SciPost Submission Page
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 | |
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| Preprint Link: | scipost_202501_00044v1 (pdf) |
| Date accepted: | 2025-02-25 |
| Date submitted: | 2025-01-23 07:37 |
| Submitted by: | Zhao, Jingyu |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| 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.
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Author comments upon resubmission
We thank the referee for the report.
Report 2
>> 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.
Yes, our results apply generally to non-commuting (gapped) Hamiltonians as well. The low entanglement excitations exist as the gapped ground state of a "defect Hamiltonian" which is everywhere the same as the original Hamiltonian except at the location of the excitation. Because the defect Hamiltonian is gapped, it is possible to quasi-adiabatically continue it from the fixed point model to generically non-commuting models.
We have added a comment at the end of the introduction to highlight this point.
>> "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?
When an LEE is obtained by condensing, for example, the e boson, then condensing the LEE does correspond to condensing the e boson everywhere. This is shown in a mathematically more rigorous way in https://arxiv.org/pdf/2403.07813. At the same time, not all LEEs come from condensation. For example, the flux loop in 3+1D gauge theories are not condensation defects. It is possible to condense such flux loops in some models and their condensation does not correspond to particle condensation. On the other hand, not all LEEs can condense. The rs and sr LEEs, which involve different anyon condensation on different sides, cannot be consistently condensed, as pointed out by the referee.
>> Does Fig 12 have an error? Should it be up/down on the left and down/up arrows on the right?
Yes, thanks for pointing that out. We have corrected this error in the figure.
>> Above eq 15, do you mean Fig. 12 rather than Fig 2?
Yes, thanks for catching the mistake, we have corrected it.
>> "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?
This is a great question. We do not know the full answer to this question. Our result is only based on the models we studied. We tried to discuss this point to the extent we can in section 3.2. We hope a more complete answer may emerge as we get a better understanding of the higher category structure of LEEs. We added a comment in the conclusion that this point was discussed in section 3.2
List of changes
1. We have added a comment at the end of the introduction to highlight the generalizability of our work.
2. Errors in Fig. 12 and above Eq 15 are corrected.
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(status: Editorial decision fixed and (if required) accepted by authors)