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Systematic Constructions of Fracton Theories

by Djordje Radicevic

This is not the current version.

Submission summary

As Contributors: Djordje Radicevic
Preprint link: scipost_201912_00043v1
Date submitted: 2019-12-11
Submitted by: Radicevic, Djordje
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Condensed Matter Physics - Theory
Approach: Theoretical


Fracton theories possess exponentially degenerate ground states, excitations with restricted mobility, and nontopological higher-form symmetries. This paper shows that such theories can be defined on arbitrary spatial lattices in three dimensions. The key element of this construction is a generalization of higher-form gauge theories to so-called $\mathfrak{F}_p$ gauge theories, in which gauge transformations of $k$-form fields are specified by $(k - p)$-form gauge parameters. The $\mathbb{Z}_2$ two-form theory of type $\mathfrak{F}_2$, placed on a cubic lattice and coupled to scalar matter, is shown to have a topological phase exactly dual to the well-known X-cube model. Generalizations of this example yield novel fracton theories. In the continuum, the U(1) two-form theory of type $\mathfrak{F}_2$ is shown to have a perturbatively gapless fracton regime that cannot be consistently interpreted as a tensor gauge theory of any kind. The compact scalar fields that naturally couple to this $\mathfrak{F}_2$ theory also show gapless fracton behavior; on a cubic lattice they have a conserved U(1) charge and dipole moment, but these particular charges are not conserved on more general lattices. The construction straightforwardly generalizes to $\mathfrak{F}_2$ theories of nonabelian two-form gauge fields, giving first examples of nonabelian higher-form theories.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 3 on 2020-1-24 Invited Report


1- Through a dual description of the X-cube model, establishes a formalism enabling the definition of fracton phases on arbitrary manifolds, going beyond the foliated fracton framework.

2- Develops a generalization of higher-form gauge theories which captures known fracton models and predicts new phases as well.


1- The discussion of fractonic theories without charge conservation is confusing.

2- Claims regarding non-Abelian higher form theories are too strong and properties of the putative non-Abelian models are not discussed in any significant detail.


This work presents a new description of fractonic theories, which is based on a dual description of the paradigmatic X-Cube model. Formulating fractonic theories as novel gauge theories, which are shown to be natural extensions of higher-form theories, the author shows that they can be placed on arbitrary spatial manifolds. Earlier work (Ref. [5] in the paper) had shown how some fracton models can be placed on foliated manifolds. The framework developed by the author lifts this restriction and hence constitutes a substantive contribution to the field. Additionally, the generalized boundary operators introduced herein hint at a novel cohomology theory underlying fracton models and may point the way towards placing these models on firmer mathematical ground.

However, there are two main points I would like the author to address before I can recommend the paper for publication:

The author claims that fractonic matter, when placed on a BCC lattice, displays all the characteristic features associated with fractons (excitations with restricted mobility and an exponential ground state degeneracy) despite lacking conserved charge or dipole moment. While I agree that higher moment conservation laws are likely not the only mechanism underlying fractonic behavior, I am not convinced that this model does not have some unusual local conservation law governing its behavior. The absence of charge conservation in particular is rather puzzling, as this would allow isolated charges to appear out of or decay into the vacuum. It could be that some non-trivial combination of charge and dipole moment is conserved or that the charge of some composite object is locally conserved. The claims made by the author should thus be tempered to account for these possibilities, unless they can be convincingly refuted.

Secondly, I am not convinced that the non-Abelian generalizations are in fact describing non-Abelian fracton phases, as no evidence is provided that these models support immobile excitations with non-Abelian braiding and fusion. While the author claims that a perturbative fracton regime can be found, this is not supported by any arguments or by illustrative calculations on a simple example - for instance, how does the $G = S_3$ model behave on a cubic lattice? Even for this simple example, one can quickly check that the Gauss' Law operators (defined in Eq. (73)) on neighboring vertices do not commute. In the absence of a commuting projector Hamiltonian, it is not at all obvious that this model is describing a phase with deconfined fractonic excitations. Establishing such properties is a tricky proposition even for solvable Hamiltonians, but two papers which do this explicitly are:
Prem et al., Phys. Rev. X 9, 021010
Bulmash and Barkeshli, Phys. Rev. B 100, 155146

