Lattice Clifford fractons and their Chern-Simons-like theory

The authors discuss a novel class of gapped fracton models in all odd spatial dimensions, which includes the Chamon model as the simplest case, and propose the low energy continuum field theory description.

(1) In various discussions it is not clear which quantities are used to define the microscopic lattice models, and which are derived quantities in the continuum field theory. See my comments in Requested Changes.

(2) Various global aspects of the continuum field theory are not discussed in detail.

While the results are interesting, various crucial aspects of the continuum field theory are not addressed, and many other discussions could still be improved. I would request a major revision before it is considered for publication.

Major changes:

(1) If I understand correctly, the T-vectors are microscopic data that enter into the definition of the lattice Hamiltonian as in equation (8). On the other hand, the matrix $K_{ab}$ are coefficients of the continuum Lagrangian. Starting from a given microscopic lattice model, there should be an unambiguous way to determine the macroscopic parameters. In section III and in appendix B, it is not clear how the matrix $K_{ab}$ is determined from the microscopic data.

Relatedly, I am confused why the authors say in the Final Remarks that "The details about an specific lattice model enter vial this matrix K...". The matrix $K_{ab}$ is introduced after the authors have defined the lattice model in equation (8).

(2) In ordinary abelian Chern-Simons theory, two different K matrices can define the same continuum field theory. What's the corresponding identification for $K_{ab}$ in the current context?

(3) In the discussion of the continuum field theory, it is not clear whether the gauge fields $A$ are $U(1)$-valued or $\mathbb{R}$-valued. In the former case, there is no need for the integer field $m^{(\alpha)}$. In the latter case, the $\mathbb{R}$-valued gauge fields $A$ are further subject to an integer gauge transformation, and the field $m^{(\alpha)}$ is the corresponding integer gauge field.

The authors are not consistent in distinguishing these two presentations. In (38), it seems that the fields $A$ are $\mathbb{R}$-valued, but then in (40) and in (43), they become $U(1)$-valued and there is no integer field $m^{(\alpha)}$. The authors could have worked with $U(1)$-valued gauge fields throughout, and comment that different charge sectors arise from nontrivial twisted sectors of the $U(1)$-valued gauge fields (rather from an independent integer field $m^{(\alpha)}$), as is standard in field theory.

(4) Starting from a microscopic lattice model, it is conceivable that the $K_{ab}$ matrix is naturally quantized in some way. This is perhaps the perspective in section IV.A. However, if one starts from the continuum field theory (38) as given, it is not clear why the matrix $K_{ab}$ has to be quantized. In ordinary Chern-Simons theory, the K matrix is quantized by the large gauge transformations. The authors should discuss what large gauge transformations quantize their $K_{ab}$.


Minor changes:

(1) The authors are not consistent with their convention for the space and spacetime dimensions. For example, in the last sentence on page 2, both "3D" and "(2+1)-dimensional" are used in the same sentence. Also, in the abstract, by "(D+1) effective theory", I assume the authors mean "an effective theory in (D+1) spacetime dimensions."

(2) Perhaps the authors can provide a more intuitive explanation why the number of qubits on each site $n$ is correlated with the spatial dimension $D=2n+1$. This is not motivated in the introduction on page 3.

(3) The indices $a,b$ are not defined in equation (1) (although it is defined later).

(4) Above equation (7), $\Lambda_o$ is not a sublattice: the sum of two odd vectors is even.

(5) Below (43), "be an integer odd".

(6) In the Final Remarks, the authors comment that the same construction applies to the Haah code, which is in 3 spatial dimensions. However, there are 2 qubits per site in the Haah code, in which case the authors' general construction would yield a fracon model in $2\times 2+1=5$ spatial dimensions. Perhaps the authors can clarify this point.

1) This work is thorough, very timely, and all
the ideas and results are presented in a clear and transparent manner.

2) The identification of continuum descriptions of fracton models on the lattice in terms of Chern-Simons-like theories gives a good way to understand the connection between the well-studied fracton lattice models and field theories.

3) The connection to hierarchical quantum Hall states is interesting and well-explained.

The authors could consider adding short discussions addressing the following:

1) It would be interesting to mention how this can be connected to the foliation paradigm present in the fracton literature.

2) How does this formalism capture the non-trivial statistics between the different subdimensional particles?

3) Can this formalism be extended to SSPT models on lattices with more exotic geometries, such as hyperbolic geometries? In this case, the group of translations results in a non-abelian group, which could potentially allow for more interesting physics.

This is a well-written and timely paper invoking a
novel approach to address an important conceptual problem related to connecting lattice fracton models with field theory descriptions. The paper is of good quality, and will make a fine contribution to SciPost.

Below, I list some optional additions (same as in the "weaknesses" section) that the authors could consider:

1) It would be interesting to mention how this can be connected to the foliation paradigm present in the fracton literature.

2) How does this formalism capture the non-trivial statistics between the different subdimensional particles?

3) Can this formalism be extended to SSPT models on lattices with more exotic geometries, such as hyperbolic geometries. In this case, the group of translations results in a non-abelian group, which could potentially allow for more interesting physics.