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Fusion of Low-Entanglement Excitations in 2D Toric Code
by Jing-Yu Zhao, Xie Chen
Submission summary
| Authors (as registered SciPost users): | Xie Chen · Jingyu Zhao |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2409.07544v1 (pdf) |
| Date submitted: | 2024-10-02 05:33 |
| Submitted by: | Zhao, Jingyu |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
On top of a $D$-dimensional gapped bulk state, Low Entanglement Excitations (LEE) on $d$($<D$)-dimensional sub-manifolds can have extensive energy but preserves the entanglement area law of the ground state. Due to their multi-dimensional nature, the LEEs embody a higher-category structure in quantum systems. They are the ground state of a modified Hamiltonian and hence capture the notions of `defects' of generalized symmetries. In previous works, we studied the low-entanglement excitations in a trivial phase as well as those in invertible phases. We find that LEEs in these phases have the same structure as lower-dimensional gapped phases and their defects within. In this paper, we study the LEEs inside non-invertible topological phases. We focus on the simple example of $\mathbb{Z}_2$ toric code and discuss how the fusion result of 1d LEEs with 0d morphisms can depend on both the choice of fusion circuit and the ordering of the fused defects.
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
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Reports on this Submission
Strengths
-clear presentation
-detailed, explicit analysis
Report
This paper provides a systematic analysis of low entanglement excitations in topological order, in particular in 2+1d and 3+1d Z2 toric code. The work gives a clear framework for understanding such objects in lattice models.
1) As mentioned at the very beginning of the paper, excitations are usually used to refer to higher energy eigenstates of the same Hamiltonian. In fixed point models, these excitations are well-defined, but usually in a lattice model we only specify the ground state and the higher energy states can be very messy. Are your results only for fixed point models, or do you expect them to generalize to other gapped (non-commuting) Hamiltonians? Quasiparticles (O(1) energy) can be quasiadiabatically continued, but I'm not sure about these excitations with extensive energy.
2) "Given their low entanglement, the LEEs are potentially ‘condensable’ such that their condensation can still have low entanglement and potentially realize a different phase" If we view LEE as obtained by condensing anyons along a submanifold, is this different from condensing everywhere rather than along the submanifold? Although this story doesn't seem to work if you have different condensations from different sides i.e. rs, sr LEEs. Can you explain this point in more detail?
3) Does Fig 12 have an error? Should it be up/down on the left and down/up arrows on the right?
4) Above eq 15, do you mean Fig. 12 rather than Fig 2?
5) "First, since the 1d circuit used to fuse 1d LEEs without morphisms are not unique, when we use different circuits to fuse 1d LEEs with nontrivial morphisms we can get different results." Is there a canonical choice for the circuit? Can you comment on a general statement of what "different results" means? i.e. given two LEEs, is there an algebraic result about the distinct possible fusion rules?
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Please respond to the comments/questions above and revise the manuscript as you feel appropriate.
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Strengths
Uncovers an interesting fusion rule of 1D Low-Entanglement Excitations and 0D morphisms in 2D and 3D toric code
Weaknesses
None
Report
The paper study the 0d and 1d Low-Entanglement Excitations in 2D and 3D toric codes. Using the language of quantum circuit, the authors discuss the fusion rule of low-entanglement excitations. In paticular, the non-commutative fusion rule in the 1d LEEs with nontrivial morphisms is very suprising and interesting. This implies a very detailed braided-fusion 2-category structure of Low-Entanglement Excitations in non-invertible topological phases.
The paper is clearly written, and contains new interesting results on algebra sturcture of 0d and 1d Low-Entanglement Excitations in 2D and 3D. Their fusion can be explicitly shown by constructing 0d unitary transformations or 1d circuits. It provides a insight into the algebra and entanglement nature of excitated states. Therefore, the referee recommends that the paper is published in SciPost.
Requested changes
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Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)