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Fermionic defects of topological phases and logical gates
by Ryohei Kobayashi
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Submission summary
| Authors (as registered SciPost users): | Ryohei Kobayashi |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2211.12394v1 (pdf) |
| Date submitted: | 2023-01-10 08:05 |
| Submitted by: | Kobayashi, Ryohei |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We discuss the codimension-1 defects of (2+1)D bosonic topological phases, where the defects can support fermionic degrees of freedom. We refer to such defects as fermionic defects, and introduce a certain subclass of invertible fermionic defects called "gauged Gu-Wen SPT defects" that can shift self-statistics of anyons. We derive a canonical form of a general fermionic invertible defect, in terms of the fusion of a gauged Gu-Wen SPT defect and a bosonic invertible defect decoupled from fermions on the defect. We then derive the fusion rule of generic invertible fermionic defects. The gauged Gu-Wen SPT defects give rise to interesting logical gates of stabilizer codes in the presence of additional ancilla fermions. For example, we find a realization of the CZ logical gate on the (2+1)D $\mathbb{Z}_2$ toric code stacked with a (2+1)D ancilla trivial atomic insulator. We also investigate a gapped fermionic interface between (2+1)D bosonic topological phases realized on the boundary of the (3+1)D Walker-Wang model. In that case, the gapped interface can shift the chiral central charge of the (2+1)D phase. Among these fermionic interfaces, we study an interesting example where the (3+1)D phase has a spatial reflection symmetry, and the fermionic interface is supported on a reflection plane that interpolates a (2+1)D surface topological order and its orientation-reversal. We construct a (3+1)D exactly solvable Hamiltonian realizing this setup, and find that the model generates the $\mathbb{Z}_8$ classification of the (3+1)D invertible phase with spatial reflection symmetry and fermion parity on the reflection plane. We make contact with an effective field theory, known in literature as the exotic invertible phase with spacetime higher-group symmetry.
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2023-3-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2211.12394v1, delivered 2023-03-16, doi: 10.21468/SciPost.Report.6911
Strengths
This work solves an important open problem about generalizing invertible topological defects in (2+1)D to include fermionic degrees of freedom along the defect
The work demonstrates, via example, how these domain walls can be used to implement different logic gates on codes such as the toric code.
The discussion is also extended to topological orders on the boundary of a three dimensional invertible phase where a gapped interface between topological orders with different chiral central charge can be constructed.
Weaknesses
The discussion is restricted to invertible fermionic defects induced by condensing a boson along a line and does not say much about the case where an emergent fermion is condensed along a line instead.
Report
In this work the author generalizes the study of topological defects in quantum spin models by allowing the defects to include physical fermions. This leads to a number of interesting new phenomena including defects that can shift self statistics and logic gates induced by sweeping the domain wall across the system.
The paper is well written and clearly laid out with nice explicit examples. I recommend acceptance.
Requested changes
It seems invertibility for the defects produced by condensing a fermion along a line with and without physical fermions is different. In the former case I think the defect is invertible, in the latter case it is said in the paper that it is not. I guess this is due to differences in condensing the emergent fermion with or without physical fermions. Could the author clarify this?
Typos:
There are some small typos throughout the work, such as on page 2:
“Topologically ordered phases of matter is characterized” -> “Topologically ordered phases of matter are characterized”
“topological defect of the theory, and their universal properties are described by modular tensor category” -> “topological defects of the theory, and their universal properties are described by modular tensor categories”
Report #2 by Anonymous (Referee 2) on 2023-3-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2211.12394v1, delivered 2023-03-07, doi: 10.21468/SciPost.Report.6862
Report
The author discusses defects with fermionic degrees of freedom in both bosonic and fermionic bulk theories. These defects host fermions which only live on the defect and do not exist in the bulk. The paper discusses various properties of fermionic defects in details.
They find the fusion rule between such invertible defects, and construct logical gates by sweeping these defects across the space. The paper also construct interface defects between different theories.
The paper includes many new results. I recommend the draft for publication.
Report #1 by Anonymous (Referee 1) on 2023-3-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2211.12394v1, delivered 2023-03-06, doi: 10.21468/SciPost.Report.6847
Report
The manuscript discusses topological domain walls in bosonic topological order, where the defect can have fermions. The domain walls can modify the gravitational response such as statistics of excitations and chiral central charge. The authors discuss applications such as logical gates in toric code.
I recommend the manuscript for publication after the following comments/questions are addressed:
- The manuscript discusses defects that are not quite topological but depend on the spin structure. If Z2f is gauged on the defects, they should give completely topological defects, can the author comment on these defects?
-Although the bulk system is supposed to be bosonic with local fermion only on the defects, some discussions in the manuscript e.g. p14 involve local fermions also in the bulk. Is the conclusion the same without bulk local fermion?