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Integrable fishnet circuits and Brownian solitons
by Žiga Krajnik, Enej Ilievski, Tomaž Prosen, Benjamin J. A. Héry, Vincent Pasquier
Submission summary
Authors (as registered SciPost users): | Žiga Krajnik |
Submission information | |
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Preprint Link: | https://arxiv.org/abs/2411.08030v2 (pdf) |
Date submitted: | March 12, 2025, 9:27 p.m. |
Submitted by: | Krajnik, Žiga |
Submitted to: | SciPost Physics |
Ontological classification | |
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Academic field: | Physics |
Specialties: |
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Approaches: | Theoretical, Computational |
Abstract
We introduce classical many-body dynamics on a one-dimensional lattice comprising local two-body maps arranged on discrete space-time mesh that serve as discretizations of Hamiltonian dynamics with arbitrarily time-varying coupling constants. Time evolution is generated by passing an auxiliary degree of freedom along the lattice, resulting in a `fishnet' circuit structure. We construct integrable circuits consisting of Yang-Baxter maps and demonstrate their general properties, using the Toda and anisotropic Landau-Lifschitz models as examples. Upon stochastically rescaling time, the dynamics is dominated by fluctuations and we observe solitons undergoing Brownian motion.
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
Author comments upon resubmission
List of changes
- Addressed points raised by the referees as specified in the responses to their comments.
- Added references.
- Minor corrections and notational clarifications.
- Fixed grammatical errors.
Current status:
Reports on this Submission
Report #3 by Vincent Caudrelier (Referee 2) on 2025-4-23 (Invited Report)
Report
The authors have very thoroughly analysed my comments, made appropriate changes to the manuscript and provided answers that did clarify the points I raised. Thank you for this. There is only one more minor change that I would recommend to avoid some confusion building up from reading this work. Strictly speaking, (2.28) is not the set-theoretic Yang-Baxter equation (which would involve the same map and not two types of maps) but an example of entwining Yang-Baxter equation, as I noted in my previous comments. Other than that, I am happy to recommend publication.
Recommendation
Ask for minor revision
Report #2 by Yuan Miao (Referee 1) on 2025-4-12 (Invited Report)
Strengths
See previous report.
Weaknesses
See previous report.
Report
I have checked the authors' replies to my previous report, which answer most of my (and other referees') comments adequately.
I think that the authors have improved the draft according to the reports and can be published as it is after a small improvement.
A comment on authors' reply of my first comment:
The authors stated that the nomenclature "fishnet circuit" is not related to the "fishnet diagram" of Zamolodchikov et al, therefore it is justified not to mention those literatures on "fishnet diagram" in QFT.
I fully agree with the authors' comment that the two things are not related. However, that is precisely my point that the authors should make a remark in the draft that those two things are not related.
But if the authors insist that they do not want to mention the papers, I would not suggest the change further. At the moment, the authors cited the papers I mentioned at the beginning of the introduction part without any explanation. Either they should follow my suggestion or delete the citation.
Two questions following authors' replies:
First, in the 13th comment, the authors mentioned that the local charges of the Toda lattice are obtained by expansion at λ→∞, thus we need to take the τ→∞ limit to obtain the continuous-time limit. In many other instances, the local charges are obtained by expansion at λ→0, is it the case then we should take the analogous τ→0 to get the continuous-time limit?
Second, about the authors' 15th comment, does it mean that the fixed-point condition give a similar analogue of the consistency condition when tracing out the transfer matrix in the quantum case? In the quantum case, one can always add a twist to the transfer matrix without destroying the integrability. Would that be related to the fixed points in the classical case?
Requested changes
See above.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report
I would like to thank the authors for clarifying most of the issues.
I am still however puzzled about the averaged propagator - for example, given by Eq. (5.5). I understand that the map Φτ in Eq. (3.5) is indeed, integrable. However, as I have written in my previous comment, I don't understand why the linear combination of these maps with different times is integrable. In fact, what does it even mean to have a linear combination of non-linear maps?
For example, does the map αΦτ mean that we evolve spins for the time τ and the obtained result rescale? at each lattice site? What happens with the condition S2x+S2y+S2z=1 in this case?
What does it mean to have a map Φτ1+Φτ2? Do we evolve with time τ1 and τ2 and the obtained results just add?
Requested changes
Please clarify the issue in the report. Namely: define the averaged propagator and argue why it is integrable.
Recommendation
Ask for minor revision