Toroidal Moments in Confined Nanomagnets and their Impact on Magnonics

Attila K\'akay; B. Mimica-Figari; Felipe Brevis; Lukas K\"orber; Pedro Landeros; Rodolfo A. Gallardo

REVIEW 4 major objections 4 minor 1 cited by

Toroidal Moments in Confined Nanomagnets and their Impact on Magnonics

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that the toroidal moment of the equilibrium magnetization of a confined nanomagnet determines the directions in which spin waves propagate nonreciprocally, and it provides an origin-independent recipe for computing this…

desk verdict A genuinely useful toroidal-moment toolbox for magnonics, but the load-bearing Im[N] ∝ τ·k bridge is asserted, not proved, and the paper's own tube equations show it is not universal. read the letter →

arxiv 2412.13309 v2 pith:T3BWPW25 submitted 2024-12-17 cond-mat.mes-hall

Felipe Brevis , Lukas Körber , B. Mimica-Figari , Rodolfo A. Gallardo , Attila Kákay , Pedro Landeros This is my paper · ORCID

classification cond-mat.mes-hall

keywords toroidalmomentmagnonnonreciprocityspinwavesconfinednanomagnetssurface-boundcurrentsgradedmagneticfilmsDzyaloshinskii-Moriyainteractiontextures

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that a single static quantity, the toroidal moment of the equilibrium magnetization, predicts the directions along which spin waves in confined nanomagnets propagate nonreciprocally, meaning that the frequencies of counterpropagating waves differ. It derives an origin-independent recipe for this moment by subtracting the average magnetization and by adding a surface term coming from surface-bound currents. Applying this recipe to graded films, bilayers, multilayers, conical-helix textures, skyrmions, and curved tubes reproduces known nonreciprocity and predicts it in settings where no analytical expression exists. The authors conclude that the direction of the toroidal moment, rather than its magnitude, is the decisive predictor, and that different definitions of the moment are always parallel to each other.

What carries the argument

The central object is the toroidal moment $\boldsymbol{\tau}$, an axial vector that measures a head-to-tail circulating arrangement of magnetization or current. For magnetization distributions the paper uses $\boldsymbol{\tau}_v=\frac{1}{2}\int dV\,\mathbf{r}\times(\mathbf{M}-\langle\mathbf{M}\rangle)$, with a companion surface term $\boldsymbol{\tau}_s=-\frac{1}{10}\oint dS\,[\mathbf{r}(\mathbf{r}\cdot\mathbf{K}_b)-2r^2\mathbf{K}_b]$ built from the surface-bound current $\mathbf{K}_b=\mathbf{M}\times\hat{\mathbf{n}}$. The argument couples $\boldsymbol{\tau}$ to spin waves through $\Delta f = \gamma\mu_0 M_s/\pi\,\mathrm{Im}[N^{(21)}]\propto \boldsymbol{\tau}\cdot\mathbf{k}$ at small $k$, where $N^{(21)}$ is the off-diagonal element of the magnetic tensor. This relation is what turns a static, origin-independent integral into a prediction of which propagation directions split in frequency.

What would settle it

Look for a counterexample in a graded film: for a linear saturation-magnetization profile the paper predicts $\boldsymbol{\tau}_v=(VL_g/24)\Delta M_s(\hat{\mathbf{g}}\times\hat{\mathbf{m}})$, so a wave vector along $\boldsymbol{\tau}_v$ must give a nonzero $\Delta f$ that grows linearly with $k$ at small $k$. A measurement or micromagnetic simulation showing $\Delta f=0$ for that geometry, or a nonzero $\Delta f$ for a texture with vanishing toroidal moment such as an antiskyrmion, would falsify the criterion.

Watch

Extended reading notes

Core claim

At the heart of the paper is the claim that, for small wave vectors, the frequency asymmetry is set by a static integral: the imaginary part of the off-diagonal element $N^{(21)}$ of the spin-wave tensor satisfies $\mathrm{Im}[N^{(21)}] \propto \boldsymbol{\tau}\cdot\mathbf{k}$, where $\boldsymbol{\tau}$ is the toroidal moment of the equilibrium magnetization. The paper makes this usable by defining the volume moment from the compensated magnetization, $\boldsymbol{\tau}_v = \frac{1}{2}\int dV\, \mathbf{r}\times(\mathbf{M}-\langle\mathbf{M}\rangle)$, which is independent of the coordinate origin, and by adding a surface contribution from the surface-bound current $\mathbf{K}_b=\mathbf{M}\times\hat{\mathbf{n}}$ that connects different definitions of $\boldsymbol{\tau}$ without time-averaging. With that bridge in place, the criterion is applied to current-induced Doppler shifts, conical helices, skyrmionic textures, curved and partially closed tubes, graded films, bilayers, and DMI films, reproducing previously reported nonreciprocal directions and predicting new ones.

