiarize with them in the context where these take over control, in the “full” solids that we discuss. Further technical details:
Technical detail 1: We actually spell this out at every relevant instance in the manuscript. It is simple: the time axis is at work dealing with gravitons, as usual probes around the “flattened” (by the crystal) background in Section IVD-G, and when delaing with the curvature fluxoid (Section VII). It appears that the cause of confusion is in the central played by the curvature of the spatial manifold in a co-moving frame which is unusual in GR.
Technical detail 2: fully agreed, and we now highlight this both in the introduction of section II and in the general introduction.
Technical detail 3: the way that {\cal W} is not intuitive, it is the bare essence of the Higgsing. One wants to change the frame arbitrarily but the presence of the crystal constraints it to passive diffeomorphisms and for reasons we spell out we only have to consider small metric fluctuations (locally “self-linearized”, the non linearities are all “collected” in the topological parts).
Technical detail 4: the referee seems to refer here to our “sloppy” dealings with covariant/contravariant indices, as the first referee did. We have added section I.I to the introduction explaining this.
Technical detail 5: For technical convenience much of our derivations revolve around writing down effective actions, from which the EOM’s (Einstein equations) can be derived. As we spell out at length, departing from the crystal the bad breaking of Lorentz invariance messes up the tensor structures. This is in Einstein equation language associated with the stress-energy tensor of the solid. This is announced in the introduction, spelled out in Section II.B, and given some substance in Section IV.G. We leave this for future work: it is not at all clear to us how this is dealt with in the “solid cosmology” literature.
Technical detail 6: again the referee is missing an essential point. As we spell out repeatedly the leading non-linearities are captured by the topological excitations. The very complicated “fracton” dynamics of the dislocations and disclinations will be at center stage in a dynamical setting: the on-going pursuit in materials science. On the other hand, one can consider higher gradient elasticity as a starting point. This is analyzed in detail in the Kleinert book but it turns out to only affect some numbers associated with the dynamics of the topological defects. For this reason we ignore it.

Also on behalf of my co-authors Balm and Beekman,

Jan Zaanen

List of changes

1. We reformulated the initial parts of the introduction, focussing the attention of the reader on the "topologization", the analogue of Abrikosov lattices in the "crystal gravity" context. For this purpose we rewrote Section I.A, added section I.B concisely reviewing the existing work on solids in GR, spelling out the relations with what we will pursue. Here we added the references to the most modern part of the literature brought to our attention by referee's 2,3. Sections 1C-1G have been subjected to mild text editing to adjust it to the new flow of arguments. We did not touch the long summary section 1H.
2. Following the suggestions of Referee 1 we added the duality overview table in section I.I and we added a very short section I.J explaining our handling of upper- and lower indices.
3. We modified the introduction of Section II, highlighting the paper arXiv:1908.01175 as brought to our attention by Referee 3.

Published as SciPost Phys. 13, 039 (2022)

Login to report or comment