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Ryota Ono, [Igor Solovyev](https://orcid.org/0000-0002-2010-9877), Sergey Artyukhin

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[Multiferroic kinks and spin-flop transition in Ni2InSbO6 from first principles](https://mdr.nims.go.jp/datasets/54c22ffa-5117-43f1-8387-fb85aed23125)

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Multiferroic kinks and spin-flop transition in Ni2InSbO6 from first principlesnpj | spintronics Articlehttps://doi.org/10.1038/s44306-024-00020-9Multiferroic kinks and spin-flop transitionin Ni2InSbO6 from first principlesCheck for updatesRyota Ono1,2 , Igor Solovyev2 & Sergey Artyukhin1Magnetoelectric multiferroics are key materials for next-generation spintronic devices due to theirentangledmagnetic and ferroelectric properties. Spiral multiferroics possess ferroelectric polarizationandareparticularly promising for electric control ofmagnetismandmagnetic control of ferroelectricity.In this work, we uncover long-period incommensurate states characterized by unique multiferroickinks in corundum nickelate Ni2InSbO6, a member of a promising family of polar magnets. Utilizing a2-orbital S = 1 model, we derive formulas for Heisenberg and anisotropic magnetic exchanges andmagnetically-induced polarization, enabling their calculations from first principles. We use theseparameters in Monte Carlo and Landau theory-based calculations to reproduce experimentallyobserved magnetic structures and polarization dependence on the magnetic field. We predictmagnetic phase transitions between flat spiral, conical spiral, canted antiferromagnetic andferromagnetic states under increasing magnetic fields. Kinks in the spiral phases repel each otherthrough aYukawa-like potential arising fromexchange ofmassivemagnons.We also find that suitablydirected electric fields can be used to stabilize the ferromagnetic and spiral states. The findings open anew pathway to predictive first-principles modelling of multiferroics and will inspire experiments andtechnological applications based on multiferroic kinks.In magnetoelectric multiferroics, a magnetic order coexists and interactswith a ferroelectric one. Several microscopic scenarios of why such coex-istence may occur and how the magnetic order can affect the electricpolarization have been established1–5 and the work is rapidly progressing inthis direction. Understanding such interactions between the magnetic andelectric degrees of freedom is of great importance from both the funda-mental and practical points of view. A special attention is paid to themutualcontrol of the magnetic structure and the electric polarization by applyingthe magnetic or electric field. For instance, the external magnetic field cancontrol the magnetic structure, while also changing the ferroelectricpolarization. Conversely, the external electricfield can be used to control themagnetic structure6.There are two main types of multiferroic (MF) materials4. In type-Imultiferroics the crystal structure itself is ferroelectric, irrespectively of themagnetism. However, the electric polarization can still be controlled bychanging the magnetic structure. In type-II MF the crystal structure iscentrosymmetric, but the inversion symmetry can still be broken by amagnetic order, which leads to the ferroelectric polarization. An interestingaspect of the type-I materials is that many of them develop chiral magneticstructure, driven by antisymmetric Dzyaloshinskii-Moriya (DM)interactions in the non-centrosymmetric crystal structure. This chirality canbe controlled by the magnetic field, presenting another interesting avenuefor magnetoelectric control in type-I materials. For instance, a very specialtype of the chiral magnetic structure is the skyrmion lattice, which has beenintensively studied in the context of MF applications in Cu2OSeO3 andGaV4S87,8.Ni2InSbO6 (NISO) is one of such chiral MFs. Its low temperaturestructure has a polar (non-centrosymmetric) rhombohedral R3 spacegroup9,10 similar to well known MF corundum derivative Ni3TeO611,12. Anumber of polar corundum derivatives have recently been synthesized andmaterials with above room temperature magnetism have been found13,14.These are promising candidates formagnetoelectric applications, in someofwhich polarization switching has been predicted 15. Previous studies16revealed an incommensurate antiferromagnetic proper-screw spiral (helix)within eachNi layer, with a long periodicity of 30 unit cells. The polarizationalong the threefold rotation axis changes quadratically with the magneticfield due to variations in the spiral order induced by the field16. In addition,recent experiments have revealed a spin-flop (SF) transition upon applyingthemagneticfield along the threefold rotationaxis17,18.However, thedetailedmicroscopic analysis of these observations is lacking. The1Quantum Materials Theory, Italian Institute of Technology, Via Morego 30, 16163 Genova, Italy. 2Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan. e-mail: ryota.ono.gm@gmail.comnpj Spintronics |            (2024) 2:17 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s44306-024-00020-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s44306-024-00020-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s44306-024-00020-9&domain=pdfmailto:ryota.ono.gm@gmail.comphenomenological mechanisms of the magneto-electric coupling con-sidered so far19–21 are not universal and are influenced by system-dependentfactors. Therefore, it is important to construct realistic models of thesematerials with interacting spins and magnetically-induced polarization,starting from the modern theory of polarization in terms of Berry phasesand Wannier centers22–24 and using model parameters obtained from first-principles calculations. According to such calculations in the generalizedgradient approximation (GGA), NISO can be regarded as an S = 1material.In the corundum structure, each Ni2+ ion is located in a distorted octahe-dron of O2− ions. Therefore, the Ni 3d states split into triply degenerate t2gand double degenerate egmanifolds. The t2g states are fully occupied and donot significantly contribute to magnetism. On the other hand, the eg statesare half-filled and form a group of narrow bands near the Fermi level, whichare mainly responsible for the S = 1 physics. Thus, we can greatly simplifyour analysis by constructing the 2-orbital model for these bands andextracting all parameters of the model from the first principles calculationsin the Wannier basis.Here we model the exchange interactions and magnetically-inducedpolarization emerging from such realistic 2-orbital eg model at the half-filling. After extracting the parameters of electronic Hubbard-like modelfrom the first principles calculations, we employ the superexchange theory,which in our case is formulated as a first-order perturbation theory for theWannier functions with respect to the hopping parameters. For theexchange interactions, the treatment is equivalent to the standard second-order perturbation theory for the magnetic energy, which in the Wannierbasis results in the expression for the spin dependent electric polarization.Our models highlight the emergence of intriguing cross-coupling phe-nomena inNISO. Specifically, we explore the SF transitions and a crossoverto the multiferroic kink array state, both induced by the external magneticfield along the threefold axis. The kinks contribute ferroelectric polarizationopposite to that of the collinear state and their energetics can be rationalizedin terms of repulsion through the Yukawa-like potential and the competi-tion between the DM and magnetic field setting their chemical potential.Additionally, using a continuous theory, we explore the possibility of cross-control of magnetic (electric) order by an electric (magnetic) field.ResultsBasic electronic structure, electronic and spin model for NISOResults of electronic structure calculations using the experimental crystalstructure, shown in Fig. 1a10, inGGA25,26 with the spin-orbit coupling (SOC)for a nonmagnetic state are illustrated in Fig. 1e. These calculations clearlyreveal two groups of the Ni 3d bands: six t2g bands (per two Ni sites)around− 1 eV and four eg bands around the Fermi level. In NISO, the t2gbands are fully occupied and nonmagnetic, while the magnetic propertiesare mainly associated with the eg bands. Therefore, we pick this group ofstates to construct a realisticHubbard-typemodel, whichwould capture themagnetic behavior of NISO. Such Hubbard model has the form:Ĥ ¼PijPabPσσ0tabσσ0ij ĉyiaσ ĉjbσ 0 þ Hon�site; ð1Þwhere a and b ( = 1 or 2) label the eg orbitals, andHon-site stands for the on-site Coulomb interactions, which are specified by the intra-orbitalinteraction U, Hund’s coupling JH and inter-orbital interactionU 0 ¼ U � 2JH27. ĉyiaσ (̂ciaσ) in Eq. (1) stands for the creation (annihilation)of an electron on the Wannier orbital a of the Ni site i with the spin σ. Theparameters of the one-electron part, tabσσ0ij , are defined as the matrixelements of GGA Hamiltonian in the Wannier basis. Since the basis iscomplete for the eg bands, these parameters perfectly reproduce the originalband structure in GGA (see Fig. 1e). The main sources of the SOC in NISOare the 5p states of the heavy In/Sb atoms. Therefore, it