Competing topological orders in three dimensions

Nice investigation of the phase-diagram of a three-dimensional X-cube+toric-code model.

At first sight, not very attractive for a reader.

Following up on previous publications by the same group on related two- [53] and three-dimensional models [39,54], the present work investigates the competition between two different topological orders in a three-dimensional X-cube+toric-code model. The total Hamiltonian can be decomposed into two commuting parts, one of which displays a self-dual spectrum. The phase diagram is derived from high-order series expansions of the ground-state energy. The central result is the ground-state phase diagram Fig. 3 for the full model that exhibits four phases. Result are reliable in the limits where one of the coupling constants vanishes while it would be good to validate the absence of further phases by other methods.

Once one sits down and reads this manuscript, this is a nice short note. However, the authors might be underselling their work, e.g., for the following reasons:
1- The title of the manuscript is a bit unspecific.
2- The presentation of the new results is quite compact and overshadowed by a long review of previous results (until the end of page 4) and a reference list that amounts to about a third of the manuscript. In particular the broadside citation of Refs. [16-40] in the second paragraph of the Introduction is not very helpful for the reader.

Beyond the above, I have a few more specific comments that I list among the “Requested changes”. Once the authors have addressed these concerns, I expect the present manuscript to be a nice piece of work that is suitable for publication in SciPost Physics.

1- The title appears a bit too generic to me. I recommend to look for one that is more specific as far as model and/or method are concerned.
2- Purge unnecessary references among [16-40] or at least make the discussion in the second paragraph of the Introduction more specific.
3- The review copy has a number of white spaces (e.g., after "where" in the line below Eq. (3)) which leads me to suspect that a certain symbol (a square?) was systematically lost. Please check and correct as appropriate.
4- First line of caption of Fig. 2: there are only "left" and "right" panels, but no "upper" and "left" ones.
5- Figure 5 is not referenced; it appears to be related to appendix A.2, but this is not stated explicitly.
6- Some details underlying the series presented in Appendix B are unclear. In particular, since ground states are supposed to be degenerate, it is not clear if these series are valid for any of them, or if they have computed with the help of a specific choice of ground state.
7- Make sure that the names "Hall", "Moore-Read", and "Ising" start with an upper-case letter in the titles of Refs. [45,46], [47], and [53], respectively.
8- The first and second author of Ref. [50] should be separated by a comma and not by "and".
9- The first word of the title of Ref. [66] seems to be misspelled.

The authors study a four-parameter Hamiltonian composed of the three-dimensional toric code Hamiltonian and the X-Cube Hamiltonian. The ground-state degeneracy and the high-order series expansion of the ground state energies are calculated in four different limits. The expressions for the energy are extended to other regions of the parameter space in order to define a phase diagram. The authors present two as of yet unreported phases arising from the competition between two known topologically ordered systems. Despite the fact that this field is not my principal area of expertise, the manuscript was accessible and clear to read. I recommend this manuscript for publication, once some structural errors in the text are corrected. These are as follows:

1) In Figure 2, the descriptions of the left and right insets appear to be switched.

2) The ‘square’ symbol in Z_square is not correctly rendered in the manuscript. (An alternative could be to simplify the notation to Z_f and Z_c, for ‘face’ and ‘cube’ operators respectively to increase legibility. )

3) In Appendix A, it seems as though the definitions of the X- and Z- phases are inverted with respect to their definitions in the main text.

I have one remark about the content of the manuscript that I would like to see addressed:

4) Appendix B lists the methods used to calculate the analytical results for four parameter limits. This section is quite concise, and lacks elaboration on how the Lowdin method is used to obtain the results. This method is not commonly used in papers on 2D or 3D systems with topological order; as such, either expanding this section, referring to specific sections of the original Lowdin paper or adding references that use this method in comparable systems would be helpful to the reader.

I also have some questions and comments for the authors, but I leave the decision of whether to use them to modify their manuscript.

5) The manuscript explores limits of a Hamiltonian composed of the 3D toric code Hamiltonian and the X-Cube Hamiltonian. The authors bring to light the existence of two additional phases of the Hamiltonian, X- and Z-phases, and their ground state degeneracies are studied. It might be more appropriate to include these results in the main text rather than relegate them to the Appendices. It is unclear to me why a more substantial discussion of these new phases is left for future work, as a clear motivation for an additional, self-contained study of these phases is not provided in the outlook section.

6) The phase diagram is constructed from four separate points in phase space, then the transition points are found by extending the analytical solutions of the energy at these points to other regions in phase space, in order to find where they intersect. Since the authors mention that there is no guarantee of a single transition point between phases, it seems premature to present a phase diagram. The assumption of a unique crossing point is a hypothesis that could be tested numerically. Therefore this task could be left to future numerical work.