Boundary Modes in the Chamon Model

We would like to address the late report we received in the previous resubmission.

Referee "I am troubled by section 3, which is reproducing a field theory that was written down quite some time ago in Ref. 9 (which is also Ref. 27), without proper acknowledgment thereof. (they appear to obtain the same field theory in a slightly different coordinate system, from a different derivation, and Ref. 9 does explicitly make the connection to the Chamon code in Section VII, which is essentially the reverse of the process that they use to derive it). — Similarly, the analysis of the edge in the continuum appears to be quite similar to the discussion in IIIB of Ref. 9. Again, in my view, this section does not properly reference the previous work that did essentially the same thing."

We start by adressing the comment regarding similarities with the work “Fractonic Chern- Simons and BF theories”. There, the authors considered Chern-Simons and BF theories of higher-rank gauge fields and what consequences the exotic gauge structure arising from the higher-order derivatives would bring. They adopted a bottom-up approach (i.e., going from the IR to UV) to make the connection to the Chamon model in a cubic lattice instead of the fcc lattice where the model was originally defined. Before addressing the main differences between the approaches, we would like to apologize for not emphasizing their work in section 3. Comments regarding their work were added to properly acknowledge what has been done and to make more explicit the important differences that we treat in our work. Now to the differences: As mentioned previously, they adopted a bottom-up approach to obtain the Chamon model in the cubic lattice. Although not necessarily problematic, this approach hides some subtleties that are evident in the top-down (UV to IR) approach that we took, for example, highlighting the role that the lattice spacing plays in the field theory. Another crucial and important difference is that the connection to the Chamon model in their work is through a higher rank Chern-Simons theory (meaning that there is only one species of gauge field) that is then discretized into the lattice to obtain the Chamon model in cubic geometry. This again is a consequence of the bottom-up approach. In this case, the layer index that we have considered, which is explicit in the lattice theory, suggests to us that the field theory description should contain two species of gauge fields, hence connecting the Chamon model with a double higher rank Chern-Simons theory (or higher rank BF theory) instead of the single Chern-Simons term obtained previously. Regarding the boundary description, we are not sure about the similarities that the referee pointed out. Given the description in terms of Chern-Simons or BF theories, it is expected that there will be non-trivial modes at the boundary due to the gravitational anomaly that appears when one puts the model into a finite manifold. In their description of the boundary, despite the level (s parameter) being dimensionful and the theory being dynamically trivial, the boundary treated in their work is not equivalent to ours. In our geometry, their case corresponds to a (011) boundary, instead of the (001) boundary we considered. In the tilted geometry that they work, obtained from requiring C3 invariance along the (111) direction, will inevitably lead to a boundary theory with mixed derivatives, which makes their construction more similar to the case of the X-cube model, explored in more detail by Prof. Karch and collaborators (also referenced in our work). In our case, the analysis should be taken with more care since the boundary we have considered contains normal derivatives of the fields that would lead to non-physical extra degrees of freedom at the boundary. Handling these normal derivative terms is an important point of our discussion in order to obtain a consistent theory at the boundary. Although subtle, this is a striking difference between our case and the work the referee mentioned. With all that considered, we believe that all these subtleties are important and constitute the main differences between what we did and what was done previously.

Referee

"The question of why it is that the edges are gappable is indeed a puzzling one when approached from this continuum theory. It’s interesting that the authors write down such gapping terms, but I do not find their analysis very satisfying. In particular, the thing that I found surprising about this gappability is that coming from 2 dimensions, it’s natural to expect that a Chern-Simons-like theory has some kind of gravitational anomaly, such that even if you are willing to break charge conservation you would still have a gapless boundary. I don’t see a discussion of this in their approach; they write down cosine terms without asking whether these should be compatible with the commutation relations of the fields. It may well be that it is and the analysis is correct, but I would certainly have appreciated a more careful justification of this."

We thank the referee for pointing this out. Perhaps the compatibility of the gapping terms was not that explicit in our presentation. Note that even in two dimensions there is the possibility of gapped boundaries. The gravitational anomaly measures the thermal Hall conductance $\kappa_H$, then of course, when one has $\kappa_H\neq 0$ then the boundary is necessarily gapless, but for $\kappa_H =0$ the boundary can be gapped given that the gapping terms obey a list of criteria (see for example Phys. Rev. X 3, 021009 (2013), Phys. Rev. B 91, 125124 (2015)). In any case, our analysis of the gappability of the boundary is in similar reasoning of those in these works we have mentioned. To start, the number of compatible cosines that can be added into the action depends on the size of the K-matrix, for an N × N matrix only N/2 cosines can be added such that gap remains stable. Adding less than that would not gap all the fields, adding more than that would lead to incompatible interactions and hence an unstable spectrum, as we have mentioned in the paragraph near Eq.(71) of the revised manuscript (Previously it was near eq.(74)). This is why we only considered one gapping term, given that our K matrix is 2×2. The “null and mutual statistics” condition (borrowing the terminology of Phys. Rev. B 91, 125124 (2015)) are encoded in our Eq.(71) of this new version, which guarantees that all possible gapping terms will commute, and hence can be pinned altogether. These conditions also ensure the stability of the gapped spectrum.

* Extended discussions in sections 3 and 4.4