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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
Submission summary
| Authors (as registered SciPost users): | Da-Chuan Lu |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202410_00004v1 (pdf) |
| Date accepted: | 2024-10-21 |
| Date submitted: | 2024-10-02 07:24 |
| Submitted by: | Lu, Da-Chuan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| 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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Author comments upon resubmission
We answered the referees' questions and updated our manuscript accordingly. We hope the modified version of the manuscript is satisfactory.
List of changes
We addressed all the referees' questions and fixed several typos in our updated manuscript. In particular,
1. We add footnote 1 on page 2 to discuss the case when gauging the non-abelian symmetry.
2. We add a discussion of $(S_3\times S_3)\rtimes Z_2$ with explicit correspondence between group elements and the mappings on the lattice Hamiltonian.
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Reports on this Submission
Report
The revised manuscript addresses most of the concerns raised by the referees. I am still confused by something about the maps in (3.13) and the related circuit in (3.15).
First, the ordering of operators is not consistent with the flow of time in the circuits. For example, around (3.12), it is mentioned that $U_\mathrm{initial}$ is applied first, which means time flows from bottom to top in (3.15). However, the authors write the associated operator as $U_\mathrm{initial} U_\mathrm{gauge}$, which means $U_\mathrm{gauge}$ is applied first. This inconsistency exists in other equations too and the authors should rectify this.
More importantly, the right-most map in (3.13) does not make sense to me. The action of the circuit $U_\mathrm{gauge}$ on the matter field $Z_i$ is given by the third map. This map already implies the action of the circuit on $Z_i Z_j$ and it gives the expected minimal coupling answer. But, in their response, the authors say that the last two maps together imply this action. Can they please explain why they need the fourth map for this?
It seems like the action of $U_\mathrm{gauge}$ on $\tilde Z_{i+\frac12}$ is given by $\tilde Z_{i+\frac12} \rightarrow \tilde Z_{i+\frac12} \tilde Z_{i+\frac32} \cdots$. What is the interpretation of this in terms of gauging?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Da-Chuan Lu on 2024-10-04 [id 4831]
(in reply to Report 1 on 2024-10-02)We thank the referee for clarifying the questions. We agree that the fourth map is unnecessary and have updated our manuscript accordingly. For the sequence of the action of U, we read it from left to right, since we previously defined the transformation of operators as $O\rightarrow O'= U^\dagger O U$.
For the action of $U_{gauge}$ on $\tilde{Z}_{i+1/2}$, $\tilde{Z}_{i+1/2}$ cannot appear alone, it must attached to a $Z_i$ operator. Because in the enlarged Hilbert space, we only introduce $\tilde{X}_{i+1/2}=1,\forall i$ but no $\tilde{Z}_{i+1/2}$. When there is a $Z_i$ operator, the gauge string will continue to infinity. As discussed around (3.26), the additional site 0 with operator $Z_0$ that labels the boundary condition of the Ising chain, under the KW transformation, $Z_0$ becomes the product of $X$ operator which corresponds to the symmetry operator in the dual model (The product of $X$ is the product of $\tilde{Z}$ after some basis transformation that is used to match with the original model).
Attachment:
triality_defect_lattice_10-4.pdf