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Composing topological domain walls and anyon mobility
by Peter Huston, Fiona Burnell, Corey Jones, David Penneys
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Submission summary
Authors (as registered SciPost users):  David Penneys 
Submission information  

Preprint Link:  https://arxiv.org/abs/2208.14018v1 (pdf) 
Date submitted:  20220915 22:44 
Submitted by:  Penneys, David 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Topological domain walls separating 2+1 dimensional topologically ordered phases can be understood in terms of Witt equivalences between the UMTCs describing anyons in the bulk topological orders. However, this picture does not provide a framework for decomposing stacks of multiple domain walls into superselection sectors  i.e., into fundamental domain wall types that cannot be mixed by any local operators. Such a decomposition can be understood using an alternate framework in the case that the topological order is anomalyfree, in the sense that it can be realized by a commuting projector lattice model. By placing these Witt equivalences in the context of a 3category of potentially anomalous (2+1)D topological orders, we develop a framework for computing the decomposition of parallel topological domain walls into indecomposable superselection sectors, extending the previous understanding to topological orders with nontrivial anomaly. We characterize the superselection sectors in terms of domain wall particle mobility, which we formalize in terms of tunnelling operators. The mathematical model for the 3category of topological orders is the 3category of fusion categories enriched over a fixed unitary modular tensor category.
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Report
Yes. I do recommend the publication of this article in SciPost.
Requested changes
listed in the attached file.
Anonymous Report 1 on 2023111 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2208.14018v1, delivered 20230111, doi: 10.21468/SciPost.Report.6502
Strengths
1. Lots of illustrative examples and figures
2. Nice physical explanations coupled with the mathematics
Weaknesses
1. Target audience sometimes unclear
2. Some notation/terminology is used extensively before being introduced
3. Some sections feel repetitive, without the link being made explicit by the authors
Report
Summary:
Anomaly free topological orders in (2+1)dimensions can be understood from two perspectives. Firstly, the UMTC $\mathcal{C}$ whose simple objects label the elementary excitations of the associated topological phase. Secondly, an input unitary fusion category $\mathcal{X}$ with Drinfeld center equivalent to $\mathcal{C}$. These two perspectives carry over to topological defects, in particular codimension 1 'domain walls', between the orders. Such domain walls are labeled by Witt equivalences between the UMTC, or equivalently, bimodule categories over the fusion categories.
In many ways, the first perspective seems more natural since it only considers the physical data of the theory. This leads to a tension: In the appropriate 4category, the composition of Witt equivalences does not decompose as a direct sum, while composition of bimodule categories can be decomposable.
The resolution to this is to extend the fusion category/bimodule category perspective to include anomalous topological orders. This is achieved by enriching everything over a UMTC $\mathcal{A}$, which labels the anomaly. Physically, this corresponds to realizing the (2+1)dimensional theory on the boundary of a (3+1)dimensional bulk with only trivial excitations. The boundary excitation are then given by the enriched center, bimodule categories must be compatible with $\mathcal{A}$, and the Witt equivalences are given by enriched endomorphism categories. The anomaly free picture is recovered by choosing $\mathcal{A}=\mathrm{Hilb}$. More generally, only the Witt class of $\mathcal{A}$ is important. This can be seen by stacking a Witt equivalence in the bulk, and fusing into the boundary.
When considering composition of parallel (noninvertible) domain walls, it is possible to decompose the result into superselection sectors. These summands are labeled by minimal projections in an algebra of 'short string operators' in the middle region.
Comments:
My general impression is that this work comprehensively deals with the question of topological defects in anomalous (2+1)dimensional topological phases (explicitly restricting to interfaces that can be localized on the boundary). For the most part, the authors do a good job of explaining concepts from both a mathematical and a physical perspective. This is aided by liberal, and effective, use of figures. Although the community has some understanding that chiral/anomalous topological orders can be realized at the surface of WalkerWang models, I am unaware of a thorough treatment of defects/interfaces and their composition. I therefore recommend publication in SciPost Physics, although I suggest some changes to help clarify the work.
In some aspects, the presentation of the work does not make the intended audience clear. Some concepts are explained extensively, including an appendix on algebra objects, but other seem to be deemed obvious. I feel that a subset of the audience, particularly in a physics journal, may benefit from some clearer distinction/direction as to whether some parts are intended for the more mathematical audience.
Remark III.13 seems to be functioning to expand on Remark II.4, however this is not made clear by the authors. Both of these deal with categorical intricacies which may be beyond many of the physically inclined readers. As such, I recommend these sections be reviewed and clarified.
A couple of times, notation/concepts are used extensively before being introduced/defined. Some particular examples include: 'free module functor', used on page 16, but defined on page 18. In example VD (pages 4647), this happens extensively.
Overall, I think the work is interesting, and accessible, to a physical audience, and should be published in SciPost physics.
Requested changes
1. The explanation of the stacking Witt equivalences on page 1213 is quite confusing. A figure should be added to clarify where everything sits.
2. An in text definition of a Witt equivalence should be given. This is central to the work, but only appears in a figure caption.
