Boundary Modes in the Chamon Model

We thank Prof. Karch for the careful reading of our manuscript and for stating that our work meets the SciPost criteria for publication.
We agree that our proposal to handle the normal derivatives in the boundary conditions should be useful to analyze other boundaries of related fracton models. In fact, it is interesting to note that the (110) boundary of the X-cube model discussed by Bulmash and Iadecola in Ref. [41] shares some properties with the (001) boundary Chamon model.
Concerning the question in the “requested changes”, indeed the boundary theory for the X-cube model derived in Ref. [43] is similar to the one for the Chamon model since both are written in terms of a two-component bosonic field with a $2 \times 2$ antisymmetric K matrix. Here it seems important to note that the BF theory for the X-cube model contains more gauge fields, but the constraints and the $S_4$ symmetry allow one to express the boundary action in terms of only two components. In the Chamon model, the two fields are associated with the layer index from the beginning. In any case, our current understanding is that different fracton models can have the same boundary theory because the anomaly of the boundary does not uniquely determine the bulk. This statement was verified for the X-cube model in Ref. [43] and it is consistent with the anomaly inflow in the presence of subsystem symmetries discussed by Burnell et al. in Phys. Rev. B 106, 085113 (2022).

We thank the referee for the careful reading of our manuscript and for raising important points which have helped us improve the presentation, emphasizing the novelty of our results. First, we would like to address some general remarks in the referee’s report:

"At the microscopic level, the work is new but the results are in some way expected, and it is not immediately clear to me that one can learn something significantly novel from them (please do correct me if I am wrong – in which case, it may help to stress is more clearly in the manuscript)."

We would like to clarify that we actually did learn something novel by analyzing the microscopic model in section II. Based only on previous work on the Chamon model with periodic boundary conditions, it was not clear whether the model with a (001) surface could have different types of gapped boundaries separated by boundary phase transitions. Here it is instructive to compare with the boundaries of the toric code in 2D (see Bravyi and Kitaev, Ref. [33]) and the X-cube model in 3D (see Bulmash and Iadecola, Ref. [41]), where smooth and rough boundaries are clearly geometrically distinct and are associated with the condensation of different types of particles. Despite some similarities with the X-cube model, the Chamon model exhibits different global constraints which have an impact on its bulk and boundary physics. In fact, it was important to go through the analysis of the microscopic model to appreciate the role of the layer index in distinguishing between the two types of gapped boundaries. Moreover, this analysis taught us that, unlike the X-cube model, there is no condensation of pairs of defects at the (001) boundary of the Chamon model; rather, the boundary allows for the conversion of a pair of defects into a single defect. This understanding developed in the exactly solvable model was crucial for our formulation of the boundary conditions in the effective field theory developed in later sections. Following the referee’s remark, we have rewritten the introduction to emphasize what we learn from the microscopic model in section II and how the results differ from the expectation based on previous results in related models.

"At the field-theoretic level, the authors’ work is definitely an impressive tour de force. It certainly deserves credit and publication. However, as I understand it, the technique that is employed was already introduced and used in earlier work (which the authors correctly and openly credit); the main innovation there is the use of a layer index (which is indeed important and significant). The approach is successful in producing a boundary theory that is consistent with expectations from results in related models (e.g., the X-cube, as the authors themselves point out). The behavior of the system inferred from field theory is consistent with the microscopic lattice model in the corresponding strong coupling regime. Therefore, the main advantage of and novel result that derives from the field theory is the ability to speculate about the weak coupling regime, where the authors argue that a stable gapless boundary phase is possible."

We agree that our prediction of a gapless boundary phase is the main advantage of the effective field theory in comparison with the exactly solvable lattice model. While we have employed field theory techniques introduced in Ref. [20], an important contribution of our work was to examine the boundary processes within this approach, for instance, what happens to the line operators that terminate at the open boundary. Here there are also important differences between our results and previous results on the X-cube model, notably the boundary conditions encoded in Eq. (38).

“I am in two minds about the general interest that this manuscript may raise and therefore I hesitate to recommend it for publication in SciPost. It should however certainly be published somewhere, and for reference, I personally would consider this a shoe-in for PRB. This is my personal conclusion. I would however be very happy for the editor to make an informed opposite decision, or for the authors to provide further arguments, and perhaps correspondingly improve the discussion/intro sections of the manuscript. With respect to the SciPost criteria: 1) "Expectations (at least one required) - the paper must:" At the moment, I would consider that the paper possibly opens a new pathway in the study of boundary behavior in fractonic models, but the case is marginal and I do not see clear potential for multipronged follow-up work.”

