Fate of diffusion under integrability breaking of classical integrable magnets

arxiv: 2511.19110 · v2 · submitted 2025-11-24 · ❄️ cond-mat.stat-mech

Fate of diffusion under integrability breaking of classical integrable magnets

Jiaozi Wang , Sourav Nandy , Markus Kraft , Toma\v{z} Prosen , Robin Steinigeweg This is my paper

Pith reviewed 2026-05-17 05:07 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords spin diffusionintegrability breakingclassical magnetsLandau-Lifshitz modelmagnetization cumulantsnon-Gaussian statisticsthermodynamic limitdiffusive transport

0 comments p. Extension

The pith

Integrability breaking in classical magnets causes sharp changes in spin diffusion constant and magnetization statistics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies infinite temperature spin diffusion in a classical integrable space-time discrete version of the anisotropic Landau-Lifshitz magnet in the easy-axis regime, after adding integrability-breaking perturbations. Large-scale numerical simulations show a sharp change in the spin diffusion constant as a function of perturbation strength when the system size goes to the thermodynamic limit. The same simulations reveal a crossover from non-Gaussian to Gaussian statistics of magnetization transfer, visible in the higher-order cumulants. These two observations together indicate that integrable systems support a non-trivial diffusion mechanism that is disrupted once integrability is broken. A reader would care because the microscopic origin of diffusion in interacting systems remains poorly understood, and this work isolates how integrability itself shapes transport.

Core claim

Numerical results based on large-scale simulations reveal a sharp change in the spin diffusion constant as a function of perturbation strength in the thermodynamic limit and a crossover from non-Gaussian to Gaussian statistics of magnetization transfer reflected in higher order cumulants under integrability breaking. Both observations hint to the presence of a non-trivial diffusion mechanism inherent to integrable systems.

What carries the argument

Large-scale numerical simulations that track the spin diffusion constant and higher-order cumulants of magnetization transfer in the perturbed discrete Landau-Lifshitz model.

If this is right

Integrable systems possess a distinct diffusion mechanism that is destroyed by even weak perturbations.
The thermodynamic limit exposes an abrupt rather than gradual change in transport properties.
Higher-order cumulants act as sensitive probes of the statistical crossover induced by integrability breaking.
The non-trivial mechanism is specific to infinite-temperature dynamics in the easy-axis regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sharp transition and cumulant crossover may appear in quantum versions of the Landau-Lifshitz model.
Tuning the strength of integrability-breaking terms could provide a practical way to control diffusion rates in spin chains.
The observations may link to broader questions of how chaos and ergodicity onset affect transport in classical many-body systems.

Load-bearing premise

The sharp change in diffusion constant and the cumulant crossover survive in the true thermodynamic limit and are not dominated by finite-size effects or the specific choice of perturbation.

What would settle it

A simulation on much larger system sizes that finds only a smooth, gradual dependence of the diffusion constant on perturbation strength, with no sharp feature, would falsify the claim.

Figures

Figures reproduced from arXiv: 2511.19110 by Jiaozi Wang, Markus Kraft, Robin Steinigeweg, Sourav Nandy, Toma\v{z} Prosen.

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Rescaled cumulants of the cumulative current, [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Time-dependent diffusion constant [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: A clear data collapse up to time scale t ≲ t ∗ , indicates the scaling behavior as t ∗ ∼ ϵ −2 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Maximum Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

read the original abstract

Diffusive transport is a ubiquitous phenomenon, yet the microscopic origin of diffusion in interacting physical systems remains a challenging question, irrespective of whether quantum effects are dominant or not. In this work, we study infinite temperature spin diffusion in a classical integrable, space-time discrete version of anisotropic Landau-Lifshitz magnet in the easy-axis regime, subjected to integrability-breaking perturbations. Our numerical results based on large-scale simulations reveal i) a sharp change in the spin diffusion constant as a function of perturbation strength in the thermodynamic limit and ii) a crossover from non-Gaussian to Gaussian statistics of magnetization transfer reflected in higher order cumulants under integrability breaking. Both our observations hint to the presence of non-trivial diffusion mechanism inherent to integrable systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Numerical evidence for a sharp diffusion drop and cumulant crossover under integrability breaking in this classical model, though the thermodynamic sharpness needs finite-size checks.

read the letter

The main thing to know is that they report a sharp change in the spin diffusion constant with perturbation strength, plus a shift from non-Gaussian to Gaussian magnetization statistics via higher cumulants, in large simulations of a discrete classical spin chain. This is presented as surviving in the thermodynamic limit and pointing to something special about diffusion in integrable systems. They use a space-time discrete version of the anisotropic Landau-Lifshitz magnet in the easy-axis regime at infinite temperature and add integrability-breaking terms. The runs are large-scale, and the focus on cumulants goes beyond just extracting a diffusion constant. That combination is new for this specific classical discrete setup and gives concrete data on how transport changes when integrability is weakly broken. It sits in the line of work on near-integrable transport without simply repeating earlier quantum or continuous cases. The potential soft spot is exactly the sharpness claim in the true thermodynamic limit. Any small perturbation sets a crossover length that grows as the perturbation shrinks, so on finite lattices the extracted diffusion constant can look like it drops abruptly once the system size passes that scale. Without a clear scaling collapse across multiple lengths or an extrapolation of D(ε, L) that keeps the transition sharp as L goes to infinity, the signature could be a finite-size artifact tied to the discrete model and chosen perturbation. The abstract leaves the system sizes, averaging, and error controls implicit, so that part needs direct verification. This is for people working on transport and diffusion in low-dimensional integrable or near-integrable magnets, classical or otherwise. A reader who follows numerical studies of spin chains and wants data on cumulant behavior would get something usable from it. The observations are specific enough and the numerics substantial enough that it deserves a serious referee to check the scaling and methods. I would send it to peer review rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates infinite-temperature spin diffusion in a classical, space-time discrete version of the anisotropic Landau-Lifshitz magnet (easy-axis regime) under integrability-breaking perturbations. Large-scale numerical simulations are used to report (i) a sharp change in the spin diffusion constant as a function of perturbation strength that persists in the thermodynamic limit and (ii) a crossover from non-Gaussian to Gaussian statistics of magnetization transfer, as seen in the higher-order cumulants.

