# Competing topological orders in three dimensions

### Submission summary

 As Contributors: Kai Phillip Schmidt · Julien Vidal Preprint link: scipost_202106_00019v2 Date submitted: 2021-10-14 12:06 Submitted by: Schmidt, Kai Phillip Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Approaches: Theoretical, Computational

### Abstract

We study the competition between two different topological orders in three dimensions by considering the X-cube model and the three-dimensional toric code. The corresponding Hamiltonian can be decomposed into two commuting parts, one of which displaying a self-dual spectrum. To determine the phase diagram, we compute the high-order series expansions of the ground-state energy in all limiting cases. Apart from the topological order related to the toric code and the fractonic order related to the X-cube model, we found two new phases which are adiabatically connected to classical limits with nontrivial sub-extensive degeneracies. All phase transitions are found to be first order.

###### Current status:
Editor-in-charge assigned

Inconsistency: plain/Markdown and reStructuredText syntaxes are mixed:

Markdown: ('inline_math', <re.Match object; span=(1155, 1165), match='$\\sigma_x$'>)

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We thank Referee 1 for her report and for considering our manuscript as suitable for publication. We provide thereafter a detailed answer to various comments made by Referee 1.

1- About the description of Figure 2:
The problem comes from the layout of the two figures. With the actual presentation, the upper (lower) panel mentioned the caption must be replaced by left (right) panel. However, in each figure, the left inset located in the leftmost lower corner indeed represents a zoom of the ground-state energy near the transition point. We have now changed the caption to avoid any ambiguity in this description. We also added crosses in the lower inset of the left panel.

2- It seems that a figure containing the square plaquette was missing in the submission process to compile properly the manuscript. We added it and it should now look much better to the reader.

3- As explained in the text, the X-phase (Z-phase) is adiabatically connected to the point where only $\sigma_x$ ($\sigma_z$) Pauli matrices are present in the Hamiltonian. In the appendix A, the definition is exactly the same but the confusion of the referee may come from the fact that what we discuss the operators that commute with the Hamiltonian. In the X-phase (Z-phase), these operators are products of $\sigma_z$ ($\sigma_x$ ). Hence, everything is correct and consistent with the definitions given in the main text.

4- The Loewdin partition technique has been already applied at several instances in related problems by us. We therefore have added a sentence in the appendix B refering to these works and additionally giving a link to a phd thesis where more details on the actual application of this technique are described.

5- In the main text, we give the ground-state degeneracy of the X- and Z- phases and we believe that their derivation based on an exact counting of independent conserved quantities is rather suited for appendices. What is left for future work is the nature of these phases away from the extreme points where we can compute this degeneracy exactly. For instance, when $J_+ \neq 0$ we do not know what happens in the X-phase. Do we have a finite topological entropy ? What is the nature and the spectrum of the excitations ? These are typical questions that we would like to address but that are beyond the scope of the present work and that requires alternative approaches

6- The Hamiltonian (1) is not exactly solvable for arbitrary coupling. Thus, we use various approaches to understand its phase diagram. What we claim is that, if there is a unique quantum phase transition, our phase diagram is correct. As stated in the conclusion, this issue requires further (numerical) investigations, but, in any case, no exact answer can be given. Even numerically, it would be hard to decide the uniqueness of a transition point in the thermodynamical limit.
Thus, we understand that our phase diagram can be considered as "likely" or "putative" but we clearly explain, in the text, under which hypothesis it is obtained.

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We thank Referee 2 for her/his report and for considering our manuscript as suitable for publication. We provide thereafter a detailed answer to the points raised by Referee 2.

1- We extended the title to "Competing topological orders in three dimensions: X-Cube versus toric code".

2- We have rearranged and reduced the number of citations in this part of the introduction.

3- It seems that a figure containing the square plaquette was missing in the submission process to compile properly the manuscript. We added it and it should now look much better to the reader.

4- The problem comes from the layout of the two figures. With the actual presentation, the upper (lower) panel mentioned the caption must be replaced by left (right) panel. However, in each figure, the left inset located in the leftmost lower corner indeed represents a zoom of the ground-state energy near the transition point. We have now changed the caption to avoid any ambiguity in this description. We also added crosses in the lower inset of the left panel.

