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Realizing triality and $p$-ality by lattice twisted gauging in (1+1)d quantum spin systems
by Da-Chuan Lu, Zhengdi Sun, Yi-Zhuang You
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Submission summary
| Authors (as registered SciPost users): | Da-Chuan Lu |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2405.14939v2 (pdf) |
| Date submitted: | 2024-07-01 19:19 |
| Submitted by: | Lu, Da-Chuan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this paper, we study the twisted gauging on the (1+1)d lattice and construct various non-local mappings on the lattice operators. To be specific, we define the twisted Gauss law operator and implement the twisted gauging of the finite group on the lattice motivated by the orbifolding procedure in the conformal field theory, which involves the data of non-trivial element in the second cohomology group of the gauge group. We show the twisted gauging is equivalent to the two-step procedure of first applying the SPT entangler and then untwisted gauging. We use the twisted gauging to construct the triality (order 3) and $p$-ality (order $p$) mapping on the $\mathbb{Z}_p\times \mathbb{Z}_p$ symmetric Hamiltonians, where $p$ is a prime. Such novel non-local mappings generalize Kramers-Wannier duality and they preserve the locality of symmetric operators but map charged operators to non-local ones. We further construct quantum process to realize these non-local mappings and analyze the induced mappings on the phase diagrams. For theories that are invariant under these non-local mappings, they admit the corresponding non-invertible symmetries. The non-invertible symmetry will constrain the theory at the multicritical point between the gapped phases. We further give the condition when the non-invertible symmetry can have symmetric gapped phase with a unique ground state.
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- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
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Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 1) on 2024-8-28 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2405.14939v2, delivered 2024-08-28, doi: 10.21468/SciPost.Report.9667
Report
This paper investigates twisted gauging of 1+1d lattice models and offers several perspectives on this procedure from conformal field theory and quantum processing. These twisted gauging generate triality and $p$-ality between lattice models and maps between different gapped phases. For theories invariant under twisted gauging, there exist corresponding triality or $p$-ality non-invertible symmetries which sometimes can be used to exclude symmetric gapped phases with a unique ground state.
Overall, the paper is well-written and represents an advancement in the study of gauging, dualities and non-invertible symmetries on lattice. The referee recommends the publication of this paper in SciPost.
Minor questions:
1. In (1.1) and (7.1), did the authors assume that the group $G$ is abelian? If not, the dual symmetry would be a non-invertible $Rep(G)$ symmetry, and it is unclear to the referee how the bicharacter factors into the gauging process.
2. In (5.15), did the authors assume $N$ is prime? If not, the inverse of $a+b$ in $\mathbb{Z}_N$ might not exist, and twisted gauging could map an SPT to a partially-symmetry-broken phase.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report #2 by Anonymous (Referee 3) on 2024-8-12 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2405.14939v2, delivered 2024-08-12, doi: 10.21468/SciPost.Report.9579
Report
The authors studied twist gauging in spin models with $\mathbb Z_N\times \mathbb Z_N$ symmetry in $(1+1)d$. They provided multiple perspectives on the twist gauging. In the lattice perspective, they define the twisted Gauss law operator to perform the twist gauging. In the quantum process perspective, they use the Pauli polynomial method and the Kraus operator for the twist gauging transformation. Through an explicit construction, they showed that twist gauging in spin models with $\mathbb Z_N\times \mathbb Z_N$ symmetry in $(1+1)d$ is equivalent to first applying the SPT entangler and then do the untwisted gauging.
With the twist gauging operation, the authors constructed the triality operators for any $N$ and the $p$-ality operators for prime number $p$. They studied the mappings of gapped phases under the triality and the $p$-ality transformation.
The triality and the $p$-ality transformation are group theoretical noninvertible. The authors studied the corresponding fusion category structures and non-invertible symmetry.
The authors used the technique from quantum process to study the twist gauging on lattice. They also extended the study of non-local mapping on lattice by introducing the triality and $p$-ality transformations. This work meets the standard of Scipost and I recommend to publish this paper.
Requested changes
1. Here are some typos:
(1) Page 4, on the RHS of the equation for $p$-ality, it should be $\mathbb Z_p\times \mathbb Z_p$.
