Exotic $\mathbb{Z}_N$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory

Dear Editor,

We would like to thank the referees for their helpful input and suggestions. We have updated our draft to address these reports.

Sincerely,
Shu-Heng Shao and Nathan Seiberg

1. The third paragraph in the introduction was added to address the first comment of Report 2.

As the referee states, this paper reproduces all the universal features of the lattice X-cube model using the field theory approach of this series of papers. Later papers by these authors use the same approach to present new models and to uncover new results. The motivation here is to make sure that the formalism works correctly, by reproducing known results.

2. Footnote 3 was added to address the third comment of Report 1.

3. The paragraph starting at the end of page 4 was updated to address the second comment of Report 1. We explain our notation of the global symmetries there.

4. The caption of figure 1 was updated to address the third comment of Report 2.

5. The third paragraph in “Outline” in the introduction and the paragraph below (4.13) were added to address the fourth comment of both Report 1 and Report 2.

The main distinction between our continuum discussion and the existing literature is a detailed characterization of the space of fields and the allowed singularities. This has important consequences, especially in relation to the various global aspects of the system, including the quantization of the level in the BF Lagrangian. Another important difference is in the careful analysis of the symmetries – both the space symmetries and the subsystem symmetries. Finally, we use dualities to relate various different presentations of the same system. This leads to new continuum Lagrangians for the long-distance behavior of the X-cube model that have not appeared in the literature.

6. Regarding the first point of Report 1, there was no typo in (2.3). But the equation and the text below it were updated to avoid confusion.

7. The paragraphs below (2.7) and below (3.8) were updated to address the second comment of Report 2.

Since the model is gapped and has no local operators, there is a neighborhood in the space of coupling constants with the same long-distance behavior. However, the high-energy behavior changes within the neighborhood. The X-cube lattice model is a particular solvable point in that neighborhood. Its high-energy spectrum includes massive charged particles – various fractons, lineons, and planons. The point we focused on is special because of two related facts. It has a larger global symmetry and does not have these charged massive particles. This larger symmetry is central to our analysis. For this reason, we think that this distinction between the two points is important. Just for comparison, as we review in Appendix C, the difference between these points is similar to the difference between a standard lattice gauge theory of a cyclic group (a la Wegner) and the toric code (a la Kitaev).