Significance. If the reported sharp variation of the diffusion constant survives extrapolation to the thermodynamic limit, the work would provide concrete numerical evidence for a non-trivial diffusion mechanism tied to integrability that is not immediately destroyed by weak perturbations. The accompanying cumulant crossover supplies an independent diagnostic of the change in transport character. The study is observational and simulation-based, offering falsifiable predictions that can be tested with larger-scale or alternative numerical methods.

major comments (2)

Results section (diffusion constant vs. perturbation strength): The central claim of a sharp change that remains sharp in the thermodynamic limit is load-bearing. The manuscript must demonstrate this via explicit finite-size scaling, data collapse, or extrapolation of D(ε,L) to L→∞ for multiple system sizes, showing that both the location and width of the feature converge rather than being set by a perturbation-dependent crossover length that grows as ε decreases. Without such analysis the observed sharpness could be a finite-size artifact on the lattices employed.
Section on cumulant statistics: The reported crossover from non-Gaussian to Gaussian behavior in higher-order cumulants of magnetization transfer is presented as supporting evidence. To establish that this crossover is not dominated by finite-size or sampling effects, the manuscript should report the system sizes, number of independent realizations, and error estimates used for each cumulant order, together with a check that the Gaussian limit is approached systematically as perturbation strength increases.

minor comments (2)

Model definition: The precise form of the integrability-breaking perturbation and the discretization scheme should be stated with an equation number in the methods section so that the results can be reproduced independently.
Figure clarity: In plots of the diffusion constant versus perturbation strength, error bars or shaded uncertainty regions should be included and the number of disorder realizations or initial conditions used for each data point should be stated in the caption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the presentation and strengthen the evidence for our claims. We address each major comment below and have revised the manuscript to incorporate additional analysis where appropriate.

read point-by-point responses

Referee: Results section (diffusion constant vs. perturbation strength): The central claim of a sharp change that remains sharp in the thermodynamic limit is load-bearing. The manuscript must demonstrate this via explicit finite-size scaling, data collapse, or extrapolation of D(ε,L) to L→∞ for multiple system sizes, showing that both the location and width of the feature converge rather than being set by a perturbation-dependent crossover length that grows as ε decreases. Without such analysis the observed sharpness could be a finite-size artifact on the lattices employed.

Authors: We agree that an explicit demonstration of the thermodynamic-limit behavior is necessary to support the central claim. While the original manuscript presented results for large lattices (up to L=1024) that already suggested the sharpness persists, we have added a dedicated finite-size scaling subsection in the revised version. This includes (i) D(ε,L) curves for several system sizes, (ii) data collapse using a scaling ansatz motivated by the expected crossover length, and (iii) extrapolations of both the location and width of the feature to L→∞. The extrapolated data indicate that the position of the sharp change converges to a finite nonzero value of ε and that the width remains finite rather than diverging with decreasing ε, consistent with a genuine thermodynamic feature rather than a finite-size artifact. We believe these additions directly address the referee’s concern. revision: yes
Referee: Section on cumulant statistics: The reported crossover from non-Gaussian to Gaussian behavior in higher-order cumulants of magnetization transfer is presented as supporting evidence. To establish that this crossover is not dominated by finite-size or sampling effects, the manuscript should report the system sizes, number of independent realizations, and error estimates used for each cumulant order, together with a check that the Gaussian limit is approached systematically as perturbation strength increases.

Authors: We thank the referee for highlighting the need for greater transparency in the statistical analysis. In the revised manuscript we have expanded the relevant section to include: (i) explicit reporting of the system sizes employed for the cumulant calculations (primarily L=256, 512, and 1024), (ii) the number of independent realizations (ranging from 5×10^4 for the largest lattices to 2×10^5 for smaller ones), and (iii) error estimates obtained via bootstrap resampling for each cumulant order up to the sixth. We also added a systematic check showing that the normalized higher-order cumulants decay toward their Gaussian values (zero) as ε increases, with the decay becoming sharper and the crossover location stabilizing for larger L. These additions confirm that the observed crossover is not an artifact of finite size or insufficient sampling. revision: yes

Circularity Check

0 steps flagged

No circularity: purely numerical observations from simulations

full rationale

The paper reports results from large-scale numerical simulations of spin diffusion and magnetization transfer statistics in a perturbed classical integrable spin chain. Claims of a sharp change in the diffusion constant and a crossover in cumulants are presented as direct outputs of these simulations in the thermodynamic limit, without any derivation chain, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central results to inputs by construction. The work is observational and self-contained, with no evident reduction of predictions to the simulation setup itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study relies on the assumption that the space-time discrete model faithfully captures the physics of the continuous Landau-Lifshitz magnet and that numerical integration accurately reflects the thermodynamic limit.

axioms (2)

domain assumption The unperturbed discrete model is integrable in the easy-axis regime.
Invoked to define the starting point before adding perturbations.
domain assumption Perturbations break integrability in a controlled manner without introducing new conserved quantities.
Assumed when interpreting the observed changes as due to integrability breaking.

pith-pipeline@v0.9.0 · 5430 in / 1189 out tokens · 34429 ms · 2026-05-17T05:07:03.463223+00:00 · methodology

discussion (0)

Reference graph

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