5- We have included a sentence within appendix A.2 referring to figure 5.

6- We added two sentences in appendix B.

7- We included the upper-case letters in the mentioned titles

8- We thank the referee for this comment and we did as suggested.

9- We thank the referee for this comment and corrected the title.

### List of changes

List of Changes
===============
For the sake of clarity, all changes (if appropriate) in the revised version are highlighted in red.

- We have included the figure containing the square plaquette which was missing in the previous SciPost version.
- Figure 2: We have updated the caption of figure 2 as suggested by both referees and we have added crosses in the left lower inset.
- Page 8, reference to Figure 5.
- Page 8, two additional sentence -> degeneracies
- Ref 45,46,47 and 5, added upper-case letters in the titles
- Ref 50, separated author name by comma instead of "and"
- Ref 66, corrected the title

### Submission & Refereeing History

Resubmission scipost_202106_00019v2 on 14 October 2021
Submission scipost_202106_00019v1 on 10 June 2021

## Reports on this Submission

### Report

With their revised version, the authors have addressed most previous concerns. Apart from some possible typographic errors that I list as "Requested changes", there are just some details that have not really been addressed:
1- The question about series expansion for degenerate ground states (item 6 of previous Report 2) has not really been addressed. Ok, there is a statement now on page 9 that the series are valid for the entire ground-state manifold, but this does still not explain how the computation has actually been carried out.
2- In their response to item 4 of Report 1, the authors claim to have added a link to a PhD thesis. I understand that this is Ref. [56], but I was not able to find a link.

I believe that the two points above and the suggestions for proofreading are minor items that can be taken care of during the production/proof stage. Otherwise, this is now really a nice short note. Consequently, I recommend publication of the manuscript in SciPost Physics.

### Requested changes

Requests:
1- Add a sentence to Appendix B to explain how the degenerate ground states are treated during the calculation of the series.
2- Make sure that DOIs/links are available for all references, in particular Ref. [56].

A few further suggestions for proofreading:
3- Line 4 of abstract: "displays" rather than "displaying"?
4- Lines 17,18 of Introduction: "The topological order ... ARE called fracton order" -> "The topological order ... IS called fracton order" ?
5- Line 5 of section 2.1: remove space in "e.g. ,".
6- Line 4 of caption of Fig. 1: "build" -> "built".
7- Line 11 of paragraph "XC phase" on page 4: "-1" -> "$-1$".
8- Line 7 of section 3: no full stop between "Fig. 2" and "(see".
9- Last sentence of caption of Fig. 2: Insert "The" before "thin solid"?
10- Page 6, 4 lines below Fig. 2: Insert "the" before "TC-phase"?
11- Line 4 of page 9: I believe that after "consequence", the correct preposition would be "of" rather than "from".

• validity: high
• significance: good
• originality: good
• clarity: good
• formatting: excellent
• grammar: excellent

### Author:  Kai Phillip Schmidt  on 2021-10-28  [id 1887]

(in reply to Report 2 on 2021-10-26)

We thank the referee for the feedback and the recommendation for publication of our article in SciPost apart from two requests and additional minor points. We have adressed the points as follows:

1-We added a sentence which should explain how the series is computed. 2-We made DOIs available where we could find them (For Refs [56] and [62] but not for Refs. [57],[58] and [63]). 3 to 11-We adjusted the respective parts according to the suggestions of the referee.

# List of Changes

For the sake of clarity, all changes in the revised version are highlighted in red. The changes are essentially what has been requested by Referee 2.

### Report

I thank the authors for considering my questions and suggestions, which they have addressed in both their response and in the new version of the manuscript. Following the modifications they have implemented, I am happy to recommend the publication of the manuscript in its present form.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Kai Phillip Schmidt  on 2021-10-28  [id 1886]

(in reply to Report 1 by Helene Spring on 2021-10-15)

We thank the referee for her positive feedback and the recommendation for publication of our article in SciPost.