We believe our work meets the SciPost criteria because it has the potential to open a new research direction and spark a follow-up investigation. To the best of our knowledge, we were the first to speculate about the possibility of a gapless boundary phase in fractonic systems using the effective theory, despite the challenges posed by UV/IR mixing in these models. To further explore this result, it is crucial to build a corresponding phase diagram using microscopic models. Studies of quantum phase transitions in the bulk of fractonic systems are relatively scarce and recent; see for instance Pretko and Radzihovsky, PRL 121, 235301 (2018), and Muhlhauser et al., PRB 101, 054426 (2020). A natural follow-up to our work is to study critical phenomena at the boundary of fractonic systems, without destroying the fracton topological order in the bulk, and to investigate whether and how the associated boundary phase transitions differ from quantum phase transitions in two-dimensional systems. In fact, we are currently working on this problem with the goal of approaching the gapless boundary phase by adding suitable boundary perturbations to the lattice model. While so far we have focused our efforts on the (001) boundary of the Chamon model, this problem is more general and can be very rich since different fracton models have different particle contents and the boundary phases may also depend on the geometry of the boundary. We have expanded the manuscript, in particular the introduction and conclusion sections, to further illustrate our results and the avenues it opens up for future investigations.

Let us now address the list of requested changes. We thank the referee for spotting the typos listed in points 1, 4, 10, and 13 of the report. We have corrected them in the revised version of our manuscript. Concerning the other points:

“shortly after Eq.(3), and throughout the paper, the authors make use of a nomenclature derived from gauge theories (namely, they talk about monopoles, dipoles, and quadrupoles). This nomenclature is suggestive (and often used in this field of re-search, and I apologize if the answer to my comment happens to be trivial), but I see no explanation in the manuscript as to why one should use it and why it is useful. For instance, one could use this terminology in the toric code, but it is not helpful and it is not done, to the best of my knowledge (it is a Z2 gauge theory). The nomenclature is on the contrary used and useful in dimer and vertex models, where the emergent theory is U(1), and indeed electromagnetic-like. Is there a notion of conserved gauge flux and charge-current continuity in this model that supports the use of such notation? If this is the case, it may help to explain it briefly and/or refer to some relevant literature. If there isn’t, then perhaps pairs and quadruples of defects are a more appropriate naming than dipoles and quadrupoles.”

Indeed we adopted the nomenclature used in the literature without much explanation. The excitations of the Chamon model were referred to as “monopoles”, “dipoles” and “quadrupoles” in Ref. [2] even before the term “fracton” was coined in Ref. [3]. As the referee correctly points out, this nomenclature is better motivated for effective field theories with emergent U(1) symmetries. This is actually the case of effective theories for fracton models, which contain conservation laws described by continuity equations with higher derivatives. In our case, if we take the action in Eq. (25) and couple the gauge fields to matter (representing excitations above the ground state manifold) respecting gauge invariance, we obtain a continuity equation of the form

\[ \partial_0\,\rho^{(l)}=D_{xz} j^{(l)}+D_{yz} j^{(l)},~~~~ \]

with $\rho^{(l)}$ and $j^{(l)}$ the corresponding charge density and currents. This equation implies conservation of charge as well as dipole moments such as $P_a^{(l)} = \int d^3r \xi^a \rho^{(l)}$. The conservation of the dipole moment is related to the mobility of pairs of defects discussed for the lattice model in Ref. [2]. The constraints of the model imply that two defects in neighboring octahedra (or dipole) cannot annihilate each other and such a pair can only move along the direction of a rigid string. On the hand, quadruples of defects (quadrupoles for short) can be created from the vacuum and have no mobility constraints. Thus, the nomenclature is helpful in the sense that it conveys some information about the mobility and decay of the composite excitations. By contrast, in the toric code, there are no mobility constraints, pairs of defects on neighboring plaquettes belong to the vacuum superselection sector, and this nomenclature is not useful. We agree with the referee that this point deserves clarification. In the revised version of our manuscript we explicitly mention that we follow the nomenclature of Ref. [2] and that the latter is further motivated by the emergent conservation laws within the effective field theory. We also added references to the reviews by Nandkishore and Hermele [Annu. Rev. Condens. Matter Phys. 10, 295 (2019)] and Pretko et al. [Int. J. Mod. Phys. A 35(06), 2030003 (2020)], where the role of dipoles in lattice models and effective field theories for fractons is discussed in more detail.

“in Fig.2 it may be helpful for the reader if the authors highlighted the sites on which the σx operators are acting to produce the resulting excitation structure.”

Following the referee’s suggestion, we have modified Fig. 2 to highlight the sites where we apply the spin operators to create the defects.

“on p.4, when local cage-net tetrahedron operators are defined and argued to commute with the Hamiltonian, I wonder if it would be helpful to add a brief paragraph discussing the residual degrees of freedom in the ground state of the system, and showing that they are fully resolved (with the addition of appropriate topological operators, I think). This would complete nicely the discussion. At the moment, the reader is left wondering whether we have identified a sufficient number of conserved quantities to resolve the residual 2N/2 GS degeneracy after Eq.(3) has been imposed.”