Load-bearing premise

The load-bearing premise is that the dynamical asymmetry is determined by a static integral: the paper assumes, without proof, that for small wave vectors the frequency shift between counterpropagating spin waves is proportional to $\boldsymbol{\tau}\cdot\mathbf{k}$, and this step is checked against only one independent simulation, a graded stripe; the bimeron, partially closed tube, and multilayer predictions remain untested.

Editorial extensions

If this is right

Nonreciprocity directions for spin waves become computable from the equilibrium magnetization alone, without solving the full spin-wave problem, for any confined texture with a nonzero toroidal moment.
The origin-independent recipe lets the prediction be applied to structures with a net magnetic moment, such as graded films, bilayers and multilayers with unequal layers, and partially closed tubes, where the old origin-dependent formula was ambiguous.
Since the different definitions of the toroidal moment are parallel, the direction of any one of them is enough to identify nonreciprocal directions; magnitude differences between definitions do not affect the prediction.
The surface contribution means the toroidal moment of a homogeneous confined structure can be estimated from the surface magnetization and the surface-bound current alone.
Known results for current-induced Doppler shifts, conical-helix textures, vortex nanotubes, graded films, and DMI films are recovered as special cases of the same $\boldsymbol{\tau}\cdot\mathbf{k}$ rule.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A practical screening tool follows implicitly: reconstructing the equilibrium magnetization, including its surfaces, of a candidate nanomagnet tells which propagation directions will be nonreciprocal before any dynamic measurement, which could speed up materials selection for magnonic diodes.
The surface-term connection suggests that surface-sensitive magnetic imaging could estimate the toroidal moment without volumetric reconstruction, a route the paper does not develop.
Extending the idea per reciprocal vector in multi-$q$ and hedgehog lattices could classify nonreciprocal directions for excitation modes that the paper does not compute.
Because the paper notes that higher-order dipolar terms can make $\Delta f(k)$ nonlinear, a natural next question is whether a generalized geometric quantity beyond the first-order toroidal moment predicts the nonlinear correction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version. This paper is worth knowing about. It derives an explicit surface toroidal moment from surface-bound currents and an origin-independent compensated volume moment, and shows how a static toroidal-moment calculation can be used as a fast symmetry predictor for spin-wave nonreciprocity across graded films, bilayers, chiral textures, and curved shells. The vector calculus in the supplementary is clean. The graded-stripe micromagnetic simulation supports the prediction. If you work in magnonics, this is a useful organizing tool.

The soft spot is exactly where the reader put it. The sentence after Eq. (5) — 'it can be shown that Im[N^(21)] ∝ (τ·k)' — is asserted with no derivation. That is the bridge that makes the toroidal moment a dynamical predictor rather than just a symmetry label. The stress-test note is right that the paper's own Eq. (8) gives an exchange-induced Δf_ex ∝ k_φ M_z for tubular curling states, which is linear in k but not proportional to the global τ·k for that state. The text acknowledges this limitation explicitly: 'the linear toroidal moment is a global property and cannot account for the exchange-induced nonreciprocity from curvature.' So the universal statement after Eq. (5) is false without qualification. A careful referee should ask the authors to either supply a derivation with the conditions under which the proportionality holds, or reframe the claim as a heuristic for dipolar and DMI mechanisms only.

Two smaller points. First, the 'always parallel' relation between the different toroidal-moment definitions is checked on a set of analytic textures, not proven for arbitrary M(r). For the intended use — predicting direction rather than magnitude — that is probably fine, but it is not a theorem. Second, validation leans heavily on previously published results, several from the authors' own group. That is acceptable, since those are the relevant benchmark calculations, but it makes the genuinely new predictions (bimeron, partially closed tube, odd/even multilayers) untested. The bimeron and tube predictions are plausible and follow the same logic; they just inherit the unproven bridge.

Verdict: send to peer review. This is exactly the kind of paper a good referee can improve by forcing the authors to state what is proven and what is heuristic. The central claim needs either a derivation or a clear scope restriction. With that, the paper is a solid contribution to magnonics and to the toroidal-moment literature.

