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Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries
by Siyuan Wang, Yanyan Chen, Hongyu Wang, Yuting Hu, Yidun Wan
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Siyuan Wang |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202406_00062v1 (pdf) |
| Date submitted: | 2024-06-28 14:04 |
| Submitted by: | Wang, Siyuan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this paper, we apply the method of Fourier transform and basis rewriting developed in [JHEP02(2020)030] for the two-dimensional quantum double model of topological orders to the three-dimensional gauge theory model (with a gauge group G) of three-dimensional topological orders. We find that the gapped boundary condition of the gauge theory model is characterized by a Frobenius algebra in the representation category Rep(G) of G, which also describes the charge splitting and condensation on the boundary. We also show that our Fourier transform maps the three-dimensional gauge theory model with input data G to the Walker-Wang model with input data Rep(G) on a trivalent lattice with dangling edges, after truncating the Hilbert space by projecting all dangling edges to the trivial representation of G. This Fourier transform also provides a systematic construction of the gapped boundary theory of the Walker-Wang model. This establishes a correspondence between two types of topological field theories: the extended Dijkgraaf-Witten and extended Crane-Yetter theories.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1. The paper presents all of the technical details of the calculations
2. The calculations appear to be technically correct
3. The results are potentially useful for advancing knowledge in the field
Weaknesses
1. The authors need to give a clearer explanation of what can be learned from the correspondence that they establish
2. There are a number of points (see below) that are poorly explained or confusing.
3. The paper presents no concrete examples. These could be useful first to clarify certain points in an accessible way (see 2), but also to illustrate more concretely what can be learned from this approach that has not been described elsewhere in the literature. For example, perhaps the authors can use their approach to identify a class of gapped boundaries for non-abelian gauge models that were not previously known.
4. The paper has no discussion or conclusion.
Report
The authors use the Fourier transform of general groups to re-write a conventional 3D gauge theory model with gauge group G in a basis where edges are instead labeled by group representations. They discuss how to do this both in the bulk and at the boundary, as well as how to make an explicit connection between the model in the representation basis and a 3D topological lattice model due to Walker and Wang. They use this to describe in general terms some gapped boundaries of these models that have not been described previously to the best of my knowledge.
The results that this paper obtains are certainly not surprising, and its main contributions to the literature are highly technical in nature, at the level of how in detail to carry out a mapping from one basis to the other. That said, this is a case where there is merit to writing out the technical details, and I could be persuaded that this work makes a sufficiently substantive contribution to the literature on 3D topological lattice models to warrant publication in Scipost. However, in its current form I find the paper to be very heavy on technical details and light on discussion of their physical interpretation and importance. I think that the authors could do a better job of:
(1) articulating what can be learned from this formulation, other than showing that two previously known models can be mapped one onto the other
(2) clarifying what their general framework says about simple examples, such as simple abelian theories, and then working out some specific examples of e.g. interesting gapped boundaries in non-abelian models.
(3) Making the connection to known models, and especially the Walker-Wang model, clearer (I have a number of specific comments about this below.).
Requested changes
ntroduction:
— I’m confused about the comparison to lattice gauge theory. Specifically: the topological models contain a different Hilbert space, in which the Gauss’ law constraint is satisfied only energetically (i.e. in the ground state). Thus the models discussed are very similar to gauge theory models with non-dynamical matter (in which charges can appear as static excitations). However, modulo this difference, the topological models are essentially a particular limit of lattice gauge theories, in which the fluctuations of the electric field are set to 0. (This is discussed, for example, in the PhD thesis of Mark De Wilde Propitious). I am similarly confused about what the authors mean when they say that lattice gauge theory has only local excitations: because of the constraint, charge excitations can live only at the ends of Wilson lines, and the constraint that the net magnetic flux passing through a closed surface must be an integer multiple of 2 pi is present in both theories. I suggest that the authors clarify this discussion.
Section 3:
— It would be helpful if the authors commented on the connections between their Fourier transform and a change of basis in a quantum gauge theory from the group basis to a basis of eigenstates of the electric field. Intuitively, I expected them to be related, but the need to introduce a braiding suggests otherwise.
— Similarly, the authors should clarify the following. In the original model, violations of the vertex term are clearly given by representations of G, while violations of the plaquette term are given by group elements. I don’t see a discussion of the excitations in the Fourier transformed model (when tails are included). This is implicit in the construction but some readers may benefit from a more detailed discussion.
— I am not sure I understand Eq. 39. In Walker and Wang’s formulation, the R factors appear in two ways. First, in the Hamiltonian, from “twisting” an edge relative to another edge with which it shares a trivalent vertex. Second, they dictate relative phases between different edge configurations, in which the configuration effectively has a twisted loop. The picture in Eq. (58) is the one relevant to evaluating the plaquette operator. Fig. 5 appears to be illustrating a situation with a crossing of the first type, in which the R matrix should be telling us about a relative phase between two configurations in a ground state. Could the authors clarify the connection between this and Eq. 38? It is not obvious to me.