3. I recommend the authors attempt to reorder/clarify some sections as mentioned above.
4. Due to the length of the $SU(2)_4$ example, I think it would benefit from explicitly reminding the reader at the end how the various categories relate to the notation $\mathcal{A}$, $\mathcal{X}$ and $Z^{\mathcal{A}}(\mathcal{X})$ used throughout.
5. Page 46, column 2, part (2). This section seems somewhat scrambled. The notation is used, the reintroduced after the equations. Some of the notation has already been used in the first column. This should be reviewed.
Very minor/Typos:
Typo page 1, column 1: 'vaccuum'
Typo page 2, column 2: 'bimdoule'
Typo page 12, column 2: Should '$K_\mathcal{B}$' be '$K_\overline{\mathcal{B}}$'?
Typo page 21, column 1: Same as above for $K_{Z(\mathcal{Y})}$
Typo page 21, column 2.5: $\mathrm{End}_{XY}(\mathcal{M})$ should be $\mathcal{A}$ enriched
Typo page 39, column 2.75: This seems to be a definition of $\mathcal{C}$
Typo page 40, column 1.99: '$[L_e , e_{m^2}]$'
Typo page 40, column 2.75: 'geometrythis'
Typo page 42, column 2: '$T_{m\to m}$'
Typo page 45, column 1.5: 'is contains'
Typo page 45, column 2: missing 1/2 factors
Typo page 46, column 1: $\rho \in \mathrm{Irr}(\mathrm{Rep}(\mathrm{Stab}_{D_n}(g)))$ should be $D_k$
Author: David Penneys on 20230324 [id 3508]
(in reply to Report 1 on 20230111)
Errors in usersupplied markup (flagged; corrections coming soon)
We would like to thank the referee for their careful reading of the manuscript and their suggestions to improve the paper. We have addressed all comments and made the requested changes in our revision. Please find our responses below.
=============================================
In some aspects, the presentation of the work does not make the intended audience clear. Some concepts are explained extensively, including an appendix on algebra objects, but other seem to be deemed obvious. I feel that a subset of the audience, particularly in a physics journal, may benefit from some clearer distinction/direction as to whether some parts are intended for the more mathematical audience.
> We have included more sign posting to indicate when certain parts are intended for a more mathematical audience, together with introducing the environment "Remark (Mathematical)" to explicitly signify this for certain remarks.
Remark III.13 seems to be functioning to expand on Remark II.4, however this is not made clear by the authors. Both of these deal with categorical intricacies which may be beyond many of the physically inclined readers. As such, I recommend these sections be reviewed and clarified.
> We have added some language to clarify the connection between these two remarks. In both cases, we have created a new environment called "Remark (Mathematical)" to emphasize that these remarks are intended for a more mathematical audience. This will serve a sign post to more physically inclined readers that they may skip these remarks. We have included text in the introduction about this new notation and our intentions for our readers.
A couple of times, notation/concepts are used extensively before being introduced/defined. Some particular examples include: 'free module functor', used on page 16, but defined on page 18. In example VD (pages 4647), this happens extensively.
> We have added Footnote 21 introducing the free module functor where it first appears, along with a reference to [BN11]. Section VD now begins with explicit sign posting to Appendix C, where this new notation is introduced and defined. We have also introduced the notation for the group algebra and the function algebra on the group here to clarify any confusion.
Requested changes
1. The explanation of the stacking Witt equivalences on page 1213 is quite confusing. A figure should be added to clarify where everything sits.
> We have now added Figure 6 to help with clarity.
2. An in text definition of a Witt equivalence should be given. This is central to the work, but only appears in a figure caption.
> It is already defined in the text in the first paragraph of IIC, where "Witt equivalence" appears in italics.
3. I recommend the authors attempt to reorder/clarify some sections as mentioned above.
> We have attempted to address this by explicitly flagging those remarks that focus on category theory as “Remark (Mathematical)." We have also tried to make sure we define our notation before using it, and we have increased sign posting to appendices when appropriate.
4. Due to the length of the SU(2)_4 example, I think it would benefit from explicitly reminding the reader at the end how the various categories relate to the notation A, X, and Z^A(X) used throughout.
> We have reminded the reader at the end of IIB about which categories correspond to A, Z(X), and Z^A(X).
5. Page 46, column 2, part (2). This section seems somewhat scrambled. The notation is used, the reintroduced after the equations. Some of the notation has already been used in the first column. This should be reviewed.
> We have included a paragraph at the beginning of Section VD making explicit reference to Appendix C, and we have introduced the notation for the group algebra and function algebra there to make things less confusing. We also have deleted repeated text in (2), which hopefully clears things up.
Very minor/Typos:
> We have corrected all the minor typos. We reply about a particular one below.
Typo page 12, column 2: Should 'K_B' be 'K_{\overline{B}}'?
> Since B\boxtimes \overline{B} is its own reverse, K_B is the same as K_{\overline{B}}. We have added a comment to this effect.
