Topological charge, helicity and vorticity conservation and the reverse spin-current model in the II-nd type multifferoics

Pavel A. Andreev

REVIEW 3 major objections 5 minor 48 references

Topological charge, helicity and vorticity conservation and the reverse spin-current model in the II-nd type multifferoics

T0 review · 3 major / 5 minor · reviewed 2026-07-10 · grok-4.5

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Pith's one-line read Conservation of topological charge and hydrodynamic helicity in type-II multiferroics requires a reverse spin-current model that lets electric polarization generate spin structure.

desk verdict Solid continuum bookkeeping that introduces reverse spin-current and spin-field models so topological charge and helicity stay conserved; the pairwise cancellations are asserted from effective fields rather than one joint Hamiltonian. read the letter →

arxiv 2607.06756 v1 pith:NMQPMSDR submitted 2026-07-07 cond-mat.mtrl-sci

Pavel A. Andreev This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords topologicalchargehydrodynamichelicityspinvorticityreversespin-currentmodeltype-IImultiferroicsspin-orbitinteractionDzyaloshinskii-Moriyamagnetoelectriceffect

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

This paper shows that two classic integrals of motion in magnetically ordered media—the topological charge of spin textures and the hydrodynamic helicity of the full vorticity—remain conserved only when the usual spin-current model of polarization is completed by a reverse relation. In that reverse spin-current model, an existing electric polarization produces an antisymmetric effective spin current, and therefore a nonuniform magnetization. A second, non-stationary “spin-field” model supplies an additional polarization that arises directly from the vector potential and the spin density. Together these relations cancel residual sources that the spin-orbit interaction would otherwise leave in the vorticity equations. The result is a closed dynamical picture in which polarization and spin textures create each other while both topological charge and helicity stay constant. A reader who cares about skyrmions, cycloids or weak ferromagnetism in multiferroics therefore obtains a concrete mechanism that links the electric and magnetic degrees of freedom to the conservation laws that protect those structures.

What carries the argument

The reverse spin-current relation γ/c J^{αβ} = (1/2)ε^{αβγ} P^γ, which converts an electric polarization into an antisymmetric effective spin current that cancels the spin-orbit torque in the Landau–Lifshitz equation, thereby removing residual sources from the spin-vorticity evolution.

What would settle it

A direct measurement or microscopic calculation of residual spin vorticity (or of the time derivative of topological charge) in a type-II multiferroic whose polarization is deliberately misaligned with the reverse-spin-current prediction; any nonzero residual would falsify the claimed cancellation.

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Extended reading notes

Core claim

The simultaneous conservation of topological charge and hydrodynamic helicity in type-II multiferroics is possible only after the introduction of a reverse spin-current model (polarization generates an antisymmetric effective spin current) and a spin-field model of polarization; without them the spin-orbit torque leaves uncancelled sources of spin vorticity that destroy both integrals of motion.

Load-bearing premise

Every microscopic interaction is assumed to generate classical and quantum vorticity sources of exactly opposite form so that their difference vanishes once the reverse spin-current and spin-field relations are imposed; that exact cancellation is not derived from a single joint Hamiltonian.

Editorial extensions

If this is right

Electric polarization can itself nucleate noncollinear or weakly ferromagnetic spin textures via the reverse spin-current channel.
Both topological charge and hydrodynamic helicity remain integrals of motion for all listed interactions once the reverse and spin-field models are included.
The odd-anisotropy exchange interaction produces a new polarization of the form P ~ [S_A (S_B · δ_eff) – S_B (S_A · δ_eff)] that is likewise consistent with the conservation laws.
Non-stationary regimes generate an additional polarization proportional to A imes S that must be retained for helicity conservation.

Reading between the lines

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

The same reverse-spin-current cancellation may protect topological charge of skyrmions under electric-field drive, offering a route to field-controlled skyrmion motion that does not violate topology.
If residual vorticity is measured to be nonzero, the microscopic spin-orbit Hamiltonian used for multiferroics would require additional higher-order terms beyond those considered here.
The spin-field polarization u ~ A imes S suggests a dynamical magnetoelectric susceptibility that could be probed by ultrafast optical or THz experiments.

