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Integrable fishnet circuits and Brownian solitons
by Žiga Krajnik, Enej Ilievski, Tomaž Prosen, Benjamin J. A. Héry, Vincent Pasquier
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Žiga Krajnik |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2411.08030v2 (pdf) |
| Date submitted: | March 12, 2025, 9:27 p.m. |
| Submitted by: | Žiga Krajnik |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| 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
Recommendation
Ask for minor revision
Report #2 by Yuan Miao (Referee 1) on 2025-4-12 (Invited Report)
- Cite as: Yuan Miao, Report on arXiv:2411.08030v2, delivered 2025-04-12, doi: 10.21468/SciPost.Report.11001
Strengths
Weaknesses
Report
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 $\lambda \to \infty$, thus we need to take the $\tau \to \infty$ limit to obtain the continuous-time limit. In many other instances, the local charges are obtained by expansion at $\lambda \to 0$, is it the case then we should take the analogous $\tau \to 0$ 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%)
We again thank the referee for their positive evaluation of our work. To avoid possible misunderstanding we have added an explanatory comment regarding our use of the name fishnet diagrams in the introduction.
Regarding the two questions:
- We believe this to be the case. Provided the local conserved quantities are obtained from the series expansion of the transfer matrix in powers of the spectral parameter, the $\tau \to 0$ limit of the discrete-time brickwork/fishnet dynamics will converge to the continuous-time Hamiltonian dynamics generated by the first (lowest order) conserved quantity in the expansion. An interesting question is how to recover the continuous-time dynamics generated by higher conserved quantities.
- So far as we are aware, taking the trace in the auxiliary quantum space does not yield additional consistency conditions. On the other hand we presently also do not see how the fixed-point condition arises in the semi-classical limit of the quantum construction. Adding a twist to the monodromy matrix, which can also be introduced in the classical construction, does not appear to be related to the fixed-point condition.
Report #1 by Anonymous (Referee 3) on 2025-3-18 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2411.08030v2, delivered 2025-03-17, doi: 10.21468/SciPost.Report.10847
Report
I am still however puzzled about the averaged propagator - for example, given by Eq. (5.5). I understand that the map $\Phi_\tau$ 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 $\alpha \Phi_\tau$ mean that we evolve spins for the time $\tau$ and the obtained result rescale? at each lattice site? What happens with the condition $S_x^2+S_y^2+S_z^2=1$ in this case?
What does it mean to have a map $\Phi_{\tau_1} + \Phi_{\tau_2}$? Do we evolve with time $\tau_1$ and $\tau_2$ 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
We thank the referee for their positive reevaluation of our work.
To avoid confusion we now give the stochastic dynamics in Eq. (5.2) and subsequently more explicitly by specifying the averaging of an observable. In practice, the average is obtained by sampling sequences of time steps $\boldsymbol{\tau}$ from $\Omega$, evolving a fixed initial configuration of degrees of freedom with the resulting propagators $\Phi_{\boldsymbol{\tau}}$ and averaging over the results. To avoid confusion we now also refer to this operation as timestep- instead of ensemble-averaging. Timestep averaging should not be confused with averaging over initial conditions (see Eq. (3.12)), but is compatible with it due to linearity of averaging.
Author: Žiga Krajnik on 2025-05-07 [id 5459]
(in reply to Report 3 by Vincent Caudrelier on 2025-04-23)We again thank the referee for their positive evaluations of the manuscript. The entwining property of the Yang-Baxter equation Eq. (2.28) is now emphasized after Eq. (2.31) to avoid confusion.