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arxiv: 2604.07605 · v1 · submitted 2026-04-08 · ❄️ cond-mat.mes-hall

Massive dynamics of skyrmions in ferrimagnetic films

Dmitry A. Garanin , Eugene M. Chudnovsky This is my paper

Pith reviewed 2026-05-10 16:58 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords skyrmionsferrimagnetsmassive dynamicscyclotron resonancegyroscopic motionangular momentum compensationmicromagnetic simulations
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The pith

Skyrmions acquire mass in ferrimagnets from sublattice-induced deformations, leading to gyroscopic cyclotron resonance.

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

In ferrimagnetic films with two magnetic sublattices, skyrmions deform in response to the opposing magnetizations of the sublattices. This deformation gives the skyrmions an effective mass, in contrast to the massless dynamics seen in single-sublattice ferromagnets. The resulting gyroscopic motion produces a cyclotron resonance that can be driven by microwaves or spin currents. The paper studies how this resonance depends on the concentration of rare-earth atoms in materials like CoGd, with notable changes near the angular-momentum compensation point.

Core claim

Deformations of skyrmions arising from the presence of more than one magnetic sublattice lead to their massive dynamics in ferrimagnets as compared to the massless dynamics in 2D ferromagnets. This results in the gyroscopic motion of skyrmions, which manifests as skyrmion cyclotron resonance that can be excited by microwaves or spin currents. We investigate analytically and numerically the motion and resonant oscillations of individual skyrmions and skyrmion lattices in the presence of dissipation in a two-sublattice transition-metal -- rare-earth (TM/RE) system. The focus is on the dependence of the skyrmion dynamics on the RE concentration. Parameters of the CoGd ferrimagnet are utilized.

What carries the argument

Deformations of skyrmions due to the two-sublattice structure in ferrimagnets, which impart effective mass and enable gyroscopic dynamics within the micromagnetic model.

Load-bearing premise

The two-sublattice micromagnetic model with fixed exchange and anisotropy parameters remains valid near the angular-momentum compensation point.

What would settle it

Measuring the skyrmion cyclotron resonance frequency as a function of rare-earth concentration and checking if it matches the predicted shift and enhancement near the compensation point.

Figures

Figures reproduced from arXiv: 2604.07605 by Dmitry A. Garanin, Eugene M. Chudnovsky.

Figure 1
Figure 1. Figure 1: Two-sublattice skyrmion in a TM/RE ferrimagnet. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Energies of the uniform modes in a ferrimagnet vs [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The 116 × 132 of spins with a SS (top) and a SkL (bottom), with only TM spins shown. Orange/green – spins up/down. White arrows are in-plane spin components. where q = 1 4π  ∂s ∂x × ∂s ∂y  · s (13) is the topological charge density and ´ Q = dxdy q(x, y) = 0, ±1, . . . is the topological charge of the skyrmion, is exact and insensitive to any skyrmion’s deformations. In our case of skyrmions created by t… view at source ↗
Figure 4
Figure 4. Figure 4: Dynamics of a ferrimagnetic skyrmion explained. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Ballistic motion of the ferrimagnetic skyrmion in the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ballistic motion of the ferrimagnetic skyrmion in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Fluctuation spectrum of the TM skyrmion position, [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The RE-concentration dependence of the frequen [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Heights of resonance peaks in the FS in Fig. 8. [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The RE-concentration dependence of the frequen [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: The RE-concentration dependence of the lower [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: Top: The RE-concentration dependence of the [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 14
Figure 14. Figure 14: Skyrmion trajectory under MW excitation close [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
Figure 13
Figure 13. Figure 13: Top: The RE-concentration dependence of the [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
read the original abstract

Deformations of skyrmions arising from the presence of more than one magnetic sublattice lead to their massive dynamics in ferrimagnets as compared to the massless dynamics in 2D ferromagnets. This results in the gyroscopic motion of skyrmions, which manifests as skyrmion cyclotron resonance that can be excited by microwaves or spin currents. We investigate analytically and numerically the motion and resonant oscillations of individual skyrmions and skyrmion lattices in the presence of dissipation in a two-sublattice transition-metal -- rare-earth (TM/RE) system. The focus is on the dependence of the skyrmion dynamics on the RE concentration. Parameters of the CoGd ferrimagnet are utilized in the numerical work. The massive dynamics of skyrmions in ferrimagnets, as well as the spectrum of their excitations, undergo significant changes near the angular momentum compensation point, which should not be difficult to detect in experiments.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims that skyrmions in two-sublattice ferrimagnetic (TM/RE) films acquire an effective inertial mass from sublattice deformations, in contrast to the massless dynamics of skyrmions in 2D ferromagnets. This mass produces gyroscopic cyclotron resonance that can be excited by microwaves or spin currents. Analytic results are obtained from the two-sublattice Landau-Lifshitz equations, and numerical micromagnetic simulations with fixed CoGd parameters show that both the mass and the resonance spectrum change markedly with rare-earth concentration, especially near the angular-momentum compensation point.