— I am also confused about how we choose R. The authors seem to be implying in the discussion below (41) that there is *a* choice of R. However, for general G, I expected there to be multiple choices. First, naively I expected that taking R to be trivial (always +1) is always allowed, and that this is the choice that would have the same spectrum as the original G theory (i.e. bulk point-like charges and line-like vortex loops). Second, I would also have expected that there are other choices of R that satisfy the hexagon equations, and that these correspond to physically different theories in which the number of point particles in the bulk is reduced. For example, when the Hilbert space on each edge is just Rep(Z_2 ) = Vec(Z_2), there are two choices for R, and these yield the 3D Toric code (with point particles and vortices) and the “3D semion model” which does not have point-like bulk excitations. The authors should address this point more thoroughly in Section 3, so that readers can understand how much freedom there is in choosing R, and how the choice of R affects the final theory. In my view this is a very central question, since it determines whether we are discussing “the” Fourier transform of the model in question, or simply “a” Fourier transform…
— Please discuss the role of R at the boundary in more detail. Again, comparing the 3D Toric code to 3D semion model, we see that the former admits two distinct gapped boundaries (which are usually thought of as flux condensing and charge condensing), while the latter apparently only admits one. Hence I would have thought that there are some stipulations on what boundary condition(s) can be chosen, and that these depend on R.
Section 4:
— In the third paragraph of Section 4, the authors should specify whether they are discussing 2+1 D or 3+1 D topological orders. They should also clarify the relationship between this paragraph and the discussion of coupled layer constructions in the following paragraph.
— Loop like excitations in 3D abelian topological orders are, however, well understood. The authors should note this and include Ref. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.165119 .
— The authors should add a discussion of the physics of these gapped boundaries. For example, given a subset of charges that can condense, what happens to loops that are brought to the boundary?
Section 5:
— I don’t understand what the authors mean when they say: “no loss of information is
“ the WW model does not contain charge and dyon excitations, just like the LW model.” Dyons in 3D theories would require point-like monopoles carrying charge — but the discrete gauge theories discussed here don’t have magnetic monopoles. So why are we talking about dyons at all?
Recommendation
Ask for major revision
Strengths
1. The paper is well-organized and clearly explains the correspondence between lattice quantum field theory and the state-sum construction of TQFT.
2. The Frobenius algebra-based boundary theory is extended to the 3d model.
3. An explicit Fourier transformation between the G-lattice gauge theory and the Walker-Wang model is provided.
Weaknesses
There are many typos that need to be revised.
Report
In this work, the author investigates the gapped boundary theory of a 3d lattice model of topological order. Similar to the 2d case [JHEP01(2018)134], the construction is based on the Frobenius algebra within the input unitary fusion category $\mathrm{Rep}(G)$. Through the Fourier transform, which reformulates the original model into the fusion basis, they argue that the 3d lattice gauge theory is equivalent to the Walker-Wang model. This generalizes the well-established result in 2d, where lattice gauge theory is equivalent to the Levin-Wen string-net model.
The results are interesting and the paper is well-written. I believe this work meets the publication criteria for Sci-Post.
Requested changes
Before publication, I believe the paper requires thorough proofreading, as there are some typos that need to be addressed, and there may potentially be more.
P 5, Line -2 of the first paragraph, "oriented cubic lattice $\Gamma$ which boundaries", "which" should be "with".
P 6, Line 2 below Eq.(7), "operators are all commute with each other".
P 7, Caption of Fig 2, "that terminat on the boundary".
P 7, Line above Eq. (11), "the duality map and its inverse has presentations".
P 10, Line -4, "Additionaly" should be "Additionally".
P 13, Caption of Fig 4, "by contracting there indices", "with an free end".
P 15, Line 2 of the second paragraph, "larger Hilber space".
P 15, Caption of Fig 5, "as are edges labeleds by".
P 27, Above Fig 9, "comparing with the LW model."
Recommendation
Ask for minor revision
Author: Siyuan Wang on 2025-01-03 [id 5081]
(in reply to Report 2 on 2024-09-10)We would like to express our sincere gratitude to the editor and the referee for their insightful comments and suggestions on our manuscript. We appreciate the time and effort taken by the referees to review our manuscript and provide valuable feedback. Below are our responses to the referee's questions.
The referee writes:
Our response:
We acknowledge that this discussion is unnecessary and misleading. Therefore, we have removed this paragraph.
The referee writes:
Our response:
The Fourier transform in the local Hilbert space of a single edge is similar to the change of basis in a quantum gauge theory from the group basis to a basis of eigenstates of the electric field. Nevertheless, if we want to define the total Hilbert space of the Fourier-transformed model, the braiding structure is needed to ensure that the Fourier-transformed model is topological.
