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Topological Defects in Floquet Circuits
by Mao Tian Tan, Yifan Wang, Aditi Mitra
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Submission summary
| Authors (as registered SciPost users): | Aditi Mitra |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2206.06272v1 (pdf) |
| Date submitted: | 2022-06-14 17:12 |
| Submitted by: | Mitra, Aditi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We introduce a Floquet circuit describing the driven Ising chain with topological defects. The corresponding gates include a defect that flips spins as well as the duality defect that explicitly implements the Kramers-Wannier duality transformation. The Floquet unitary evolution operator commutes with such defects, but the duality defect is not unitary, as it projects out half the states. We give two applications of these defects. One is to analyze the return amplitudes in the presence of "space-like" defects stretching around the system. We verify explicitly that the return amplitudes are in agreement with the fusion rules of the defects. The second application is to study unitary evolution in the presence of "time-like" defects that implement anti-periodic and duality-twisted boundary conditions. We show that a single unpaired localized Majorana zero mode appears in the latter case. We explicitly construct this operator, which acts as a symmetry of this Floquet circuit. We also present analytic expressions for the entanglement entropy after a single time step for a system of a few sites, for all of the above defect configurations.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-8-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2206.06272v1, delivered 2023-08-07, doi: 10.21468/SciPost.Report.7617
Strengths
1. The writing is clear.
2. The results presented are original and innovative.
3. The study of topological defects in Floquet circuits presented in this work could offer essential information for the understanding and classification of Floquet symmetry-protected topological phases.
Weaknesses
Sec. V reads a bit too technical.
Report
In this work, the authors investigated Floquet circuits with topological defects, which obey non-trivial fusion rules and can be deformed in space and time without changing the physics. Circuits with topological defects for the Floquet Ising model are constructed under different boundary conditions. Specially, the duality-twisted boundary condition allows the system to a host an isolated Majorana zero mode, which was found to manifest in the auto-correlation function.
Overall, this manuscript is clearly written and the results presented are innovative. This detailed study of topological defects in Floquet circuits could offer indispensable information for the understanding and classification of Floquet symmetry-protected topological phases. I think this manuscript could be accepted by Scipost Physics for publication after the following questions are addressed.
Requested changes
1. A section or subsection could be added to discuss how to realize the proposed Floquet circuits in experiments and how to detect the related topological properties. Specially, are there any suggestions for the realization of the duality-twisted boundary conditions and the isolated Majorana zero modes in any real experimental setups?
2. One unique feature of the Floquet Ising-Majorana chain is to hold Majorana edge modes with the quasienergy pi. Is there any possibility to find a single unpaired localized Majorana pi mode in the Floquet circuits considered by the authors? Under which conditions could we obtain Floquet pi modes in Floquet circuits with topological defects? The authors are suggested to comment on these issues.
Report #1 by Anonymous (Referee 3) on 2023-8-4 (Contributed Report)
- Cite as: Anonymous, Report on arXiv:2206.06272v1, delivered 2023-08-04, doi: 10.21468/SciPost.Report.7612
Strengths
1. Introducing novel quantum field theory notions into non-equilibrium systems.
2. Paper is well written. Through explicit construction, the idea of categorical symmetry is introduced in a pedagogical way.
Weaknesses
The motivation and physical implementation needs to be further clarified.
Report
In this paper, the authors introduced novel quantum field theory notions like categorical symmetry and duality defect to the design of Floquet circuits. In previous studies of Floquet systems, circuits based on the transverse field Ising model played a major role in the discovery of for example Floquet time crystal. In the equilibrium transverse field Ising model, it is well known that the model is self-dual and the duality transformation has been studied recently as an example of categorical symmetry in the system. By introducing such symmetry and related defect into the design of the Floquet circuit, the authors were able to observe the same `topological' nature of the symmetry as in equilibrium systems, reproduce the same categorical fusion structure, and introduce Majorana zero modes into Floquet systems. There are a few questions that I hope the authors can address.
1. The duality transformation is not unitary. Maybe I missed it, but it is not entirely clear how to implement the transformation or to insert the twisted boundary in the Floquet circuit. Is measurement a necessary part of the protocol? Does one need to post-select based on the measurement result?
2. While introducing topological defects into Floquet systems induces interesting features, most of the added features seem to reproduce what one expects in the equilibrium version of the system. Is this the general expectation? Could topological defect in Floquet system have new properties than their equilibrium counterpart?
3. It is not entirely clear whether the features describes in this paper are stable against errors in the implementation of the circuit.
If these questions are adequately addressed, I think this paper is worth publishing in SciPost Physics as it connects two very different fields and introduces some new ideas.