An additional possibility is that the non-Abelian models considered here are similar to those in arXiv:1908.07601, which describe symmetry-protected fractons but do not have a topologically protected exponential ground state degeneracy, and are hence distinct from models such as the X-Cube. Finally, the claim that the models constructed here provide the first examples of non-Abelian higher form theories is incorrect. See e.g.,
H. Pfeiffer, Annals of Physics 308: 447 (2003)

I can certainly appreciate that an in-depth exploration of the non-Abelian models is beyond the scope of this paper, but then it should be clearly stated that, at this stage, the fractonic nature of these models is a conjecture.

Requested changes

1- Clarify discussion of models without charge and dipole moment conservation.

2- Address issues regarding non-Abelian generalizations.

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

Author Djordje Radicevic on 2020-03-10
(in reply to Report 3 on 2020-01-24)
answer to question
reply to objection

I thank the referee for the insightful comments and suggested references. I have just posted a revised version of the manuscript, which I hope addresses the points raised by this report. I would like to briefly comment on these points here, and to explain how I addressed them.

1) I have significantly expanded the section on models on BCC lattices. As the referee suggests (and as the previous version of the manuscript underemphasized), there are indeed new symmetries that replace conservation of the naive charge and dipole moment in the models under consideration. I hope the explicit calculations here will remove any confusion on this point.

2) I have also expanded the section on nonabelian models. Even though the Gauss operators on different sites need not commute, it is still possible to define gauge-invariant operators and a gauge-invariant Hilbert space. This discussion was not very explicit in the previous version, but now it is presented in considerable detail.

3) The referee's point that nonabelian higher-form symmetries have appeared in the literature is well taken. I have added the reference to the original work by Baez on this subject. However, I also point out that these higher-group gauge theories seem to necessarily involve multiple gauge fields of different rank (the paper by Pfeiffer mentioned by the referee indeed has fields on both plaquettes and links). I have modified my paper to stress that the higher-rank nonabelian theories I present are pure, i.e. they do not need to be coupled to any theories of other ranks.

4) Finally, I have addressed the referee's last remark about the "fractonic nature" of the theories in question. When talking about theories of fractons, I simply refer to the existence of some regime where, at least perturbatively, it makes sense to talk about an effective field theory of excitations (either gapped or gapless) satisfying certain fracton properties. I have edited the introduction to label these properties F1-3, and I subsequently refer to them more explicitly when discussing specific examples; I explicitly state that these are the only hallmarks of fracton physics I will look for. Even in the previous version of the manuscript, I never claimed that these were novel phases of fractons, and I have tried to be more explicit about the distinction, both in the introduction and later on when discussing the Coulomb regime of $\mathfrak{F}_2$ theories.

By drawing an analogy to arXiv:1908.07601, I think the referee has already intuited that this is what I may be doing. I think there is an even more apt analogy here. The situation is similar to talking about photons in (2+1)D Maxwell theory, or about gluons in (3+1)D Yang-Mills: these excitations are gapped, but there are regimes (at weak enough coupling, at scales below the confinement scale) in which it really does make sense to talk about these gluons or photons as gapless, perturbatively defined objects that are treated the same way as genuinely gapless photons that appear in (3+1)D Maxwell. Studying such theories has been a topic of considerable interest for decades now: a particularly fascinating example is the appearance of gravity as the "double copy" of perturbative gluon theories (arXiv:1004.0476). This is why I believe that studying perturbatively fractonic theories with their exotic kinematics may be similarly worthwhile, even if these are not necessarily genuine phases of matter. As the referee says, in-depth exploration of such issues is beyond the scope of this paper, but I hope that the explanations I have given remove the impression that I necessarily talk about novel phases, whether Abelian or nonabelian.

