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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 | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2411.08030v2 (pdf) |
| Date submitted: | 2025-03-12 21:27 |
| Submitted by: | Krajnik, Žiga |
| 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
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 $\Phi_\tau$ 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