The referee writes:
Our response:
In 3D GT models with gauge group $G$, charge excitations are labeled by irreducible representations of $G$, which can be seen explicitly from the Fourier-transformed expression of the vertex operator. Nevertheless, the classification of flux excitations in the 3D GT model is very tedious when $G$ is non-Abelian (only when $G$ is Abelian violations of the plaquette term are given by group elements). A detailed discussion of excitations in these models will be referred to in future work.
The referee writes:
Our response:
The two appearances of the $R$-matrix are exactly the same. Generally, a category is said to be braided if for arbitrary two objects $X$ and $Y$ we have an isomorphism $c_{X,Y}:X\otimes Y\to Y\otimes X$. As the tensor products of the group representations are commutative up to isomorphism, the category $\operatorname{Rep}(G)$ is always braided. The morphism $c_{\mu,\nu}$, where $\mu,\nu\in L_G$ then enables us to define the state with crossing edges in the graph. For example, the ket in Eq.(38) should be viewed as the morphism $c_{\mu,\nu}$. In order to evaluate the inner product in Eq.(38), we further need to define the tensor representation of the morphism, which naturally introduces the $R$-matrix as shown in Eq.(39). The $R$-matrix fully determines the isomorphism between two morphism spaces $\operatorname{Hom}(\rho,\mu\otimes\nu)$ and $\operatorname{Hom}(\rho,\nu\otimes\mu)$, as shown in Eq.(40). Eq.(40) enables us to “twist” an edge relative to another edge with which it shares a trivalent vertex and evaluate the plaquette operator.
The referee writes:
Our response:
We are very grateful that you have pointed out the confusion of how to choose $R$, which is indeed important and unclarfied in our paper. In fact, for any $R$-matrix which satisfied those conditions discussed in Section 3, the Fourier transform can be well defined, and hence we will get a series of Fourier-transformed models differed by the choices of the $R$-matrix. Nevertheless, all of these models are physically equivalent. We have added some discussion on the choices of the $R$-matrix in Section 3. In Section 5 we also mention that WW models with the same $F$-symbol and different $R$-matrices are also equivalent.
3D toric code model and 3D semion model are actually differed by $F$-symbols. The 3D toric code model is equivalent to the 3D GT model with input data $\mathbb Z_2$, while the 3D semion model is equivalent to the 3D twisted GT model with input data $\mathbb Z_2$ and the non-trivial 4-cocycle $\omega\in H^4(\mathbb Z_2,U(1))$. As the Fourier transforms of cocycles are really hard, in this paper we do not discuss twisted GT models or twisted boundary.
The referee writes:
Our response:
As explained earlier, given bulk input data $G$ and boundary input data $K$, we have a series boundary theory of the Fourier-transformed models differed by the choices of the $R$-matrix. Similarly, these boundaries are physically equivalent.
The referee writes:
Our response:
From the Fourier transform introduced in this paper, we can see that, in three-dimensional topological order, the charge condensation at the boundary is completely described by the boundary condition. This statement can also be understood through the layer construction, and thus we add a paragraph to discuss it. To avoid confusion, we have moved the discussion of layer construction to the end of the section and added an explanation.
The referee writes:
Our response:
We appreciate your reference to this article. We have therefore made changes to reflect the study of loop-like excitations in this article. Nevertheless, they only studied the braiding statistics of particle-like and loop-like excitations in 3D gauge theories with finite, Abelian gauge group, while generally the input data of the 3D GT model discussed in our paper could be non-Abelian. As far as I know, the classification of loop-like excitations in 3D gauge theories with finite non-Abelian gauge groups is very complicated (see Ref. 48-49 in the revised version). Describing these excitations is beyond the scope of this work. Moreover, loop-like excitations in the Fourier-transformed model, i.e., the WW model, are much less well understood. We believe that the Fourier transform constructed in this work will be helpful in studying loop-like excitations in the WW model.
The referee writes:
Our response:
Loop condensation on the boundary of the 3D GT model with a finite Abelian gauge group is obvious and has already been discussed in many publications. Nevertheless, study loop condensation on the boundary of the 3D GT model with a finite non-Abelian gauge group is extremely complicated. Therefore, these results will be reported in future work.
The referee writes:
Our response:
We appologize for the confusion here. In fact, we just want to explain that the Hilbert space $\tilde{\mathcal H}_0^\text{GT}$ can be identified with the Hilbert space of the WW model with no charge excitations, which is the Hilbert space where the plaquette operators of the WW model are actually defined. To make the discussion here more accurate, we have made corresponding changes in the article.
We are grateful for the opportunity to improve our manuscript and are willing to make additional changes if necessary. We hope that our responses and revisions will address the concerns raised.