Anonymous Report 2 on 2020-1-19 Invited Report


1) Includes a nice review of much type-I fracton physics from the perspective of a high energy theorist.
2) Defines a new model using an interesting formalism.


1) Few results are given about the new model. No examples with new fracton physics are discussed.
2) Many of the claims seem to be incorrect.


The author reviews the X-cube model, some dualities to theories with matter, and generic type-I fracton physics from the perspective of a high energy theorist. Although many brief reviews of the X-cube model exist in the literature, I'm not aware of any reviews from a high energy perspective.

The author then defines a new F_2 model which generalizes the X-cube model. The new model is essentially defined by the gauge constraint in Eq (32), and gauge invariant terms are implicitly defined as discusses around Eq (37). A large K limit for the Z_K gauge group version of the model is studied and is claimed to exhibit a Coulomb regime. However, I think a previous work (0801.0744) showed that the Coulomb regime is unstable to monopole proliferation. The F_2 model on the BCC lattice is considered, but evidence is given which seems to suggest that the model is equivalent to the 3+1D toric code. Coupling to matter and a nonabelian generalization of the model are also discussed. However, I do not think the nonabelian model is gauge invariant.

I do not think the paper accomplishes its goals or justifies many of the claims in the abstract, introduction, and conclusion. I do not think a systematic construction of fracton theories on arbitrary lattices was given since it is not clear when the F_2 model results in a fracton model (eg a model with any of the three features at the beginning of section 1.1). Indeed, the BCC lattice example seems to yield toric code instead of a fracton model. Furthermore, as previously mentioned, my understanding is that the U(1) generalization is unstable to monopole proliferation, and the nonabelian model is not gauge invariant.

Requested changes

Before publication, I think these issues must be resolved by weakening claims, removing incorrect results, or by resolving my possible confusion:

1) One of the selling points of this work is defining fracton models on arbitrary lattices. However, no example is given which shows that the new F_2 model produces any new fracton models beyond the X-cube model on a cubic lattice. The BCC lattice example in section 4.3.2 appears to just be 3+1D toric code. (This result vaguely reminds me of the last paragraph of the main text of Ref [9].) A possible direction is discussed in the last two sentences of section 4.3.2, but it is too brief for me to understand the resulting physics. Furthermore, the X-cube model was already generalized to arbitrary lattices in appendix A of [14]. However, their construction relied on the input of a foliation structure, which I guess the author is trying to avoid.

Regarding this point:
"all known examples ... live on spatial lattices that are cubic or that can be decomposed into subdimensional layers. Even in examples formulated in the continuum, known fracton theories always rely on the Euclidean structure of the underlying manifold in some fundamental way"
These two papers discussed fracton models in curved spaces:

2) I don't understand what is meant by comment #3 just before section 1.2? What part of the paper does this pertain to?

3) The paper stresses that using the dual X-cube model was essential for defining it on a more general lattice. However, that does not seem to be the case to me. Eq (32) and the paragraph around Eq (37) are generalizations of the X-cube operators shown in figure 1. Perhaps the author just meant "dual lattice", but this text
"Unlike the X-cube model itself, its dual has a natural generalization to arbitrary lattices and to all gauge groups"
seems to clearly suggest "Kramers-Wannier dual" as it appears just after a paragraph about the Kramers-Wannier dual.

4) Section 4.3 discusses the Coulomb regime of the F_2 theory. But I think Xu and Wu showed that the Coulomb phase should be confined at arbitrarily small g^2 due to monopole proliferation in
Ref [7] also mentions this in figure 1 of their paper. I think this paper should not hide this fact.

Also, below Eq (53), an exponential degeneracy is claimed to result from the photon sector. But then the author admits that this degeneracy could be interpreted as unphysical, just like the the third polarization in standard U(1) gauge theory. So it seems that this should not be thought of as exponential degeneracy since it's just a gauge redundancy.

