{"id":"ddd27159-2e5d-4c49-beea-ed99bd00db6d","arxiv_id":"2412.13309","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An origin-independent toroidal moment criterion, including a new surface term, predicts the direction of magnon nonreciprocity across current-driven, textured, graded, and DMI-based magnetic systems.","lead":"Spin waves, the magnetic ripples in nanomagnets, often travel differently depending on direction. This paper argues that a single quantity, the toroidal moment, predicts which magnetic patterns will show this one-way behavior, and offers a recipe for computing it in confined structures.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central proportionality Im[N^(21)] ∝ (τ·k) after Eq. 5 is asserted without derivation and is contradicted by the exchange-curvature term in Eq. 8; the paper's predictions inherit this unproven bridge.","rationale":"The strongest claim of the paper is the predictive bridge between the static toroidal moment and the linear-k part of the spin-wave frequency asymmetry. If this bridge were established, the paper would provide a genuinely general design rule. But it is not: the sentence after Eq. 5 is the only statement of it, and no derivation or reference is supplied. The supplementary material derives the toroidal moment formulas for textures but never computes the dynamic matrix N^(21)(k). The reader's conditional verdict correctly identifies this. I add a concrete internal tension: Eq. 8 shows a linear-in-k nonreciprocity from exchange in curved shells that is not of the form τ·k for the global toroidal moment, and the authors explicitly admit this. This proves the unqualified 'it can be shown' sentence is not a theorem of the microscopic equations, and it makes the precise domain of the proportionality unclear. The other supporting claim—that the different toroidal-moment definitions are 'always parallel'—is checked only on the specific analytic models in the supplement. Since the paper recommends using any definition to compute τ·k, a counterexample with non-parallel definitions would break the prediction rule. Both issues are testable: the first by a direct first-order derivation, the second by random-texture numerics. Because the core framework may still be useful for dipolar/STT/DMI systems where the proportionality can be proven case-by-case, I do not move the verdict beyond the reader's CONDITIONAL; the paper should either supply the missing derivation or narrow the claim to the demonstrated examples.","tokens_in":26860,"tokens_out":8713,"duration_ms":82319,"concrete_test":"Independently derive the first-order-in-k imaginary part of the spin-wave tensor N^(21)(k) from the linearized Landau-Lifshitz equation for an arbitrary confined equilibrium M(r), retaining only dipolar and exchange, and test whether it factors as C (∫ d^3r r×(M−⟨M⟩))·k with a texture-independent constant C. The derivation must reproduce the nanotube curling-state result (Eq. 7) as a limit. If the coefficient depends on k direction or on surface orientation beyond the integral, the bridge after Eq. 5 is invalid and all untested predictions inherit that failure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that a static, origin-independent toroidal moment predicts the directions of magnon nonreciprocity at small k. The bridge is the sentence 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).' No derivation is given: the supplementary material only evaluates τ for specific textures, never the dynamic matrix. This matters because the proportionality is stated as a general first-order theorem, yet the paper's own Eq. 8 gives an exchange-induced Δf_ex ∝ k_phi M_z for tubular curling states, which is linear in k but is not proportional to the global τ·k (which is ∝ M_phi k_z for the same state). The text acknowledges this: 'the linear toroidal moment is a global property and cannot account for the exchange-induced nonreciprocity from curvature.' Thus the universal statement after Eq. 5 is false without qualification, and the paper does not delimit the precise conditions under which the proportionality holds. Every untested prediction—bimeron, partially closed tube, graded multilayer—inherits this unproven bridge. The supplementary 'always parallel' check also covers only hand-picked analytic textures, so definition-independence is not established for arbitrary M(r).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":27114,"tokens_out":2181,"duration_ms":23592,"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":[{"comment":"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":"Section II, after Eq. (5)"},{"comment":"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":"Section II C, Eqs. (7)-(9)"},{"comment":"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":"Section III and Supplementary Material, 'always parallel' claim"},{"comment":"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.","section":"Section II B and II C, new predictions"}],"minor_comments":[{"comment":"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.","section":"Figure 6 and its caption"},{"comment":"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":"Eq. (10) and surrounding text"},{"comment":"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.","section":"Section II, 'Supplementary material' reference"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper builds heavily on earlier work from the same groups (Refs. 36, 40, 41, 101, and related), and the central proportionality is not derived from those works either. The main risk is not novelty but the scope of the claimed theorem; a focused revision that turns the asserted bridge into a qualified, provable statement, and that adds at least one independent spin-wave validation for a new texture prediction, would make the paper suitable for publication. The current mismatch between the universal claim after Eq. (5) and the acknowledged exception in Eq. (8) is the key technical issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nTwo 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.\n\nVerdict: 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.","headline":"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.","tokens_in":27667,"tokens_out":2263,"would_cite":true,"duration_ms":19228,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["toroidal moment","magnon nonreciprocity","spin waves","confined nanomagnets","surface-bound currents","graded magnetic films","Dzyaloshinskii-Moriya interaction","magnetic textures"],"falsifier":"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.","tokens_in":26650,"feed_emoji":"🧲","tokens_out":8236,"duration_ms":72822,"temperature":0.7,"pith_summary":"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.","feed_headline":"Toroidal moment predicts one-way spin waves","feed_subtitle":"In confined nanomagnets, a static vector from the equilibrium magnetization marks the directions where magnon frequencies split","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the multipole expansion of the vector potential from which the toroidal-moment definition and its surface contribution follow.","marker":"[24]"},{"why":"Provides the compensated-magnetization construction that makes the volume toroidal moment independent of the origin.","marker":"[48]"},{"why":"Gives the general formula for the frequency asymmetry as an imaginary part of the off-diagonal spin-wave tensor element.","marker":"[44]"},{"why":"Establishes the vortex-nanotube case where a toroidal moment along the tube axis accompanies axial spin-wave nonreciprocity.","marker":"[15]"},{"why":"Defines the alternative toroidal-moment convention that the surface term reconciles with the volume definition.","marker":"[19]"},{"why":"Predicts spin-wave nonreciprocity in magnetization-graded films, the class of systems the paper generalizes.","marker":"[40]"},{"why":"Provides the antiparallel-bilayer result and analytic tensor formula that the toroidal-moment criterion reproduces in the small-wave-vector limit.","marker":"[41]"},{"why":"Reports conical-helix spin-wave nonreciprocity that the texture calculation matches.","marker":"[37]"},{"why":"Supplies the bond-magnetic-toroidal-dipole description that extends the criterion to DMI films.","marker":"[31]"}],"fun_headline_variants":["Toroidal moment criterion for nonreciprocal magnons","How toroidal moments dictate magnon directionality","One-way spin waves from toroidal moment in nanomagnets","Toroidal moment: the rule for spin wave nonreciprocity","Static toroidal moment sets magnon frequency splits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Toroidal moment criterion for nonreciprocal magnons","How toroidal moments dictate magnon directionality","One-way spin waves from toroidal moment in nanomagnets","Toroidal moment: the rule for spin wave nonreciprocity","Static toroidal moment sets magnon frequency splits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000691,"raw_usage":{"total_tokens":3147,"prompt_tokens":983,"completion_tokens":2164,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":599,"completion_tokens_details":{"reasoning_tokens":2084}},"tokens_in":599,"tokens_out":2164,"duration_ms":15558,"temperature":1.0,"reasoning_tokens":2084,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:15:28.240703+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}