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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_00062v2 (pdf) |
| Date submitted: | 2025-01-03 15:34 |
| 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
Author comments upon resubmission
List of changes
1. On page 3, the third paragraph of subsection 1.2, a sentence is added to explain the phrase ``UBFC $\mathrm{Rep}(G)$'', as in general literature the category $\mathrm{Rep}(G)$ is not required to be braided.
2.On page 6, the first paragraph of section 3 is completely rewritten in order to emphasize that the Fourier transform introduced in section 3 is determined by the data $\{C_{\mu\nu\rho},R^\rho_{\mu\nu}\}$.
3.On page 16, at the end of subsection 3.2, a discussion on the choices of the $R$-matrix is added.
4.On page 21, the fourth paragraph of subsection 4.2, we revised our discussion of the classification of loop-like excitations in 3D topological orders to include recent progress. References 47-49 are also added.
5.On page 21, subsection 4.2, we have moved the discussion of layer construction to the last paragraph and added an explanation.
6.On page 22, subsection 5.1 is rewritten 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. The confusing discussion about dyon is removed.
7. On page 23, below Eq.(59), a sentence is added to explain the necessity of the braiding structure in evaluating the Fourier-transformed plaquette operator.
8.On pages 24-25, the last paragraph of subsection 5.2, we add the conclusion that the three-dimensional GT model with input data $G$ is equivalent to the WW model with input data $\mathrm{Rep}(G)$, and different WW models, each with input data $\mathrm{Rep}(G)$ but equipped with distinct braiding structures, are also equivalent.
Current status:
Reports on this Submission
Strengths
The authors have made some improvements to clarity, and to the possible applications of this formalism.
Weaknesses
The paper is still extremely technical and of interest to a fairly narrow audience. The authors could do more (see requested changes below) to both situate this work in the general literature on lattice gauge theories, and to clarify the physics of the models that they are presenting.
Report
The authors have addressed some of my comments. In particular, the clarifications about the R matrices are helpful, and the introduction has been clarified.
However, I do not feel that the authors have addressed all of the recommended changes, and some things still require attention. Of particular importance are those comments aimed at increasing the general interest of the article.
Requested changes
— I feel strongly that the authors should address in the text the connection between what they are doing, and non-abelian lattice gauge theory written in the electric/ magnetic bases, as described in classic work by e.g. Kogut. I recommend adding a couple of sentences in the introduction about the relationship between the Fourier transform and this change of basis at the Hamiltonian level, and then adding a more detailed discussion in section 3. Situating this work in the context of the large historical literature on lattice gauge theories will make it accessible to a much wider audience.
— Something I didn’t notice last time: In the discussion of gapped boundaries circa Eq. 5, it is not clear that the gapped boundaries introduced are a complete set, or simply some of the gapped boundaries. The authors should clarify this point. Certainly in 2 dimensions there is also a choice of cocycle involved in picking a gapped boundary.
— The authors should give more detail in terms of what they mean when they say that they believe that the elementary excitations in 3DTO’s are not fully understood at the end of section 2. Please add some specific open questions here that are not resolved in the literature. There has certainly been recent progress on defects in 3DTO’s, but I would not normally call these excitations.
— I do not feel the authors have clarified Eq. 39. When comparing to the work of Walker and Wang, it is not clear to me which edges acquire R symbols. Is the statement that one picks a projection of the 3D lattice onto 2D, and that any edges that cross in this projection will pick up an R matrix when evaluating a particular state?
— A physical interpretation for the meaning of R must be given — I find this point very confusing in the present version! If all of the models are equivalent, is the choice of R essentially a gauge choice? If so, why is it not possible to simply choose R to be trivial? For abelian gauge theories, it would seem that I can always make this choice. Is there something different in the non-abelian case? This would potentially be a place where discussing examples would be helpful. The authors also are claiming, without proof, that these models are physically equivalent; they should at least indicate at the end of section 3.3 what the argument is (or where in the text it can be found).
— In discussing the boundary, it is natural to ask whether R has to be the same at the boundary as in the bulk, or whether there is some freedom. Can the authors clarify this?
— The authors should add to the text some discussion of the excitations in the Fourier transformed model. If providing a general construction of the membrane operators that create vortex loops, and the string operators creating charges, is beyond the scope of the present paper, at least a basic discussion including where the different kinds of excitations are found (plaquettes, vertices, tails) and which ones correspond to charges/ which to fluxes in the gauge theory should be added.
— The authors should clarify the role of the tails in the Hilbert space. In a Walker-Wang model (or generalized 3D Toric code), such tails are not needed to capture all of the excitations of the discrete gauge theories.
— My previous question about the fate of loops at the boundaries is not a request to investigate other boundary types. Rather, I simply think the authors should clarify, for the boundaries that are already described in their paper, what happens to various bulk excitations when they come to the boundary. I presume that loops can break open, but the end-points remain linearly confined, but this is not discussed at all in the text.