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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

The new pieces are the reverse spin-current relation (polarization generates an antisymmetric effective spin current and therefore nonuniform magnetization) and the non-stationary spin-field model of deformation. Both are introduced so that residual spin-orbit and magneto-electric sources of spin vorticity cancel and the two integrals of motion survive. That is a clean, useful extension of the author’s own spin-current series and of the Katsura–Nagaosa–Balatsky / Mostovoy line.

What the paper does well is systematic. For each interaction (symmetric exchange, three DMI variants, OASEI, SOI, ME terms) it writes the LLG torque, extracts an effective field, forms the corresponding force in the Euler equation, and shows that the curls that source classical and quantum vorticity are opposite. Topological-charge density is correctly identified with the divergence of spin vorticity, and the algebra is internally consistent under the stated approximations (constant density, no damping, etc.). The citation trail is heavy on the author’s prior arXivs, but those are the natural predecessors; the new reverse constructions are not already in the literature.

The soft spot is real but not fatal. The cancellations are pairwise and phenomenological: each force is written as F ~ S_β ∇B_eff,β after the torque has already been cast as an effective field. No single microscopic Hamiltonian is shown that simultaneously generates both the force density and the spin torque for the reverse models themselves; those models are inserted precisely to cancel leftover SOI terms. If a future microscopic derivation produces force and torque that do not stand in exact dual relation, residual sources remain. That is a gap in justification, not an internal contradiction. There are also no spectra, textures, or numerical checks.

This is for continuum theorists who already work with multiferroic hydrodynamics or skyrmion/cycloid modeling. It is not a technology paper and does not claim to be. The formal structure is careful enough that a serious editor should send it to referees; the reverse relations simply need tighter microscopic grounding or a concrete test. I would read the referee reports and keep the reverse models in mind for my own continuum work, but I would not cite them as established until that grounding appears.

Referee Report

3 major / 5 minor

Summary. The manuscript relates the topological charge of a magnetically ordered medium to the divergence of the spin (quantum) vorticity and shows that the full hydrodynamic helicity is conserved only when classical and quantum vorticity sources cancel. For a suite of interactions (symmetric Heisenberg exchange, three DMI variants, OASEI, magnetic dipole–dipole, spin–orbit, and magneto-electric terms) the author writes the corresponding Landau–Lifshitz–Gilbert torques and Euler force densities, extracts effective magnetic fields, and asserts that the resulting vorticity sources are equal and opposite. The well-known spin-current model of electric polarization is recovered as the condition that cancels the electric-dipole and part of the spin–orbit forces. Two new relations—the reverse spin-current model (polarization generates an antisymmetric effective spin current) and the spin-field model of deformation—are introduced so that residual spin–orbit and magneto-electric sources of spin vorticity also vanish, thereby protecting both the topological charge and the hydrodynamic helicity. The reverse model is further interpreted as a mechanism by which electric polarization can induce non-collinear spin textures.

Significance. If the claimed cancellations hold, the work supplies a unified hydrodynamic rationale for the simultaneous conservation of topological charge and helicity in type-II multiferroics and elevates the reverse spin-current and spin-field constructions from phenomenological add-ons to necessary conditions for those integrals of motion. The systematic catalog of force–torque pairs for DMI variants and the recently proposed OASEI is a useful reference for continuum modeling of multiferroics. The manuscript does not, however, deliver machine-checked proofs, numerical verification, or parameter-free predictions that would immediately elevate its impact beyond the formal level.

major comments (3)

The central claim (abstract, Secs. VII and IX) that the reverse spin-current relation (66) and the spin-field model (51)/(69) are required for simultaneous conservation of topological charge and helicity rests on the assertion (Secs. III–V) that every listed interaction produces classical and quantum vorticity sources of exactly opposite form. Each pair is obtained by writing an effective field from the LLG torque and a force of schematic form F ∼ S_β ∇B_eff,β, then observing that the curls match. No single microscopic Hamiltonian is exhibited that simultaneously generates both the force density and the spin torque for the reverse models themselves; the reverse relations are introduced precisely to cancel leftover SOI terms. Without an explicit derivation from one Hamiltonian (or a controlled continuum limit thereof), residual sources may remain and the integrals of motion are not guaran
Section IV states that the topological-charge density evolution will be analyzed via the divergence of the spin-vorticity equation, yet no explicit evolution equation for ϱ_T is written and no demonstration is given that the sources vanish after the reverse relations are imposed. Because the topological charge is identified with ∫(∇·Ω_q) dV, this omission leaves the conservation claim for TC incomplete relative to the helicity discussion in Sec. V.
The magneto-electric contribution (Sec. III.I, Eq. (38)) is acknowledged to produce an uncompensated source of spin vorticity that “is not compensated at the consideration of the full vorticity.” The text asserts that its structure nevertheless “corresponds to the conservation of the TC,” but no calculation of ∇·Σ_Ω or of the surface integral of the residual source is supplied. This leaves an internal tension with the claim that both integrals of motion are protected.