(2) In the first equation of Eq (6.15) (and below Eq(6.16)), $G$ in the subscript should be $\mathbb Z_N\times\mathbb Z_N$.
(3) In Eq(7.9) and (7.10), is $\mathsf{KW}^{e,o}$ equal to $\mathsf{KW}^e\mathsf{KW}^o$? In the early text (below Eq(4.28)), $\mathsf{KW}$ has been used for the product (untwisted gauging).
I also have some questions. It will be helpful if the author can address them.
2. In section 4.1, the author claimed that the untwisted gauging, the twisted gauging and $\mathsf{KW}^e$ generates the $(S_3\times S_3)\rtimes \mathbb Z_2$. Can the author write down the group structure in terms of the generators explicitly? What is the generalization of this action for the case of $\mathbb Z_N \times \mathbb Z_N$?
3. In page 33, the non-invertible triality operator $\mathcal{Q}$ appears without definition. Is it equal to $TC^e\times K_{TG^1}$, where $K_{TG^1}$ is defined in Eq(5.14)? After defining $\mathcal{Q}$, can the authors show the derivation of the fusion rule, for example the triple product of $\mathcal{Q}$, explicitly?
3. In page 33, the author said "the $\mathsf{KW}^e\mathsf{KW}^o$ effectively adds 1 lattice site which correspond to quantum dimension $N$". The quantum dimension of $\mathsf{KW}^e\mathsf{KW}^o$ should be $N^2$ according to the fusion rule
\[
(\mathsf{KW}^e\mathsf{KW}^o)(\mathsf{KW}^e\mathsf{KW}^o)=(\sum_{k=1}^{N}U_e^k)(\sum_{k=1}^{N}U_o^k)
\]
where $U_{e/o}$ is the $\mathbb Z_N$ operator on the even/odd sites. Why it contributes to quantum dimension $N$?
4. The authors showed that the triality operator can be constructed from twist gauging with diagonal bicharacter Eq(7.7). According to Eq(7.10), the triality operator can also be constructed from twist gauging with off-diagonal bicharacter Eq(7.8). I am just curious. Does it means that starting from different bicharacters, we will have the same triality (and $p$-ality) operator?
Recommendation
Publish (meets expectations and criteria for this Journal)
Author: Da-Chuan Lu on 2024-10-02 [id 4819]
(in reply to Report 2 on 2024-08-12)
We thank the referee for carefully reading our manuscript and expressing a positive evaluation of its contribution to the field. We appreciate that the referee generally agrees that our work deserves publication in SciPost.
-
We thank the referee for pointing out the typos, we have changed them accordingly. For the last one, we change to use $KW^{eo}$ to denote the untwisted gauging of both even and odd symmetries.
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We thank the referee for raising the question. We updated the manuscript to clarify the point. In particular, the twisted gauging TG corresponds to the order 3 element in the $S_3$, and $KW^{eo}$ corresponds to the order 2 element in the $S_3$. $KW^e$ corresponds to the order 2 element that exchanges the first and second $S_3$. However, we should mention that all the elements are not on equal footing. The SPT entangler and $KW^{eo}$ are both order 2 elements but they are implemented by quantum circuits with different depths. More severe, they have different quantum dimensions. For the case of $Z_N\times Z_N$, the actions form a group corresponds to the anyon permutation symmetry of $Z_N\times Z_N$ toric code, which is the split orthogonal group ${A\in GL(4,\mathbb{Z}_N)| A\begin{pmatrix} 0 & I \ I & 0\end{pmatrix}A^T=\begin{pmatrix} 0 & I \ I & 0\end{pmatrix}}$.
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We thank the referee for pointing out the confusing point. We updated the manuscript accordingly. The operator $Q$ acting on the low energy continuum theory corresponds to the $Tri =T C^e TG^1$ acting on the lattice. Based on the field theory calculation, the $Tri$ is an order 3 action and maps the partition function to the symmetric sector, therefore, the corresponding fusion of $Q^3$ gives the projection to the symmetric state. It is also interesting to explicitly check the fusion by simplifying the sequential quantum circuits which act on the lattice.
3'. As the referee mentioned, $KW^{eo}\times KW^{eo}=(\sum_k U_k^e)(\sum_k U_k^o)$, the left hand side $dim(KW^{eo})^2$ and the right hand side is $N\times N$, so $dim(KW^{eo})=N$.
