SciPost Submission Page

Field Theories for type-II fractons

by Weslei B. Fontana, Pedro R. S. Gomes, Claudio Chamon

Submission summary

As Contributors: Weslei Fontana
Arxiv Link: (pdf)
Date submitted: 2021-04-22 14:26
Submitted by: Fontana, Weslei
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


We derive an effective field theory for a type-II fracton starting from the Haah code on the lattice. The effective topological theory is not given exclusively in terms of an action; it must be supplemented with a condition that selects physical states. Without the constraint, the action only describes a type-I fracton. The constraint emerges from a condition that cube operators multiply to the identity, and it cannot be consistently implemented in the continuum theory at the operator level, but only in a weaker form, in terms of matrix elements of physical states. Informed by these studies and starting from the opposite end, i.e., the continuum, we discuss a Chern-Simons-like theory that does not need a constraint or projector, and yet has no mobile excitations. Whether this continuum theory admits a lattice counterpart remains unanswered.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2021-7-3 (Invited Report)


This paper proposed a continuum field theory description of a type-II fracton model, the Haah's cubic code. The novelty of the proposal is that the gauge field theory has a topological term similar to Chern-Simons theory, which differs from previous constructions.


1. The proposed derivation of the field theory from the lattice model starts from rewriting the Haah's code in terms of anti-commuting gamma matrices. Then a key step (Eq. (4)) is to represent the gamma matrices using a new variable A, which then becomes the dynamical field in the continuum limit. Given that this is the crucial passage from the lattice to field theory, it is unsatisfactory that the nature of this field A is never discussed. Naively A should be a discrete variable since gamma squares to 1, but from later discussions (especially from the discussion about excitations on page 10), I believe the authors treated A as a U(1) variable. This should be justified, or at least explicitly stated around Eq. (4).

2. The introduction of the capital Gamma's is somehow not well motivated. Why is it necessary to have the Gamma's in place of gamma's to obtain the field theory? Or perhaps it reflects freedom in identifying the microscopic spin operators with the continuum field variables? It would be useful to have more explanations for this step.

== The following are not really "weaknesses", but rather questions/comments that I hope the authors can address ==

3. Are the three conditions for the T vectors mutually exclusive, so they can never be satisfied simultaneously? This is the impression I got, and if so perhaps it should be explicitly stated.

4. Is the choice for the T vector given in Eq. (10) unique? If not, how does the choice affect the resulting field theory?

5. It is stated below Eq. (14) that the constraint leads to gauge structure, playing the role of Gauss's law in usual gauge theory. If this is the correct interpretation, then Eq. (14) should generate the gauge transformations given in Eq. (18), at least the time-independent ones. It would be good to clarify this point.

6. A somewhat unusual feature of the field theory is that there are two "flavors" of the time component $A_0^{(\alpha)}$, $\alpha=1,2$, but not for the spatial components. I can understand the two flavors corresponding to two kinds of excitations (violations of the X term and the Z term) as $A_0$ is coupled to the charge density, but this interpretation would imply that there are two kinds of currents, so the spatial components should also have the flavor index. More discussions about this issue would help clarify the physical meaning of the gauge fields.

7. The authors mentioned that the field theory is gapped. This is not entirely obvious from the continuum action given in Eq. (8). Perhaps an explicit calculation of the photon spectrum would be helpful.

8. The constraint in Eq. (14) was required because in the lattice model, the eight operators on the corners of a cube multiply to identity. Since the constraint is crucial in establishing the immobility of excitations, one would wonder whether it is similarly significant in the lattice model.


This work proposed an interesting solution to an important open problem in the theory of fracton topological order, namely continuum field theory for gapped type-II fracton model. A detailed derivation of the field theory was provided, and the authors showed that the field theory does have the right mobility structure for excitations. So I believe the paper meets the expectation for acceptance.

Requested changes

Please see above.

Minor issues:

1. In the condition (ii) on page 5, "sistematically" should be "systematically".

2. Below Eq. (5), the meaning of the index $\alpha$ should be explained.

3. Near the end of the paragraph below Eq. (14), does the author mean "i.e. $c_{\vec{x}}^{(\alpha)}=-1\rightarrow q_{\vec{x}}^{(\alpha)}=1$"?