Referee Report

4 major / 4 minor

Summary. The paper develops a toroidal-moment framework for predicting the direction of magnon nonreciprocity in confined nanomagnets. It derives a surface contribution to the toroidal moment from surface-bound currents, proposes an origin-independent volume toroidal moment based on the compensated magnetization M−⟨M⟩, and then applies these definitions to a broad set of systems: conical-helix textures, skyrmions and bimerons, curved shells and partially closed tubes, graded films, bilayers and multilayers, and DMI films. The central predictive claim is that, at small wave vectors, the imaginary part of the off-diagonal spin-wave matrix obeys Im[N^(21)] ∝ τ·k, so that a static toroidal moment identifies propagation directions along which spin-wave frequencies are nonreciprocal. The paper validates parts of this claim against a micromagnetic simulation of a graded stripe and against previously reported analytic and experimental results for several other geometries, and it leaves new predictions for bimerons, partially closed tubes, and certain multilayer configurations untested.

Significance. If the central proportionality holds, the paper provides a simple, parameter-free criterion for identifying nonreciprocal magnon propagation directions from the equilibrium magnetization alone, which would be a useful design rule for magnonic devices. The derivation of the surface toroidal moment and the origin-independent volume expression are clean and constitute a genuine contribution. The manuscript is also commendable for not fitting any parameters to data and for including an explicit micromagnetic test of one prediction. However, the paper's main bridge from the static toroidal moment to the dynamic frequency shift is asserted rather than derived, and one of the paper's own equations contradicts the unqualified version of that assertion. The 'always parallel' relation among toroidal-moment definitions is likewise checked only on selected analytic textures. These issues limit the current strength of the universal claims, although they do not invalidate the framework for the specific systems where the proportionality is explicitly verified.

major comments (4)

[Section II, after Eq. (5)] The sentence 'If Δf is expanded up to first order in the wave vector, then it can be shown that Im[N^(21)] ∝ (τ·k)' is the load-bearing bridge of the paper, but no derivation is provided either in the main text or in the supplementary material. The supplementary evaluates toroidal moments for specific textures; it never derives the small-k expansion of the dynamic matrix. Because all subsequent predictions (bimeron, partially closed tube, graded multilayer) inherit this statement, the manuscript must either prove the proportionality under clearly stated conditions or substantially weaken the scope of its claims.
[Section II C, Eqs. (7)-(9)] The paper's own example of the curling-state nanotube shows that a linear-in-k nonreciprocity can exist without being proportional to the global τ·k. For the curling state, τ∝M_ϕ z, so τ·k∝M_ϕ k_z, while Eq. (8) gives Δf_ex ∝ k_ϕ M_z. The text acknowledges this by stating that 'the linear toroidal moment is a global property and cannot account for the exchange-induced nonreciprocity from curvature.' That acknowledgment directly contradicts the unqualified statement after Eq. (5). The paper needs to delimit precisely when the proportionality holds—e.g., which energy terms, which mode branches, and which boundary conditions—so that readers do not extend the criterion beyond its valid domain.
[Section III and Supplementary Material, 'always parallel' claim] The abstract and conclusions claim that the different toroidal-moment definitions 'are always parallel' for confined magnetic structures. The supplementary demonstrates this only for the particular analytic textures analyzed (conical helices, skyrmions, bimerons, merons, tube states). Since the paper explicitly states that the magnitude discrepancies are irrelevant because only the direction matters, the 'always parallel' property is itself load-bearing. The manuscript should either provide a general proof or replace the universal statement with a clearly limited empirical observation for the treated textures.
[Section II B and II C, new predictions] The predictions for bimeron strings, partially closed tubes, and graded multilayers are not independently tested against spin-wave calculations or experiments. The lone micromagnetic test, the graded stripe in Fig. 6, is consistent with the toroidal-moment prediction. Given that the central proportionality is not derived, the untested predictions inherit the unproven bridge. At minimum, the authors should add a validation for one of the new texture classes, or explicitly mark these as conjectures that rely on the assumed proportionality.

minor comments (4)