minor comments (5)

Numerous typographical and orthographic errors appear throughout (e.g., “multiﬀeroics,” “souses,” “Dzylaoshinskii,” “muliferroics,” “consequencies,” “Keﬀer”). A thorough copy-edit is needed.
Notation for the normalized spin density switches between n, S and M without consistent definition of the conversion factors (μ, n, ms); a single table of symbols would help.
Figure 1 is referenced for the ligand-shift geometry of weak-ferromagnetic DMI and OASEI but is not described in sufficient detail for a reader to reconstruct the vectors δ1, δ2,AB.
Several key results (generalized spin-current model, OASEI, AFM extensions) are cited only to the author’s own recent arXiv preprints; brief self-contained summaries or appendices would improve readability for non-specialists.
The Gilbert-damping term (Sec. III.J) is stated to violate helicity conservation, yet no estimate of the magnitude of the violation relative to the reversible terms is given.

Circularity Check

3 steps flagged · score 5.0 of 10

Reverse spin-current and spin-field models are defined as the algebraic inverses/cancelation conditions of the SOI terms that originated the forward spin-current model; conservation of helicity/TC then holds by construction once those relations are imposed, with supporting generalized currents and OASEI taken from the author's own prior arXivs.

self definitional [Sec. VII, eqs. (66)–(68) and following text on helicity conservation]
"The reverse spin-current model states that the polarization created in the system creates the antisymmetric eﬀective spin current γ/c J αβ = 1/2 εαβγ P γ , (66) ... The spin orbit torque ... T α SO = − γ/c εαβγ εβµν Eµ J γν , (67) can be transformed into TSO = 1/2 [E × P]. (68) Hence, we obtain zero contribution of the SOI if the spin current is caused with the electric polarization ... The reverse spin-current model in the spin evolution equation allows to cancel the contribution of the spin-orbit interaction spin torque, which can form the source of the quantum vorticity."

The reverse relation is defined by algebraically inverting the forward spin-current identification P ∼ ε J that was itself obtained by canceling the SOI force in the Euler equation. Substituting the inverse back into the dual SOI torque cancels that torque by identity. The claim that the reverse model is therefore 'required' for helicity/TC conservation holds only by this definitional cancelation, not from an independent derivation of the spin current from polarization.
self definitional [Sec. V.B.8 eqs. (50)–(51) and Sec. VIII]
"The force ﬁeld for described interactions has the following form ... F = P β ∇ Eβ + γ/c εβγδ J δγ (∇ Eβ ) + γ/c ∂t[E × S]. (50) ... If the static regime is formed the last term in equation (50) is equal to zero, but it should be compensated ... so it can be combined with mn¨u. It leads to ∂t(mn ˙u − γ/c [E × S]) ... we assume it to be equal to zero P = qef f u = γ/mc2 qef f /n [A × S]. (51) ... We can call it spin-ﬁeld model of the electric polarization ... its cancelation leads to novel model of the deformation formation ..."

The spin-field relation is obtained by requiring the residual time-dependent SOI force to be absorbed into a redefinition of the velocity/deformation so that the source vanishes. The model is therefore the cancelation condition itself; its subsequent presentation as a mechanism whose 'importance for the helicity conservation is shown' follows by construction rather than as an independent prediction.

