arxiv: 2605.07435 · v1 · submitted 2026-05-08 · ❄️ cond-mat.mes-hall

Recognition: 2 theorem links

· Lean Theorem

Point-gap topology of damped magnon excitations in skyrmion strings

Yusuke Koyama , Yuki Kawaguchi

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:56 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords non-Hermitian skin effectmagnon excitationsskyrmion stringspoint-gap topologyGilbert dampingwinding numberLandau-Lifshitz-Gilbert equation

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The pith

Damped magnons in skyrmion strings exhibit point-gap topology that produces the non-Hermitian skin effect even without nonlocal damping.

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

The paper studies magnons with finite lifetimes induced by Gilbert damping inside a skyrmion-string lattice. It shows that the non-Hermitian skin effect appears from local damping alone and that, when nonlocal damping acts along a single direction, the winding number of any band possessing a unique energy minimum equals the sign of the wave number at that minimum. These topological features are obtained by combining spin-wave theory with perturbation theory on the Landau-Lifshitz-Gilbert equation and are confirmed by simulating how magnetic-field pulses launch spin waves whose propagation direction changes from band to band according to the calculated winding numbers.

Core claim

By incorporating the spin-wave theory and perturbation theory for the Landau-Lifshitz-Gilbert equation including nonlocal damping terms, we analytically evaluate the spectral winding number for point gaps, which indicates the existence of the non-Hermitian skin effect. We find that the NHSE can occur even in the absence of nonlocal damping. In the presence of nonlocal damping along one direction, we show that the winding number for an energy band with a unique minimum is determined from the sign of the wave number at the band minimum. We demonstrate these results using a model that hosts a skyrmion-string lattice as a steady state and further investigate spin-wave propagation dynamics.

What carries the argument

The spectral winding number of point gaps in the complex magnon energy bands under local and nonlocal Gilbert damping, which diagnoses the non-Hermitian skin effect.

If this is right

The non-Hermitian skin effect appears in magnon systems that contain only local Gilbert damping.
Unidirectional nonlocal damping fixes the winding number of a band solely by the sign of the wave number at its energy minimum.
Spin-wave propagation direction changes sharply from band to band according to the presence or absence of local and nonlocal damping.
The topological winding numbers correctly predict the observed asymmetry in magnon transport under different damping conditions.

Where Pith is reading between the lines

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

Damping parameters could be engineered to select preferred propagation directions for magnons in skyrmion-based devices.
Analogous point-gap topology may govern damped excitations in other periodic spin textures or bosonic lattices.
Edge-resolved spectroscopy on finite skyrmion-string samples would provide a direct test of the predicted skin-mode localization.

Load-bearing premise

The skyrmion-string lattice remains a stable steady state of the system so that perturbation theory on the damped Landau-Lifshitz-Gilbert equation correctly describes the finite-lifetime magnon modes.

What would settle it

Direct observation that spin-wave packets launched in a skyrmion-string sample with unidirectional nonlocal damping do not reverse propagation direction when crossing a band minimum whose wave number changes sign would falsify the winding-number rule.

Figures

Figures reproduced from arXiv: 2605.07435 by Yuki Kawaguchi, Yusuke Koyama.

**Figure 3.** Figure 3: FIG. 3. Magnon complex spectra at [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Spin-wave propagation excited by a left-circularly rotating magnetic-field pulse [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Spin-wave propagation excited by a circularly rotating magnetic-field pulse with frequency [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

read the original abstract

We theoretically study the non-Hermitian topology of magnons with finite lifetimes due to Gilbert damping. By incorporating the spin-wave theory and perturbation theory for the Landau-Lifshitz-Gilbert equation including nonlocal damping terms, we analytically evaluate the spectral winding number for point gaps, which indicates the existence of the non-Hermitian skin effect (NHSE). We find that the NHSE can occur even in the absence of nonlocal damping. In the presence of nonlocal damping along one direction, we show that the winding number for an energy band with a unique minimum is determined from the sign of the wave number at the band minimum. We demonstrate these results using a model that hosts a skyrmion-string lattice as a steady state. We further investigate spin-wave propagation dynamics excited by a magnetic-field pulse and show that the propagation direction changes drastically from band to band depending on the presence of local and nonlocal damping, consistent with the nontrivial winding numbers.