[Figure 6 and its caption] The text refers to 'Fig. 6(c)' and 'Fig. 6(d)', but the figure shows panels (a)-(d) with the caption describing (b-c) and (c) inconsistently; the panel labels and references should be reconciled.
[Eq. (10) and surrounding text] The symbol δ_{l,0} in Eq. (10) is not defined in the main text; please define the Kronecker delta and clarify the meaning of the azimuthal mode index l for readers.
[Section II, 'Supplementary material' reference] Reference [51] is given as 'URL_will_be_inserted_by_publisher'; the authors should replace it with a working link or a proper citation to the supplementary material.
[Throughout] Several typos and minor English issues remain, e.g., 'conﬁned' is used inconsistently, and the sentence 'A more general magnetization conﬁguration for thin cylindrical shells can be considered here' is a fragment; a careful proofread would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction found; the key 'Im[N^(21)] ∝ (τ·k)' bridge is an unproven assertion rather than a tautology or a fit.

full rationale

Walking the derivation chain, the toroidal moments τv and τs (Eqs. 2–4) are obtained from Eq. (1) by vector identities and are definitions/rewritings with no hidden dynamic input. The dynamical side is Eq. (5) (from Ref. 44) and the small-k bridge statement after Eq. 5: 'If Δf is expanded up to first order in the wave vector, then it can be shown that Im[N^(21)] ∝ (τ·k).' This sentence is asserted without derivation or citation; it is a physical claim about the dynamic matrix, not an identity with the static integral τ, so it is an omitted proof rather than a circular reduction. The paper later narrows its scope by saying 'the linear toroidal moment is a global property and cannot account for the exchange-induced nonreciprocity from curvature' (Section II C), which is an honest limitation of the asserted proportionality. The applications are checked against independent dynamical results and external experiments: the bilayer dynamic tensor Eq. (18) comes from Ref. 41; the nanotube dipolar/DMI shifts in Eqs. (7)–(9) come from prior tube analyses; and the graded-stripe prediction is tested by the paper's own TetraX micromagnetic simulations in Fig. 6. The self-citations (Refs. 15, 40, 41, 101, 58, 64) provide parameter-free dynamic-matrix results or prior dispersions, not fitted parameters renamed as predictions. One unpublished self-citation, Ref. 64, is a caveat for the STT tensor formula, but the conclusion Δf_stt ∝ (τ·k) follows algebraically from τ ∝ Jf and the cited Im[N^(21)_stt] ∝ Jf·k, so it is not circular. The supplementary claim that τv and τs are 'always parallel' is verified only for selected analytic textures, which is an overgeneralization risk but not a circular step. Overall, no load-bearing step reduces by construction to its own inputs; score 1 reflects the unproven central proportionality and the notable reliance on self-authored prior validations, without constituting circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There are no parameters fitted to nonreciprocity data; all inputs are standard material constants or analytically specified texture parameters. The main imported assumptions are the symmetry criterion, the Ederer-Spaldin subtraction, and the unproved small-k proportionality. No new physical entities are postulated.

assumptions (5)

domain assumption Space-time symmetry breaking indicated by τ·k≠0 is sufficient to estimate magnon nonreciprocity.
Invoked throughout Section II; based on Neumann's principle and Refs. [10,11,15,30], not derived in this paper.
domain assumption The part of the toroidal moment from the uniform magnetization ⟨M⟩ cannot induce frequency asymmetry in a homogeneous centrosymmetric lattice.
Used in Section II to justify subtracting ⟨M⟩ in Eq. (4); imported from Ref. [48].
ad hoc to paper For small wave vectors, Im[N^(21)(k)] ∝ τ·k.
Stated after Eq. (5) without proof; this is the load-bearing bridge connecting the static toroidal moment to the dynamical frequency shift.
domain assumption Magnetization textures can be represented by the stated analytic models (conical helix, skyrmion/bimeron/meron, tube modes) for the purpose of computing toroidal moments.
Required for all closed-form results in Section II and the supplementary.
ad hoc to paper The various toroidal-moment definitions are always parallel for confined magnetic structures.
Asserted in Section III and the abstract; demonstrated only for the specific textures in the supplementary, not proven in general.