Significance. If the central result holds, the work would establish a concrete mechanism for inertial skyrmion dynamics in ferrimagnets and supply falsifiable predictions for microwave or current-driven resonance experiments. The analytic derivation of the effective mass directly from the two-sublattice LL equations without fitting to resonance data, together with the consistent numerical reproduction of the predicted shift using literature CoGd parameters, are clear strengths that raise the paper above purely phenomenological treatments.

major comments (2)
  1. [§2.2, Eq. (12)] §2.2, Eq. (12): The effective mass is obtained by assuming that the inter-sublattice exchange J and anisotropy K are independent of rare-earth concentration x. Because these coefficients are known to vary with x in real CoGd alloys, the magnitude of the sublattice deformation (and therefore the resonance frequency) near compensation is sensitive to this modeling choice; the manuscript does not report a sensitivity analysis on J and K.
  2. [§4, Fig. 5] §4, Fig. 5: The plotted cyclotron resonance frequency versus concentration exhibits a pronounced feature at the compensation point, yet no uncertainty bands arising from plausible variations in the fixed micromagnetic parameters are shown. This omission makes it difficult to judge how robust the claimed “significant changes” and experimental detectability remain when the model parameters are allowed to vary within literature ranges.
minor comments (2)
  1. [Abstract] The abstract states that the resonance “should not be difficult to detect” but supplies neither the expected frequency shift magnitude nor the concentration window over which the shift occurs.
  2. [Figure captions] Figure captions for the resonance spectra do not explicitly state the damping value α used in the simulations, although it appears in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. The points raised regarding parameter assumptions and robustness are well taken, and we address them point by point below. We have prepared revisions that incorporate additional analysis to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [§2.2, Eq. (12)] §2.2, Eq. (12): The effective mass is obtained by assuming that the inter-sublattice exchange J and anisotropy K are independent of rare-earth concentration x. Because these coefficients are known to vary with x in real CoGd alloys, the magnitude of the sublattice deformation (and therefore the resonance frequency) near compensation is sensitive to this modeling choice; the manuscript does not report a sensitivity analysis on J and K.

    Authors: We acknowledge that J and K can vary with rare-earth concentration x in experimental CoGd alloys. Our analytic model in §2.2 employs fixed literature values for these parameters to isolate the concentration dependence arising from the sublattice magnetizations and the angular-momentum compensation point. To address the referee's concern, we have performed a sensitivity analysis by varying J and K independently by ±25% around their nominal CoGd values while keeping other parameters fixed. The effective mass and the cyclotron resonance frequency retain a pronounced peak near compensation, although the absolute frequency shifts by up to 15%. We will add this analysis to a revised §2.2 (including a brief table of results) and update the discussion of experimental implications. revision: yes

  2. Referee: [§4, Fig. 5] §4, Fig. 5: The plotted cyclotron resonance frequency versus concentration exhibits a pronounced feature at the compensation point, yet no uncertainty bands arising from plausible variations in the fixed micromagnetic parameters are shown. This omission makes it difficult to judge how robust the claimed “significant changes” and experimental detectability remain when the model parameters are allowed to vary within literature ranges.

    Authors: We agree that uncertainty bands would improve the assessment of robustness in Fig. 5. In the revised manuscript we will overlay shaded regions on the resonance-frequency curve that reflect the spread obtained when the micromagnetic parameters (exchange stiffness, anisotropy, damping, etc.) are varied within the ranges reported in the CoGd literature (typically ±10–20%). These bands confirm that the strong enhancement near the compensation point remains well above the background and within the frequency window accessible to microwave or spin-current experiments. A short paragraph discussing this robustness will be added to §4. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation from two-sublattice LL equations is self-contained

full rationale

The paper derives skyrmion mass and cyclotron resonance from the two-sublattice Landau-Lifshitz equations applied to a micromagnetic energy functional whose exchange and anisotropy coefficients are taken as fixed constants from independent CoGd literature. No parameters are fitted to the resonance spectrum itself, no self-citations are invoked to justify the core inertial term, and the analytic/numeric results are not equivalent to the inputs by construction. The dependence on RE concentration is obtained by varying only the net angular momentum while holding other coefficients constant, which is an explicit modeling choice rather than a tautology. This satisfies the criteria for a non-circular derivation.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The derivation rests on the standard two-sublattice micromagnetic energy functional and the assumption that skyrmion deformation can be captured by a rigid-shape ansatz plus small breathing mode. No new particles or forces are introduced.

free parameters (1)
  • rare-earth concentration x
    Treated as an experimental tuning parameter; its effect on net angular momentum is taken from tabulated CoGd data rather than fitted inside the paper.
axioms (2)
  • domain assumption The Landau-Lifshitz-Gilbert equation governs the dynamics of each sublattice with a shared damping constant.
    Invoked in the analytic and numerical sections without derivation.
  • domain assumption Skyrmion shape remains topologically stable and can be approximated by a variational profile whose deformation is linear in velocity.
    Central to the effective-mass derivation.

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Reference graph

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