Author: David Penneys on 20230324 [id 3509]
(in reply to Report 2 on 20230131)Errors in usersupplied markup (flagged; corrections coming soon)
We would like to thank the referee for their careful reading of the manuscript and their suggestions to improve the paper. We have addressed all comments and made the requested changes in our revision. Please find our responses below.
=============================================================
1. The notation “2D bulk” in Fig. 1 is confusing. This “2D” must be space dimension, right? but 2+1D is clearly a spacetime dimension. Maybe a declaration of the convention is needed.
> We have now included our conventions for this figure in its caption below.
2. Page 2, the mathematical term “Morita 4category UBFC of unitary braided fusion categories” appears suddenly. I think that it is better to explain its definition. Otherwise, I do not see why “objects in individual UMTCs, do not appear as higher morphisms in the 4category UBFC.”
> We include a more detailed explanation of UBFC here along with an explanation of why the anyons do not appear at the correct categorical level in UBFC to describe topological order.
3. Page 2, the notion of a module tensor category over a braided tensor category also appeared in Definition 2.6.1 in [KZ18a].
> Here we are only citing the original reference that we are aware of, although we acknowledge the same notion has appeared in many interesting articles since.
4. Page 7, what does the term “\Omega tensor” mean? Is it the \Omega in Eq. (3)? If so, it is better to refer to (3) here.
> The \Omegatensor is the 6j description of the halfbraiding as described in Equations (42) and (43) of [LLB21]. We have changed the sentence by including "the \Omegatensor (3) giving the 6jdescription" to make this more clear.
5. Page 8, is C in Remark II.5 a typo?
> Fixed!
6. Page 11, “also that Witt equivalent UMTCs can appear as surface topological orders of the same invertible bulk.” I have a question here. Is that true that Witt inequivalent UMTCs can not realized as surface topological orders the same invertible bulk? If so, the invertible 3+1d TQFT are classified by the Witt group of UMTC, right? Two surface topological orders associated to two Witt inequivalent UMTCs can be connected by 1+1D gapless domain wall. Does this gapless wall produces a 2+1D ‘relative bulk’ in 3+1D?
> This is a very important and interesting question that we specifically wanted to avoid in this article, as it is beyond the scope of the present work. We were careful to only focus on topological domain walls and commuting projector local Hamiltonians.
7. Page 12. is the “UBFC” in Definition II.7 a typo for UMTC? Appendix B only discuss UMTC.
> Yes, since we are claiming K_A is Lagrangian. Fixed!
8. Page 13, the “M” at the bottom of the left column should be a typo.
> Fixed!
9. Page 14, A in the footnote is a typo.
> Fixed!
10. Page 15, “we now turn to the question of central interest: namely, what can happen when we compose two or more domain walls by stacking them? This question is very important, since any domain wall can be obtained by such compositions.” The reason provided here does not sound very convincing. In addition to the reasons from condensed matter physics, maybe the authors can also add that computing the “fusion rules” of topological defects in various dimensions is becoming a fashion especially after the new wave of studying the noninvertible symmetries from the high energy community.
> We have added to the wording here to explicitly discuss noninvertible symmetries, along with including many new references.
11. Page 20, I am not sure if the notation (M \boxtimes N)_{S_Y} has been explained anywhere in the paper. Is it the category of left or right S_ymodules in (M \boxtimes N)? If so, this notion is not so obvious to physics oriented journal. Maybe an explanation is needed.
> We now include Remark B1 in the appendix defining this, and we include a forward reference to this remark.
12. Page 20, “In this alternative definition, a y string emanating from the left (X,Y) boundary cannot end in the bulk, and must cross to the right (Y, Z) boundary.” This sentence is hard to visualize without a picture.
> We have revised our explanation of the Deligne product in this context and also included a picture.
13. Page 22, Example III.12 and more examples can be found in Table 1 in arXiv:2205.05565.
> We have included a citation to this reference.
14. Page 4142, some mathematical details in the discussion of double Ising might have appeared in arXiv:1903.12334.
> We have added a citation to this reference, as well as an additional new reference.
Anonymous on 20230403 [id 3536]
(in reply to David Penneys on 20230324 [id 3509])Strangely, after many attempts, I am not able to see the new version of the paper. But I can see the reply from the authors. See a short remark below. Except this minor point, I do recommend the acceptance of the paper.
Referee's reply I: Page 2, the notion of a module tensor category over a braided tensor category also appeared in Definition 2.6.1 in [KZ18a].
Author's reply: Here we are only citing the original reference that we are aware of, although we acknowledge the same notion has appeared in many interesting articles since.
Referee's reply II: It depends on how you define the term "original". [KZ18a]=arXiv:1507.00503, [HPT16]=arXiv:1509.02937.
Author: David Penneys on 20230404 [id 3544]
(in reply to Anonymous Comment on 20230403 [id 3536])Thank you for pointing out the arXiv number on [KZ18a]; we were unaware how close these were posted to the arXiv given the publication dates. We apologize for this oversight. We will update the manuscript with this reference, and we will make sure future work also cites this article.