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Cite this review

Pith. "Pith review of Toroidal Moments in Confined Nanomagnets and their Impact on Magnonics." pith.science (2026). https://pith.science/paper/T3BWPW25

@misc{pith2026241213309,
  author       = {Pith},
  title        = {Pith review of: Toroidal Moments in Confined Nanomagnets and their Impact on Magnonics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T3BWPW25}},
  note         = {Machine review of arXiv:2412.13309}
}

read the original abstract

The nonreciprocity created by dipolar coupling, electric currents, and Dzyaloshinskii-Moriya interactions is discussed in cases where the magnon propagation direction has a component parallel to the toroidal moment. A criterion for calculating the toroidal moments is established, addressing the issue of correct origin selection by considering compensated and uncompensated magnetization distributions. This criterion is then applied to various nonreciprocal magnetic systems, with the calculations consistent with those reported in the literature and predicting the existence of nonreciprocity in a more general manner. These results broaden the physical significance of the toroidal moment and facilitate the identification and estimation of nonreciprocity in magnonic systems. This work also clarifies the interrelations between different definitions of the toroidal moment for confined structures, where a surface term arising from surface-bound currents connects these definitions without the need for time-averaging. Comparing these definitions of the toroidal moment applied to different magnetic textures demonstrates that they are always parallel but may differ in magnitude and sign. The discrepancy in the different definitions is deemed irrelevant since its direction, rather than its magnitude, primarily predicts the existence of magnon nonreciprocity.

Figures

Figures reproduced from arXiv: 2412.13309 by the authors.

**Figure 2.** FIG. 2. Conical-helix textures in a thin film with bulk (left) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Skyrmion, bimeron, and meron magnetic textures. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. Illustration of three possible conical-helix magnetic [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. (a) Transversal cross-section of an incomplete tube [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) Graded ferromagnetic stripe with [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. (a) Asymmetric ferromagnetic bilayer having thick [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Bond magnetic toroidal moment for a square crystal [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 1.** Figure 1: FIG. 1. (a) Nonzero components of the toroidal moments [PITH_FULL_IMAGE:figures/full_fig_p017_1.png]

**Figure 2.** Figure 2: FIG. 2. Relevant components of the volume toroidal moment as a [PITH_FULL_IMAGE:figures/full_fig_p018_2.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

In the absence of free currents, and for nonuni- form magnetization textures, Eqs

is suit- able to connect the toroidal moment with nonrecipro- cal wave phenomena when only free currents are present (J = Jf ). In the absence of free currents, and for nonuni- form magnetization textures, Eqs. ( 3-4) are used to cap- ture the proper relation between the nonreciprocal wave propagation and the toroidal moment [ 56]. Nonreciprocal propagati...

[2] [2]

On the other hand, τ s = − 1 10 ˛ ∂V dS [ r (r · Kb) − 2r2Kb ] (3) is due to surface-bound current Kb = M × ˆn, with ˆn a unit vector normal to the local surface

[ 48]. On the other hand, τ s = − 1 10 ˛ ∂V dS [ r (r · Kb) − 2r2Kb ] (3) is due to surface-bound current Kb = M × ˆn, with ˆn a unit vector normal to the local surface. No- tice that τ s, which has not been reported before, is analogous to Eq. ( 1) but with a closed surface inte- gral and opposite sign. Nonetheless, there is an al- ternative deﬁnition fo...

[3] [3]

Kodera, D

T. Kodera, D. L. Sounas, and C. Caloz, Magnetless nonreciprocal metamaterial (mnm) technology: Appli- cation to microwave components, IEEE Trans. Microw. Theory Tech. 61, 1030 (2013)

2013

[4] [4]

( 4) does not capture since τ v = 0

Nonetheless, the surface toroidal moment ( τ s) is also nonzero for the n = 1 state [ 51] something that Eq. ( 4) does not capture since τ v = 0. Thus, based on the toroidal moment calcu- lations, only the states n = 0, ±1 could exhibit frequency nonreciprocity of dipolar origin, whereas states such as the hedgehog conﬁguration ( n = 0, qz = 0 and ψ = 0) ...

[5] [5]

Y. Shi, Z. Yu, and S. Fan, Limitations of nonlinear op- tical isolators due to dynamic reciprocity, Nat. Photon. 9, 388 (2015)

2015

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Jiang, J

Z. Jiang, J. Lim, Y. Li, W. Pfaﬀ, T.-H. Lo, J. Qian, A. Schleife, J.-M. Zuo, V. Novosad, and A. Hoﬀmann, Integrating magnons for quantum information, Appl. Phys. Lett. 123, 130501 (2023)

2023

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Camley, Nonreciprocal surface waves, Surf

R. Camley, Nonreciprocal surface waves, Surf. Sci. Rep 7, 103 (1987)

1987

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Caloz, A

C. Caloz, A. Al` u, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-L´ eger, Electromagnetic nonreciprocity, Phys. Rev. Appl. 10, 047001 (2018)

2018