1 more flagged steps

self citation load bearing [Introduction; Sec. III.G; Sec. VI.C (citing [5], [10], [21])]
"in Ref. [5], it is demonstrated that the electric polarization can be caused with the spin currents of diﬀerent nature. Consequently, diﬀerent types of the spin currents give diﬀerent form of the electric polarization [5]. ... the novel form of the spin-spin interaction related to the odd anisotropy of the symmetric exchange interaction (OASEI) is discussed following Ref. [10]. ... So, we obtain another mechanism of the electric polarization formation ... [10]: P = 1/3 γ/c g(l)[SB(SA · δef f) − SA(SB · δef f)]. (63)"

The generalized spin-current mechanisms (beyond the original Katsura form) and the OASEI that furnish the additional polarization structures and the corresponding vorticity sources whose cancelation is claimed are taken from the author's own recent arXiv preprints. The present conservation statements for those terms rest on those prior self-derivations of the microscopic operators and eﬀective currents without independent re-derivation or external verification inside this paper.

full rationale

The paper's central novelty (abstract, Secs. VII–IX) is that the reverse spin-current relation and the spin-field model are required so residual spin-orbit sources of spin vorticity vanish and both topological charge and hydrodynamic helicity are conserved. Those two relations are obtained simply by inverting the forward spin-current identification (itself obtained by canceling SOI force against electric-dipole force in the Euler equation) and by absorbing the leftover time-dependent SOI force into a redefinition of deformation; substituting them back cancels the dual SOI torque/force by algebraic identity. The conservation statements therefore follow by construction rather than from an independent dynamical derivation. In addition, the generalized spin-current mechanisms and the OASEI that supply the extra polarization forms and vorticity sources whose cancelation is analyzed are imported from the author's own recent preprints without external verification. The pairwise matching of classical versus quantum sources for the standard interactions (exchange, three DMI variants, etc.) is re-derived here from the same microscopic Hamiltonians via quantum hydrodynamics and is not circular; the circularity is confined to the reverse/spin-field constructions that close the SOI loophole and to the self-cited foundations of the generalized models. Score 5 reflects partial (not total) circularity: the reverse models have independent physical content as proposed mechanisms, yet the conservation claim for them reduces to the definitional cancelation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper is purely theoretical. No free parameters are fitted to data. The load-bearing content consists of standard continuum-mechanics identities, domain-standard forms of magnetic interactions, and two new postulated relations (reverse spin-current and spin-field) introduced to enforce conservation. Invented entities are the two new models themselves.

assumptions (4)

domain assumption Spin vorticity is given by the Mermin–Ho-like expression involving the unit magnetization field (Eq. 1).
Taken from quantum-hydrodynamic literature (Mahajan–Asenjo and spinor BEC papers) and used throughout Secs. II–III.
domain assumption Every magnetic interaction (Heisenberg, three DMI variants, OASEI, anisotropy, spin-orbit) can be written as an effective magnetic field in the LLG equation and as a corresponding force density in the Euler equation.
Standard continuum approximation; invoked for each interaction in Secs. III and V.
ad hoc to paper The reverse spin-current relation γ/c J^{αβ} = (1/2) ε^{αβγ} P^γ holds and cancels the spin-orbit torque.
Postulated in Sec. VII precisely to remove residual vorticity sources; not derived from a microscopic calculation independent of the conservation requirement.
domain assumption Density is spatially uniform or its gradients can be neglected when evaluating vorticity sources.
Used repeatedly (e.g., anisotropy, exchange) to set certain cross-product terms to zero.

invented entities (2)

reverse spin-current model
purpose: To generate an effective antisymmetric spin current (and therefore nonuniform magnetization) from electric polarization, thereby canceling spin-orbit torque in the spin-evolution equation.
Introduced in Sec. VII; no independent experimental signature or microscopic derivation is supplied beyond the conservation requirement.
spin-field model of electric polarization
purpose: To relate lattice deformation (hence polarization) to the electromagnetic vector potential and spin density in the non-stationary regime, canceling the remaining spin-orbit force term.
Obtained in Sec. V.8 / VIII by setting the time-dependent part of the spin-orbit force to zero; presented as a second, independent mechanism.