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

The paper shows that non-Hermitian skin effect appears in magnons even without nonlocal damping, and with it the winding number for a band with one minimum equals the sign of the wave number at that minimum.

read the letter

The main takeaway is that non-Hermitian skin effect shows up in these damped magnon systems even when there's no nonlocal damping at all, and when nonlocal damping is present in one direction, the winding number for a band with one minimum just equals the sign of the wave number at that minimum. They get this by combining spin-wave theory with perturbation on the Landau-Lifshitz-Gilbert equation. The skyrmion-string lattice serves as a steady-state background, and they compute the point-gap topology analytically for the bands. Then they look at how a magnetic field pulse excites spin waves and see the propagation flip direction from band to band based on the damping type, which lines up with the winding numbers. What works is the direct link between the topological invariant and the observable propagation. It's a nice, specific result for this setup rather than a generic extension. The analytical evaluation of the winding number is explicit, and the model demonstration keeps things grounded. The soft spot is around the validity of the linear perturbation for finite-lifetime magnons. The stress-test concern is real: if the background skyrmion lattice drifts or if magnon-magnon interactions kick in at the relevant damping strengths, the extracted winding numbers could shift. The paper uses standard methods, but without explicit checks against the full nonlinear dynamics or estimates of the approximation range, it's hard to know how far the claims extend beyond the linear regime. This paper is for researchers in magnonics and non-Hermitian physics who want a worked example with skyrmions. A reader interested in topological control of spin waves would get something concrete out of it. The thinking is clear and it engages the literature properly, so it should go to peer review rather than a desk reject. I'd want to see the full derivations and any supplementary checks on the perturbation accuracy before deciding on the strength of the conclusions.

Referee Report

3 major / 2 minor

Summary. The manuscript theoretically studies non-Hermitian topology of finite-lifetime magnons arising from Gilbert damping in skyrmion strings. Combining spin-wave theory with perturbation theory on the Landau-Lifshitz-Gilbert equation that includes nonlocal damping terms, the authors analytically compute the spectral winding number of point gaps to establish the non-Hermitian skin effect (NHSE). They report that the NHSE appears even in the absence of nonlocal damping and that, when nonlocal damping acts along one direction, the winding number of a band possessing a unique minimum is fixed by the sign of the wave number at that minimum. These analytic results are illustrated in a model that supports a skyrmion-string lattice as a steady state, and they are further supported by numerical simulations of spin-wave propagation excited by a magnetic-field pulse, which show damping-dependent direction reversal consistent with the computed winding numbers.

Significance. If the central claims hold, the work meaningfully extends non-Hermitian topology into the magnonics domain by showing that ordinary Gilbert damping alone can generate point-gap topology and skin effects. The explicit relation between winding number and the sign of k at a band minimum supplies a simple, falsifiable diagnostic that could guide both theory and experiment. The analytic evaluation together with the concrete skyrmion-string demonstration and the pulse-propagation numerics constitute clear strengths that make the predictions testable in existing magnon platforms.

major comments (3)

[Model definition and steady-state verification] The perturbation treatment of the LLG equation around the skyrmion-string background assumes that the lattice remains an exact steady state once the chosen local and nonlocal damping terms are included. No explicit substitution of the background configuration into the full damped LLG equation is shown to confirm that the damping contributions identically vanish or balance, which is required before linearization can be trusted for the subsequent winding-number calculation.
[Perturbation theory and spectral winding number] Both the claim of NHSE without nonlocal damping and the rule that the winding number equals the sign of k at the band minimum rest on the first-order perturbative spectrum. No quantitative error estimate, validity window in damping strength or magnon amplitude, or side-by-side comparison with direct numerical integration of the nonlinear LLG dynamics is provided; such checks are necessary to establish that the extracted point-gap topology survives at the finite lifetimes considered.
[Spin-wave propagation dynamics] The numerical pulse-propagation results are stated to be consistent with the analytic winding numbers, yet no quantitative metrics (extracted group velocities, decay lengths, or skin-localization lengths) are reported that would allow a direct, falsifiable comparison between the simulated dynamics and the predicted non-Hermitian spectrum.

minor comments (2)