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

Anonymous Report 1 on 2021-6-5 (Invited Report)


This work constructs an effective field theory for the Haah code, a prototypical example of type-II fracton models. It makes an interesting proposal that in order to capture the physics of the lattice model, the effective field theory has to be supplemented with an additional constraint that projects the Hilbert space to certain subspace. Without imposing the constraint, the effective field theory includes mobile dipoles and hence cannot describe a type-II fracton model.


1. It is not clear that after restricting the Hilbert space, the effective field theory is a consistent theory.
2. The global aspects of the effective field theory are not discussed.


This work proposes an interesting idea on the important questions of constructing effective field theories for fracton model. However, I have a few concerns/questions/comments that I hope the authors could address before this work is published.

1. Is the method used in this work applicable only to $\mathbb{Z}_2$ theories? Can it be generalized to $\mathbb{Z}_N$ theories, such as $\mathbb{Z}_N$ X-cube model or Haah code? It might be worth adding a few sentences commenting on this issue.

2. Is the equation (2) and (3) correct? Could there be an overall minus sign missing? If so, does it affect the analysis afterward?

3. Do the commutation relations in equation (6) reproduce $(\gamma^I)^2=1$?

4. In equation (12), should it be $d_1=\frac{1}{2}(\partial_x^2+\partial_y^2+\partial_z^2)$?

5. Is the condition in equation (14) gauge invariant? If so, it might be worth stressing this. I am also concerned about the consistency of the theory on this restricted Hilbert space. Usually, given a consistent theory, naively restricting the Hilbert space to a subspace leads to an inconsistent theory.

6. If I am not mistaken, the Haah code should have two types of excitations for each flavor label $\alpha$. For example, we can create excitations that only excite $C^{(1)}$ using $Z$ operator that can either act on the first spin or the second spin on a site. I assume that the charge configurations in figure 2(a) and 2(b) are related by coordinate transformation. Please correct me if my understanding is incorrect. If so, then the continuum field theory seems to only capture one type of excitations as shown in figure 2(b). What happens to the other excitations?

7. In (A.4) should the last term in $\mathcal{D}_a^{(2)}$ be $(\partial_x+\partial_y+\partial_z)^j$ instead of $(-\partial_x-\partial_y-\partial_z)^j$?

8. In (A.9) should the second equation be $T_a^{(4,1)}K_{ab}\left(T_b^{(1,2)}+ T_b^{(2,2)}\right)=0$ instead?

9. In (B2) should $D_{uv}=\partial_u^2+3\partial_v^2$?

Requested changes

I hope the authors could make appropriate changes in the draft addressing some of questions I raised above. In addition to the questions above, I would also suggest the following changes:

Major changes:

1. Various global aspects of the action in equation (18) is not discussed. For example, is the gauge field compact or non-compact? If the gauge field is compact, what is the periodicity and how it affects the operator spectrum? The authors should acknowledge that the global properties of the effective field theory are not discussed.

2. In the last paragraph of page 9, the author claims that there exist line operators in the full Hilbert space. It would help the reader if the line operators were written down explicitly.

3. It is not clear to me how the differential operator $d_2$ is discretized to obtained the charge arrangement in figure (2). It would help if the authors could elaborate on this procedure, and clarify whether this procedure has any ambiguity or not.

4. In the paragraph below (D5), the author claims that there is a local operator that can create eight defects. It would help the readers if the authors can write down which local operator explicitly.

Minor changes:

1. In the first paragraph of introduction, “This dependence on the lattice details signals a sort of infrared/ultraviolet (IR/UV) mixing …”. I believe the standard terminology is “UV/IR mixing”.

2. In figure 1, add $\dagger$ on the appropriate $\gamma$ operators as in equation (2) and (3).

3. On page 5, “… can be read sistematically from …” -> “… can be read systematically from …”.

4. In the paragraph below (14), $q^{(\alpha)} \mod 2q^{(\alpha)}$ -> $q^{(\alpha)} \mod 2$.

5. In the second paragraph of page 9, the variables $l, u, v$ are first mentioned but they are not defined.

6. Below (A12), “diferent” -> “different”.

7. In appendix D, $T^{I,\alpha}_a$ has index $I$ ranging from 0 to 7 instead of 1 to 5 as in the main text. This might confuse the reader. Also, it is not explained how the index $I$ and vector $n_I$ are labeled around a cube.

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

Login to report or comment