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Holographic tensor network for double-scaled SYK
by Kazumi Okuyama
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Kazumi Okuyama |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2503.23003v2 (pdf) |
| Date submitted: | June 15, 2025, 11:11 p.m. |
| Submitted by: | Kazumi Okuyama |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We construct a holographic tensor network for the double-scaled SYK model (DSSYK). The moment of the transfer matrix of DSSYK can be mapped to the matrix product state (MPS) of a spin chain. By adding the height direction as a holographic direction, we recast the MPS for DSSYK into the holographic tensor network whose building block is a 4-index tensor with the bond dimension three.
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
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2025-7-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.23003v2, delivered 2025-07-30, doi: 10.21468/SciPost.Report.11672
Strengths
- Clear presentation
- Intruiging explicit connection between DSSYK and a tensor network
Weaknesses
- No holographic interpretation of tensor network is explained
- Proofs of key equations are not presented
Report
The explicit realization of matrix elements of the transfer matrix and other operators in DSSYK as a tensor network is very interesting. It opens up the potential to make some of the famous statements about the direct gravitational interpretation of tensor networks (in the context of AdS/CFT) precise (or to explain in which sense they are not precise, we will see). The presentation of the paper is very nice and clear.
I have a few minor remarks 1. Equation (4.11) is one of the main equations, probably the most important one. It's validity is not shown via an explicit calculation. I am not doubting whether the equation is correct, but it may be useful to present a few more details, thus saving the reader some work. The paper is short and pedagogical, so I do not think that a few extra lines would disrupt the work as a whole too much, personally. 2. There is not much "AdS" in this work. It would have been nice to have attempts at relating the tensor network directly gravitational descriptions of DSSYK. Can we literally view the tensor network as building up bulk spacetime? Is the number of tensors as function of h some indication of the size of the bulk spacetime as we go inwards?
These are but very minor remarks, barely complains. I recommend publication.
Requested changes
None.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
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Eq.(4.11) is easily understood from the examples, such as eq.(4.4). The green arrows only touch the bottom edge of the rectangle and hence the bottom indices correspond to spins, while the other edges of the rectangle are not crossed by the green arrows and thus assigned the index 0.
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AdS interpretation of this tensor network is not clear at the moment. It is a future problem.
Author: Kazumi Okuyama on 2025-08-20 [id 5742]
(in reply to Report 2 on 2025-08-14)Thank you for carefully reading the manuscript and making valuable comments. Following the suggestion in the referee report, I have made the following changes:
I added eq.(2) for the variance of the random coupling. But we set \mathcal{J}=1 for simplicity.
Z_k is not determined by <P|P>=1, but by the condition eq.(23) coming from the probabilistic interpretation of the Markov process.
I added more explanations on the q-dependence around eq.(44). I stressed that when q is non-zero, the non-crossing pairing is not equal to the chord diagram in general.
I added discussions on the multi-particle chord Hilbert space and the bulk dual of DSSYK at the end of section 5. I cited the references listed in the referee report.
I hope the above changes answer the request in the referee report.