[Notation and damping terms] The notation distinguishing the local Gilbert damping term from the nonlocal damping tensor should be introduced with explicit component-wise equations early in the methods section to avoid ambiguity when the winding-number formulas are derived.
[Figures] Figure captions for the dispersion and propagation plots would benefit from explicit labels indicating the location of each band minimum and the sign of the associated winding number, facilitating immediate visual verification of the analytic claims.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive suggestions. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Model definition and steady-state verification] The perturbation treatment of the LLG equation around the skyrmion-string background assumes that the lattice remains an exact steady state once the chosen local and nonlocal damping terms are included. No explicit substitution of the background configuration into the full damped LLG equation is shown to confirm that the damping contributions identically vanish or balance, which is required before linearization can be trusted for the subsequent winding-number calculation.

Authors: We acknowledge that an explicit verification was omitted. The model is constructed such that the skyrmion-string lattice is an exact steady state of the undamped LLG equation, and the damping terms (local and nonlocal) are added as perturbations that vanish identically on the static background by symmetry. In the revised manuscript we will add an explicit substitution of the background magnetization into the full damped LLG equation, demonstrating that all damping contributions cancel, thereby rigorously justifying the linearization step. revision: yes
Referee: [Perturbation theory and spectral winding number] Both the claim of NHSE without nonlocal damping and the rule that the winding number equals the sign of k at the band minimum rest on the first-order perturbative spectrum. No quantitative error estimate, validity window in damping strength or magnon amplitude, or side-by-side comparison with direct numerical integration of the nonlinear LLG dynamics is provided; such checks are necessary to establish that the extracted point-gap topology survives at the finite lifetimes considered.

Authors: We agree that additional validation of the perturbative spectrum is desirable. The first-order correction in the Gilbert damping α is controlled by the small-α expansion standard in magnonics; the leading error is O(α²) and vanishes in the experimentally relevant limit α ≪ 1. In the revision we will (i) state the validity window explicitly, (ii) supply a quantitative error bound derived from the perturbation series, and (iii) present a direct numerical comparison of the perturbative eigenvalues with the spectrum obtained by exact diagonalization of the linearized damped LLG operator for representative α values. Full nonlinear LLG integration lies outside the linear spin-wave regime that defines the point-gap topology under study; we will add a brief remark clarifying this scope limitation while noting that the linear approximation is the appropriate framework for the reported winding numbers. revision: partial
Referee: [Spin-wave propagation dynamics] The numerical pulse-propagation results are stated to be consistent with the analytic winding numbers, yet no quantitative metrics (extracted group velocities, decay lengths, or skin-localization lengths) are reported that would allow a direct, falsifiable comparison between the simulated dynamics and the predicted non-Hermitian spectrum.

Authors: We accept that quantitative metrics would improve the falsifiability of the comparison. In the revised manuscript we will extract and tabulate group velocities, exponential decay lengths, and (where relevant) skin-localization lengths from the pulse-propagation simulations. These numbers will be placed alongside the analytic predictions obtained from the winding numbers and the non-Hermitian dispersion, enabling a direct, quantitative test of consistency. revision: yes

Circularity Check

0 steps flagged

No circularity: results follow from standard spin-wave and LLG perturbation theory

full rationale

The derivation applies established spin-wave theory and perturbative linearization to the LLG equation with damping terms to compute spectral winding numbers analytically for a skyrmion-string lattice model. No parameters are fitted to data within the paper and then relabeled as predictions; the winding numbers and NHSE claims emerge directly from the linearized non-Hermitian spectrum without self-definitional loops or load-bearing self-citations. The skyrmion lattice is posited as a steady state by construction of the model, but this is an input assumption rather than a derived output that feeds back into the topology calculation. All steps remain independent of the target results and rest on external, standard methods.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no explicit free parameters, axioms, or invented entities can be extracted. The work relies on the standard Landau-Lifshitz-Gilbert equation and spin-wave approximation, which are treated as background.

pith-pipeline@v0.9.0 · 5460 in / 1165 out tokens · 30947 ms · 2026-05-11T01:56:54.233899+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We find that the NHSE can occur even in the absence of nonlocal damping... winding number... determined from the sign of the wave number at the band minimum.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using the nondegenerate perturbation theory, we show that the presence of damping gives rise to an imaginary part in the magnon eigenenergy.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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