The cage-net operators indeed commute with the Hamiltonian, but not necessarily among themselves. Thus, finding the ground state degeneracy through the cage-net operators is not so obvious. In Ref. [2] the ground state degeneracy of the Chamon model in the 3-torus geometry was determined by counting the number of independent line operators of the form

\[ W^{(k)}_{IJ} = \prod_{\mathbf{r}\in \gamma}\sigma_{\mathbf{r}}^{(k)} \]

where $\gamma$ is a rigid closed line, in the $IJ$-plane (with $I\neq J\neq k$), that winds around the system. These operators commute with the Hamiltonian and obey the constraint $W^x_{yz} W^y_{xz} W^z_{xy} = 1$. Following the referee’s suggestion, we have added a discussion regarding the ground state degeneracy and the line operators below Eq. (3).

“in addition to the boundary operator in Eq.(6), one could also define in principle another broken stabilizer centered at $r=(i,j,1)$. This would be made up of a single $\sigma_{r-e_3}^z$ operator, I think. Is there a reason why such additional boundary stabilizers are not considered?”

We cannot add the single-spin operator suggested by the referee because each $\sigma^z_{r−e_3}$ anti commutes with four stabilizers on the same plane, meaning that the resulting model would no longer be a stabilizer code. Moreover, the model that we consider already contains one stabilizer for each spin on the boundary. A more physical way to understand the boundary stabilizers is to imagine that we start from the model in the thermodynamic limit and apply a strong magnetic field in the z direction in the half-space $z > 0$. The effective Hamiltonian is then obtained by projecting the stabilizers in the z ≤ 0 region onto the sector where all spins in the z > 0 region are polarized with $\sigma^z_r = +1$. On the boundary plane, this projection reduces the standard six-site stabilizers in Eq. (1) to the five-site stabilizers in Eq. (6). This way of obtaining the Hamiltonian in half space in terms of a projection of the stabilizers was mentioned in section 4.4. Motivated by the referee’s question, we have decided to present this argument right after Eq. (6) to clarify our definition of boundary stabilizers.

“incidentally, the choice $r-e_3$ in Eq.(6) suggests that the z<=0 half-space is occupied by the system, and $z>0$ is empty. I think it would be good to state this choice explicitly in the text (some readers may instinctively put the system in the z>=0 half-space).”

That was indeed our choice. We now state it explicitly above Eq. (6).

“on p.5 the authors introduce the concept of boundary fracton. Could this concept be explained in a little more detail? Is a boundary fracton different from a bulk one, or can it be simply thought of as a fracton that happens to have reached the boundary? Also, in relation to my point 2) above, it is not obvious to me that the notion of boundary fracton can be made consistent with a gauge charge interpretation of the defects. I presume it is so; in which case adding a few lines to explain and clarify the point would help, I think.”

By boundary fracton we mean a defect of the five-site stabilizers defined on the boundary. Like a bulk fracton, a boundary fracton cannot move without creating additional excitations. However, there are processes involving the boundary fracton that cannot happen in the bulk, for instance, the conversion of a dipole into a boundary fracton illustrated in Fig. 5. This type of process can be consistently incorporated into the effective field theory in terms of the boundary condition in Eq. (38), which connects the bosonic field with its derivative in the direction perpendicular to the boundary. In section 4.3 we used this boundary condition to show that the line operators that have dipoles at their endpoints in the bulk (see our response to point 2 above) actually create a fracton when they reach the boundary. We have added the precise definition of boundary fracton below Eq. (6).

“around Eq.(17) it may be worth clarifying to the reader that the dimensionless vector $\vec{r}$ has been replaced with $a_s\vec{r}$. It is a trivial point, but worth stating, in my opinion.”

We thank the referee for noticing this point. We have decided to include the factor of as in the original definition of the lattice vectors above Eq. (1) to avoid the change in notation.

“Eq.(26) is argued to follow from Eq.(25) in the absence of matter. Could this be explained in greater detail / referenced if need be? The fact that the notion of matter has not been introduced at all in the paper suggests that a certain level of expertise and familiarity with gauge theories is required to derive (26) from (25).”

Here the absence of matter means that we are dealing with the free equations of motion in the gauge theory that describes the ground state subspace. More generally, we could couple the gauge fields to matter currents; see our response to point 2) above. We have changed the sentence to “From the equation of motion...” in order to avoid any confusion.

“in Eq.(27), I wonder if a condition $I\neq J\neq K$ is needed, or if the expression works in any case.”

The condition $I\neq J\neq K$ is indeed needed. We now state this explicitly in Eq. (27).