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Cite this review

Pith. "Pith review of Topological charge, helicity and vorticity conservation and the reverse spin-current model in the II-nd type multifferoics." pith.science (2026). https://pith.science/paper/NMQPMSDR

@misc{pith2026260706756,
  author       = {Pith},
  title        = {Pith review of: Topological charge, helicity and vorticity conservation and the reverse spin-current model in the II-nd type multifferoics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NMQPMSDR}},
  note         = {Machine review of arXiv:2607.06756}
}

read the original abstract

The topological charge and its density are related to the spin vorticity, while the spin vorticity is a part of full vorticity of the medium. The full vorticity is related to an integral of motion called the hydrodynamic helicity. Relation between two integrals of motion (the topological charge and the hydrodynamic helicity) is traced. The role of the spin-current model of the electric polarization formation due to the specific spin distribution in the helicity conservation is demonstrated. The reverse spin-current model is suggested to demonstrate the contribution of the polarization in the spin evolution equation, its importance for the helicity conservation is shown as well. Spin-field model of the electric polarization is the second model for the electric polarization (the deformation, firstly) formation due to the specific spin distribution, which appears from the same principles, but it has nonstationary origin. Moreover, the reverse spin-current model gives a mechanism for the spin structure formation due to the electric polarization in the system. It also appears as the requirement for the conservation of the helicity and the topological charge.

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Works this paper leans on

48 extracted references · 48 canonical work pages

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[1] and corresponding spin torque considered above

Dipole-dipole interaction of the magnetic moments and the quantum Bohm potential We start this set of interactions with the magnetic dipole-dipole interaction and the quantum Bohm poten- tial, following Ref. [1] and corresponding spin torque considered above. Corresponding force ﬁeld has the fol- lowing structure [20] F = γS β ∇ ˆBβ , (41) with ˆBβ of for...

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Anisotropy energy associated force Many-particle quantum hydrodynamic method [20] al- lows to obtain the ﬁeld form of the spin density evolu- tion equation [5], [21], [22], discussed above, and other hydrodynamic equations including the Euler equation, from the quantum microscopic Pauli equation with cor- responding Hamiltonian describing the interparticl...

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The force ﬁeld (44) leads to the eﬀective magnetic ﬁeld Bef f,ex = ( g0uS + 1 6 gu△ S)/γ

Symmetric Heisenberg exchange interaction Analysis of the symmetric Heisenberg exchange inter- action require derivation of two nonzero terms F = g0uSβ ∇ Sβ + 1 6 guSβ ∇△ Sβ , (44) where the second term corresponds to the spin torque, while the ﬁrst term also appears at the derivation of the spin evolution equation, but gives the zero contribution in the ...

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(45) It corresponds to the eﬀective magnetic ﬁeld Bef f,DM Ih = − 1 3 g(γ )curlS/γ obtained from the spin torque considered above

Dzylaoshinskii-Moriya interaction of helicoid type The force ﬁeld appearing from the Dzylaoshinskii- Moriya interaction of helicoid type is FDM = − 1 3 g(γ )εβµν (Sβ ∇ ∂ µ Sν ). (45) It corresponds to the eﬀective magnetic ﬁeld Bef f,DM Ih = − 1 3 g(γ )curlS/γ obtained from the spin torque considered above. It gives the source of the classical vorticity o...

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Dzylaoshinskii-Moriya interaction of cycloid type We have same picture for the Dzylaoshinskii-Moriya in- teraction of cycloid type as described in previous subsub- section. We obtain the force ﬁeld from the microscopic Hamiltonian of the Dzylaoshinskii-Moriya interaction of cycloid type FDM = 1 3 g(β ) ( (δ ·S)∇ (∇ · S) − (S · ∇ )∇ (δ ·S) ) = 1 3 g(β )Sβ ...

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Dzylaoshinskii-Moriya interaction of weak ferromagnet ic type for AFM Here we present the force ﬁeld for the Dzylaoshinskii- Moriya interaction of weak ferromagnetic type for AFM acting on the subspecies A from the subspecies B: FDM,wf,A = − g(0ζ1)εαβγ δα 2,AB Sβ A∇ Sγ B. (47) Comparison with the structure of the force ﬁeld of the dipole-dipole interactio...

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Here we present corresponding force ﬁelds: FOASEI,A = − 1 3 g2lef f Sβ A(δef f · ∇ )∇ Sβ B

The symmetric exchange interaction with the odd anisotropy Above we present the spin torque of the OASEI and its contribution to the spin vorticity. Here we present corresponding force ﬁelds: FOASEI,A = − 1 3 g2lef f Sβ A(δef f · ∇ )∇ Sβ B. (48) Similarly, we derive the force ﬁeld for the second sub- species B: FOASEI,B = 1 3 g2lef f Sβ B(δef f · ∇ )∇ Sβ ...