For 4. Sorry for causing the confusion. The triality $Tri=T C^e TG^1$ involves translation by 1 site, therefore it always corresponds to twisted gauging with off-diagonal bicharacter. The diagonal one is not order 3 since $(TG^1)^3 :Z[A_1,A_2] \rightarrow Z[A_2,-A_1]$, which is not an identity action.
Report #1 by Anonymous (Referee 2) on 2024-8-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2405.14939v2, delivered 2024-08-05, doi: 10.21468/SciPost.Report.9539
Report
This paper constructs triality and $p$-ality maps on the lattice using lattice twisted gauging, which involves introducing link variables (gauge fields) with minimal coupling, and imposing a twisted version of Gauss law. The twisted Gauss law operator defined in this paper neatly packages the information of the discrete torsion. The authors analyse the action of the triality and $p$-ality maps on various gapped phases with $\mathbb Z_p \times \mathbb Z_p$ symmetry. They also construct Kraus operators that implement the above maps. Finally, they provide the conditions under which a trivial gapped phase has a non-invertible triality or $p$-ality symmetry.
The paper definitely meets the criteria for acceptance in SciPost Physics. The presentation is clear and the content is well organized. I did find some minor typos listed below. Otherwise, I am happy to recommend its publication.
Requested changes
1- The last mapping in (3.13) should be corrected.
2- Below (5.3), the Hamiltonians $H_\text{SYM}$ and $H_\text{SSB}$ should have $+h.c.$.
3- What is the abelian group $A$ above (7.13)?
4- What is $p$ below (7.16)?
5- Perhaps the authors could include the fusion of $\mathcal Q$ and $\overline{\mathcal Q}$ in (6.15-6.16) and (7.13-7.14)? It can be inferred from the given fusion rules but I think it is more fundamental than the triple fusion of $\mathcal Q$'s.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Da-Chuan Lu on 2024-10-02 [id 4817]
(in reply to Report 1 on 2024-08-05)
Reply
We thank the referee for carefully reading our manuscript and expressing a positive evaluation of its contribution to the field. We appreciate that the referee generally agrees that our work deserves publication in SciPost.
- We thank the referee for pointing this and we modified the manuscript for a better explanation. The circuit is used to generate the minimal coupling between the matter field and the gauge field. If the circuit meets a matter field $Z_i$ then it generates a string of gauge field to infinity $Z_i \rightarrow Z_i Z^t_{i+1/2}Z^t_{i+3/2}\cdots$. When the gauge string meets another matter field, then the mapping does nothing $\cdots Z^t_{j-1/2} Z_j \rightarrow \cdots Z^t_{j-1/2} Z_j$. Therefore, the circuit will map, for example, $Z_i Z_j$ to $Z_i Z^t_{i+1/2}\cdots Z^t_{j-1/2}Z_j$ which coincides with the ordinary minimal coupling procedure. $Z^t$ is \tilde{Z}, there is technical problem on the webpage.
2, 3 and 4. We thank the referee for pointing out the typo. We have changed accordingly. $A$ is referring to $\mathbb{Z}_N\times \mathbb{Z}_N$. And the automorphism group of $\mathbb{Z}_N\times \mathbb{Z}_N$ should be $GL(2,\mathbb{Z}_N)$ when N is a prime number.
For 5. We thank the referee for the suggestion, and we have updated the manuscript accordingly.
Author: Da-Chuan Lu on 2024-10-02 [id 4818]
(in reply to Report 3 on 2024-08-28)We thank the referee for carefully reading our manuscript and expressing a positive evaluation of its contribution to the field. We appreciate that the referee generally agrees that our work deserves publication on in SciPost.
We made corresponding clarifications in the manuscript to address the questions raised by the referee. In particular, 1. We assume $G$ is abelian. Indeed, once gauging the non-abelian G symmetry, the dual symmetry becomes non-invertible Rep(G) symmetry. It is possible to write down the partition function with the insertion of Wilson lines in a regular representation of G as an analogous formula. But the more accurate treatment is using the torus partition function with the insertion of the topological defect lines. 2. Yes, we assume N is prime, sorry for causing the confusion. We added the clarification in the updated manuscript.