5) My understanding is that the Lagrangian in Eq (70) is supposed to be the matter content of the F_2 theory on the BCC lattice. But as mentioned above, I expect F_2 on the BCC lattice is just an ordinary vector gauge theory. I think phi_c is just a trivial local degree of freedom which primarily just affects the energetics (and not the universal physics). At low energy, I think phi_c approaches a constant. Then, one can use the symmetry (where c is a constant)
phi -> phi + c (x + y + z) + Lambda
phi_c -> phi_c - c
with c=phi_c to effectively set phi_c=0. Then the resulting Lagrangian just looks like the matter for ordinary vector gauge theory with an ordinary global symmetry phi->phi+Lambda.

I also do not think Eq (71) is a symmetry of (70).

6) I'm not certain, but I think the nonababelian model in section 4.5 is not gauge invariant. That is, I can't figure out how to make Eq (73) and (74) commute. Perhaps I'm just confused about a sign structure. It would be extremely helpful if a cubic lattice example could be written out explicitly for the four operators: one from (73) and three from (74) due to the three orientations. For example, something like G = L_{f_1} L_{f_2} L_{-f_3} ... and similar for (74) where the faces f_i are labeled in a figure would be extremely helpful.

7) Some minor/optional suggestions/comments/questions:
7a) I think a figure for Eq (15) would be very helpful.
7b) In Eq (20), I think it would be more clear to write sigma_ik instead of sigma_i, and similar for Eq (22).
7c) I find Eq (40) hard to understand. Is it supposed to be a generalization of the straight line operators on the right of figure 2?
7d) In higher dimensions, are the F_p theories related to the higher dimensional models the following paper?

  • validity: low
  • significance: ok
  • originality: ok
  • clarity: good
  • formatting: excellent
  • grammar: perfect

Author Djordje Radicevic on 2020-03-11
(in reply to Report 2 on 2020-01-19)
reply to objection

I thank the referee for the detailed comments. I have just uploaded a new version that addresses the referee's points. I would like to further comment on some of these issues here. My bullet points will follow the referee's.

1) In general, there is a distinction between the statement that a theory has a regime which satisfies the fracton properties listed in the introduction (now labeled F1-3) and the statement that a theory is in a fracton phase. In the former case, excitations satisfying F1-3 may do so perturbatively, for instance only when considered above some energy scale. [I will have more to say about this in response to referee's point 4.]

The previous version of the manuscript did not immediately highlight this distinction. In the new version I stress this issue in the introduction, and also later when discussing the Coulomb regime of $\mathfrak{F}_2$ theories. While my main interest is in effective theories that satisfy F1-3, at the end of sec. 4.3.2 I did suggest that after sufficient tuning one may obtain a fracton regime with a ground state degeneracy that is generically different from the X-cube one. I have now clarified and expanded upon this point a bit; I agree with the referee that this presents an interesting direction for further research.

The referee further states that "the X-cube model was already generalized to arbitrary lattices in appendix A of [14]. However, their construction relied on the input of a foliation structure, which I guess the author is trying to avoid." As the referee says, the foliation structure was important in the construction presented in [14]. The set of lattices with a foliation structure is of measure zero in the set of all lattices; on a generic lattice, different sites will not have the same coordination numbers, and there will be no way to assemble the links into well-behaved leaves of a foliation. This is why the generalization presented in this paper is not trivial, and why I do not think that it is fair to say that the X-cube model has already been generalized to arbitrary lattices. However, I do appreciate that the paper can always be made clearer, and I have changed the introduction to make a mathematically more precise statement about the novelty of the models in this paper.

2) I have streamlined this part of the introduction to make its connection with other parts of the paper clearer.

3) I have made a minor change in wording that should resolve the confusion here. The referee is certainly right that the two theories related by Kramers-Wannier duality are the same. However, my point was that their presentations (field content, constraints) differ greatly, and that only in one duality frame it is immediately clear how to generalize to arbitrary lattices.

4) The referee states that "Section 4.3 discusses the Coulomb regime of the F_2 theory. But I think Xu and Wu showed that the Coulomb phase should be confined at arbitrarily small g^2 due to monopole proliferation in Ref [7] also mentions this in figure 1 of their paper. I think this paper should not hide this fact."