— The authors should add at least one example — perhaps a dihedral group. A concrete example would make it much easier for readers to understand how the actual data is obtained, and also what the potentially new boundary conditions are. The paper’s main application is supposed to be to construct new types of boundaries, so an example of concretely which boundaries it realizes would be extremely helpful.
Recommendation
Ask for major revision
Author: Siyuan Wang on 2025-03-18 [id 5297]
(in reply to Report 1 on 2025-02-09)We would like to express our sincere gratitude to the editor and the referee for their continued time and effort in reviewing our manuscript. Below are our responses to the referee's questions.
The referee writes:
Our response:
The required discussion is added. See the changing list for detail.
The referee writes:
Our response:
Generally, for a GT model with input data $G$, the gapped boundary is specified by a subgroup $K\subseteq G$ and a 3-cocycle $\alpha\in H^3[G, U(1)]$, which is mentioned in the introduction. Therefore, there do exists a choice of 3-cocycle involved in picking a gapped boundary. Nevertheless, in our paper we do not discuss any twist, and hence only some of the gapped boundaries, that is, those boundaries with trivial $\alpha$, are discussed in our paper, which has been clarified in the revised version.
The referee writes:
Our response:
For example, a general construction for the membrane operators generating loop excitations in 3DTO's is still unkown. Moreover, our understanding of nontrivial braidings of loops (three-loop braiding) in 3DTO's is limited to the abelian case (see for example, https://doi.org/10.1103/PhysRevB.99.235137). Nevertheless, in order to avoid dispute, we have already modified our discussion in the revised version.
The referee writes:
Our response:
Yes. During the Fourier transform, firstly one picks a projection of the 3D lattice onto 2D, and any two edges that cross in this projection will pick up an $R$ matrix when evaluating a particular state, i.e., evaluating the inner product Eq.(38), which has been clarified in the revised version.
The referee writes:
Our response:
Discussing the physical meaning of $R$ is beyond the scope of our work. We just find that in order to define the Fourier transform of the 3D GT model, one have to fix a set of $R$-symbols. For the abelian cases, we can always choose $R$-symbols to be trivial. Nevertheless, when the input data $G$ is non-abelian, then in most cases $R$-symbols cannot be chosen to be trivial. The simplest example is the dihedral group $G=D_3$. Appendix C is added to list the data for $G=D_3$.
We also state that models with different $R$-symbols are physically equivalent. Although it is hard to write down a rigrous proof, we have the argument as follows:
Starting with a GT model with input data $G$ (denoted by $\text{GT}_G$), given the UFC $\text{Rep}(G)$ equipped with a set of $R$-symbols, we can define a Fourier transform, which maps the Hamiltonian of the GT model to the Hamiltonian of a WW model term by term. This transformation (denoted by $\text{FT}_R$) is a unitary linear transformation that does not affect the spectrum of the Hamiltonian and the ground state degeneracy. Therefore, the resulting WW model (denoted by $\text{WW}_{\text{Rep}(G),R}$) is physically equivalent to the original GT model $\text{GT}_G$. Nevertheless, if we choose another set of $R$-symbols denoted by $R'$, we will get another WW model $\text{WW}_{\text{Rep}(G),R'}$ via the transformation $\text{FT}_{R'}$, which is also physically equivalent to the original GT model $\text{GT}_G$. Therefore, the two WW models $\text{WW}_{\text{Rep}(G),R}$ and $\text{WW}_{\text{Rep}(G),R'}$ are related through the transformation $\text{FT}_{R'}\circ\text{FT}_R^{-1}$, and hence must be physically equivalent. The argument above is added in the revised version.
The referee writes:
Our response:
By our construction, it is natural to choose $R$ to be the same at the boundary as in the bulk, because boundary edges may have crossing with bulk edges. Although there may be some special cases where one can choose a different set of $R$-symbols than in the bulk, these cases are beyond our discussion.
The referee writes:
Our response:
The required discussion for charges is added. See the changing list for detail. In Section 5.2, we have already discussed where flux excitations (i.e., the loop-like excitations) are found.
The referee writes:
Our response:
In fact, to capture all of the charge excitations in the WW model, those tails are needed, which is analogous to the tails in the extended LW model discussed in http://dx.doi.org/10.1103/PhysRevB.97.195154. This reference is added in the revised version.
The referee writes:
Our response:
Physically, we do agree that loops can break open with linearly confined end points at the boundary. Nevertheless, to study these string excitations with two end points attaching to the boundary concretely, we still need to construct the membrane operator which generates such excitations, which is beyond the scope of our work.
The referee writes:
Our response:
Two examples: abelian group $\mathbb Z_n$ and dihedral group $D_3$ are discussed in Appendix C.
We are grateful for the opportunity to improve our manuscript again and are willing to make additional changes if necessary. We hope that our responses and revisions will address the concerns raised.