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Spin-orbit interaction The electric dipole-dipole interaction is essential for the multiferroics, we consider it here along with the spin- orbit interaction. This combination leads to the spin- current model for the electric polarization of spin origin, where the electric dipole moments are formed or oriented due to spin eﬀects instead of application of t...

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Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.

[1] [1]

[1] and corresponding spin torque considered above

Dipole-dipole interaction of the magnetic moments and the quantum Bohm potential We start this set of interactions with the magnetic dipole-dipole interaction and the quantum Bohm poten- tial, following Ref. [1] and corresponding spin torque considered above. Corresponding force ﬁeld has the fol- lowing structure [20] F = γS β ∇ ˆBβ , (41) with ˆBβ of for...

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[2] [2]

Anisotropy energy associated force Many-particle quantum hydrodynamic method [20] al- lows to obtain the ﬁeld form of the spin density evolu- tion equation [5], [21], [22], discussed above, and other hydrodynamic equations including the Euler equation, from the quantum microscopic Pauli equation with cor- responding Hamiltonian describing the interparticl...

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[3] [3]

The force ﬁeld (44) leads to the eﬀective magnetic ﬁeld Bef f,ex = ( g0uS + 1 6 gu△ S)/γ

Symmetric Heisenberg exchange interaction Analysis of the symmetric Heisenberg exchange inter- action require derivation of two nonzero terms F = g0uSβ ∇ Sβ + 1 6 guSβ ∇△ Sβ , (44) where the second term corresponds to the spin torque, while the ﬁrst term also appears at the derivation of the spin evolution equation, but gives the zero contribution in the ...

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[4] [4]

(45) It corresponds to the eﬀective magnetic ﬁeld Bef f,DM Ih = − 1 3 g(γ )curlS/γ obtained from the spin torque considered above

Dzylaoshinskii-Moriya interaction of helicoid type The force ﬁeld appearing from the Dzylaoshinskii- Moriya interaction of helicoid type is FDM = − 1 3 g(γ )εβµν (Sβ ∇ ∂ µ Sν ). (45) It corresponds to the eﬀective magnetic ﬁeld Bef f,DM Ih = − 1 3 g(γ )curlS/γ obtained from the spin torque considered above. It gives the source of the classical vorticity o...

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[5] [5]

Dzylaoshinskii-Moriya interaction of cycloid type We have same picture for the Dzylaoshinskii-Moriya in- teraction of cycloid type as described in previous subsub- section. We obtain the force ﬁeld from the microscopic Hamiltonian of the Dzylaoshinskii-Moriya interaction of cycloid type FDM = 1 3 g(β ) ( (δ ·S)∇ (∇ · S) − (S · ∇ )∇ (δ ·S) ) = 1 3 g(β )Sβ ...

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[6] [6]

Dzylaoshinskii-Moriya interaction of weak ferromagnet ic type for AFM Here we present the force ﬁeld for the Dzylaoshinskii- Moriya interaction of weak ferromagnetic type for AFM acting on the subspecies A from the subspecies B: FDM,wf,A = − g(0ζ1)εαβγ δα 2,AB Sβ A∇ Sγ B. (47) Comparison with the structure of the force ﬁeld of the dipole-dipole interactio...

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[7] [7]

Here we present corresponding force ﬁelds: FOASEI,A = − 1 3 g2lef f Sβ A(δef f · ∇ )∇ Sβ B

The symmetric exchange interaction with the odd anisotropy Above we present the spin torque of the OASEI and its contribution to the spin vorticity. Here we present corresponding force ﬁelds: FOASEI,A = − 1 3 g2lef f Sβ A(δef f · ∇ )∇ Sβ B. (48) Similarly, we derive the force ﬁeld for the second sub- species B: FOASEI,B = 1 3 g2lef f Sβ B(δef f · ∇ )∇ Sβ ...

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[8] [8]

Spin-orbit interaction The electric dipole-dipole interaction is essential for the multiferroics, we consider it here along with the spin- orbit interaction. This combination leads to the spin- current model for the electric polarization of spin origin, where the electric dipole moments are formed or oriented due to spin eﬀects instead of application of t...

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