Xu and Wu have indeed provided evidence that the U(1) theory of the kind I discussed is gapped, and I have now added the reference to their work in my paper. I thank the referee for this pointer. However, I must stress that the previous version of the manuscript was already clear about this possibility. Section 4.3 began with a reference to the analogous phenomenon in ordinary Maxwell theory in (2+1)D, where it was stressed that despite any possible confinement at low energies the fracton properties F1-3 can still be established in the Coulomb regime, where gapless photons provide a good description of the physics (e.g. at energies above the confinement scale, or at distances below the inverse of the gap). This is precisely why I used the phrase "Coulomb regime" and not "Coulomb phase." This possibility was also mentioned in the abstract ("the U(1) two-form theory of type $\mathfrak{F}_2$ is shown to have a perturbatively gapless fracton regime", emphasis added) and on the first page of the paper ("at least at energies above any confinement scale"). Therefore I must respectfully reject the claim that I was trying to hide this fact. I do acknowledge that the distinction I explained in point 1) of this reply was not made as forcefully as it could have been, and I have tried to make it much more apparent in the new version.

I now appreciate that the reference to gauge redundancy below eq. (53) was confusing. The point is that this degeneracy is only superficially similar to a gauge redundancy; this was appreciated already by Xu and Wu in their discussion of the same dispersion relation. I believe that this degeneracy is lifted by nonperturbative effects; it would be nice to understand this better. In the meantime, I have removed the confusing statement from the paper.

5) I have made a significant overhaul of the section devoted to U(1) theories on BCC lattices. The referee's analysis is based on a formula which was only given schematically, in order not to overburden the paper with notation; I have now included all the relevant details, and have explained in what sense the properties F1-3 are obeyed on BCC lattices. I hope that this will resolve the confusions here.

6) I have also added many more details in the analysis of nonabelian theories, and in particular I have explicitly proven the gauge invariance of all operators in the Hamiltonian, just as the referee suggested.

7) I have done my best to address these issues and have added the requisite pictures/explanations, though I have decided to keep the notation in point 7b) in order to keep the formula free from additional clutter.

I hope that these added explanations will help resolve the referee's doubts, and that they will reconsider their assessment about the validity of the results in this paper.

Anonymous Report 1 on 2020-1-5 Invited Report


1 - Allows for the definition of fracton phases on arbitrary manifolds, without the need for a foliation structure, by taking advantage of duality.
2 - Provides nice mathematical structure to the theory of fractonic gauge theories.


1 - Claims are too strong in several places, for example regarding the consistency of symmetric tensors.
2 - I was very bothered by the discussion of fracton theories without even conservation of charge.


In this work, the author proposes a new way of mathematically defining fracton theories which allows them to be placed on arbitrary manifolds. This provides a notable extension of the work of Shirley et al., which showed how to define fracton models on manifolds with a foliation structure, which the present framework does not require. Along the way, the author develops a fairly nice mathematical structure for fracton gauge theories, making this a solid contribution to the literature. I do have a few complaints, however, which I would like to see addressed before this paper proceeds to publication.

To summarize the positive aspects, this paper makes use of a dual formulation of the X-cube model to define it on arbitrary manifolds, introducing a novel type of cohomology theory to do so. The author provides a fairly rigorous way of defining a generalized boundary operator which returns the corners of a membrane operator. This provides some badly needed mathematical teeth to some previously quasi-known things in the fracton literature.

As a question of curiosity, has the author thought about how to incorporate the "diagonal" operators commonly encountered in the U(1) theories, where the "boundary" is not four corners of a square, but rather points all in a line? It would be interesting if these results could be extended to that case.

As a matter of pedantry, I'm not sure why the author refers to the fields used as "higher-form." The higher-form terminology tends to imply antisymmetric behavior, whereas (as the author indeed notes) these fields could have been defined to have been either symmetric or antisymmetric (or neither).

On that note, the author is incorrect that there is no consistent symmetric tensor description of the U(1) theories at the level of the equations of motion. The author specifically looks at the hollow U(1) theory studied first by Xu and Wu, and later by Ma et al. and Bulmash and Barkeshli. The trick to formulating things consistently is to regard the three magnetic quantities as the diagonal components of a rank-2 tensor: Bxx, Byy, and Bzz. In this language, both sides of Equation 51 become manifestly symmetric. However, this is a minor issue and did not detract too much from the rest of the paper.

One other complaint I would like to make is about the claims regarding the absence of conservation laws. The author claims that dipole conservation might not be required for fracton behavior, which I could be persuaded to believe, but then goes on to say that some models don't even have conservation of charge! This to me seems problematic. If there is no charge conservation, or some other local conservation law, what is stopping a fracton from just decaying directly into the vacuum? The author mentions the process of dipoles contracting into single charges. Maybe there is some unusual conservation law involving both charge and dipole moment? I'm not really sure. But surely there must be some guiding principle in terms of some generalized local conservation laws.

Otherwise, the contents of the paper look good to me. Once the author has addressed these issues, I will be prepared to recommend the paper for acceptance.

Requested changes

1 - Address issue of consistency of symmetric tensors
2 - Address issue of charge non-conservation
3 - Consider other optional questions, such as extension of coboundary operator to diagonal elements

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

Author Djordje Radicevic on 2020-01-08
(in reply to Report 1 on 2020-01-05)
answer to question
reply to objection

I thank the referee for the thoughtful comments. I would like to take this opportunity to provide quick replies to several points raised in their report:

1) Regarding symmetric tensor theories, I acknowledge that the third bullet point in sec. 1.1 was phrased ambiguously. I certainly do not believe that all symmetric rank-2 theories with U(1) gauge group are inconsistent. The idea that should have been conveyed was merely that the $\mathfrak{F}_2$ theories are not consistent if naively regarded as yet another example of symmetric rank-2 theories. More generally, I wanted to stress that demanding the rank-2 tensor be symmetric is not necessary to get fracton behavior. I believe this paragraph is the only place where such an unclear statement is made, and this will be easy to fix in a future version. The analysis in section 4.3.1 is not meant to contradict the analysis of existing rank-2 theories -- and I do not think there are any claims that can be construed to mean that there is a contradiction; I merely point out the similarities along the way, e.g. below eqs. (46-47).

2) Regarding the "diagonal" operators the referee mentions, these seem to be naturally included in my framework by thinking of them as coming from square plaquettes in which two diagonally opposite vertices have been identified. I currently do not have anything specific to say about them, but this is an interesting suggestion and I will certainly make an effort to mention it in the future.

3) I agree with the referee that "higher-form" is perhaps an unfortunate usage of the term, especially when we think about fields in the continuum. In high-energy theory, "higher-spin" would be the more appropriate term, but I think this would make things more confusing when thinking about the lattice. Perhaps "higher-rank" will be an acceptable compromise.

4) Finally, I must say I was glad that the referee was bothered by the lack of charge conservation in some of the fracton theories I presented. I mean absolutely no disrespect here: I merely think that such a reaction is a testament to a novel and surprising point being made. I have already presented a rather detailed example of that in eq. (70), which is a theory with all the requisite fracton properties, but without a global shift symmetry that would correspond to charge conservation. As the referee says, there should be some more exotic symmetry that stops the fractons from decaying into the vacuum, and indeed there is: the global symmetry is given in eq. (71). This means that only specific point charge -> dipole processes are allowed. One way of thinking about it is to say that, on the BCC lattice, there are conserved quantities given by linear combinations of the total charge on the sublattice made of cube centers and of the total dipole moments on the sublattice made of cube vertices. While these dipoles/charges on each sublattice are not conserved, together they add to something conserved. I would be happy to make this point more prominent in the manuscript.

I will incorporate these changes into the manuscript after hearing from the editor. If there is another unclear point in the manuscript that the referee had in mind, I am happy to address that too.

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