is important toinclude the SOC before the wannierization, at the level of regular GGAcalculations. Then, although theWannier functions are formally associatedwith theNi eg states, the SOCof the heavy In/Sb atomswill still contribute tothematrix elements tabσσ0ij , which canbediagonal aswell as off-diagonalwithrespect to the spin indices.Fig. 1 | Crystal structure and electronic structureof NISO. aHexagonal cell of NISO. bNi ions in thehexagonal cell. Only the closest neighbors coupledwith exchange constants J1 and J2 are shown withyellow and gray lines, respectively. c J1 and J2 bondsfrom side view and top view around a Ni ion.d Definition of DM vector parameters. This figureexplicitly depicts parameters for α = 1 bond type.e Electronic structure of a rhombohedral unit cell ofNISO around Fermi level calculated within GGA(solid black line) and 2-orbital model constructed byMLWFmethod (cyan dashed line). The inset showsa schematic of the k-path in the Brillouin zone forthe rhombohedral unit cell.https://doi.org/10.1038/s44306-024-00020-9 Articlenpj Spintronics |            (2024) 2:17 2Next, we map this low energy electronic model onto the spin modelemploying for these purposes the superexchange theory. In the atomic limit,the ground state corresponding to the high-spin S = 1 state at half-filling isdescribed by the single Slater determinant. The same holds for the one-electron and one-hole eg states emerging in the superexchange theory in theprocess of virtual excitations. Therefore, here we essentially deal with theone-electron theory. This results in the spin Hamiltoninan:Hs ¼X<ij>�Jij~ei �~ej þ ~Dij � ð~ei ×~ejÞ þ~ei � ⃡Γij~ej� �; ð2Þwhere~ei stands for the classical spin vector at site i, the first term is theisotropic interaction, the second term is the anisotropic Dzyaloshinskii-Moriya interaction and the last term is the (traceless) symmetric exchangeanisotropy28.Similarly, the expression for the magnetically-induced polarization isgiven by~Ps ¼X<ij>~Pij~ei �~ej þP⃡ij � ð~ei ×~ejÞ þ~ei � Π⃡ij~ej� �; ð3Þwhere the first term describes isotropic exchange striction, while the fol-lowing terms originate form the antisymmetric and the (traceless) sym-metric anisotropy29–31. This model is derived in the framework of themodern theory of polarization in solids22–24, using perturbation expansion ofthe Wannier functions.Since the R3 group has only one three-fold rotation axis (along z in thechosen coordinates), all bonds with the same distance should be trans-formed into each other by the Ĉz3 rotation. To illustrate this, consider thebonds surrounding aNi ion labeledas 0 in Fig. 1c.We index itsNi neighborsin the layer above as j = 1, 2, 3, which are classified as bond type α = 1.Conversely, the Ni ions in the layer below, indexed as j = 4, 5, 6 and cate-gorized as bond type α = 2, are at a slightly longer distance. The vectorsconnecting ion 0 with ions j = 1, 2, 3 are expressed as~ϵ0j ¼ ðϵk0j cosð2πj=3Þ; ϵk0j sinð2πj=3Þ; ϵ?0jÞ, where ϵk0j and ϵ?0j are the lengthsof the vector components parallel and perpendicular to the Ni layer,respectively (see Fig. 1d). For j = 4, 5, 6, similar formula applies, with thesame ϵk0j, but a slightly different ϵ?0j and with the arguments of sin and cosincremented by π. Then, the DM vectors are as follows:~Dα0j ¼ dkα cos θ0j;α; dkα sin θ0j;α; d?α� �; ð4Þwhere θ0j;α ¼ 2πj=3þ θα, d?α and dkα are bond-dependent parameters. TheDM vectors are antisymmetric: ~Dαj0 ¼ �~Dα0j. Similarly, contributions fromthe isotropic (Heisenberg) exchange to the magnetically-induced polariza-tion (first term of Eq. (3)) are as follows:~Pα0j ¼ pkα cos θ0j;α; pkα sin θ0j;α; p?α� �; ð5Þwith symmetric~Pαj0:~Pαj0 ¼~Pα0j.An additional single-ion term is allowed in systems S > 1/2. However,in the subsequent discussion, we neglect this term, expecting its effect to beminor since theorbital angularmomentummatrix elements vanishbetweeneg orbitals, making SOC inactive. Indeed, in our 2-orbital model, the energysplitting due to SOC inside S = 1 triplet results in very tiny ΔE ≈ 7 μeV (seeSupplementary Note 2). Turning to the magnetically-induced polarization,we calculate the single-ion term by the formula given in ref. 32. However, wealso neglect this term as it is spin-independent in this 2-orbital model.We derive analytical formulas for the symmetric and antisymmetricexchange constants in (2) and magnetically-induced electronic componentof the polarization in (3) from the 2-orbital model following the strategyused in refs. 30,31 (see “Methods” for the derivation). For exchangeinteractions we have:Jij ¼ 13ΔEP1;2αβ�3 t0αβij� �2þ Trðtαβij � tαβij Þ� �;~Dij ¼ 2ΔEP1;2α;βt0αβij tαβij ;⃡Γij ¼ 2ΔEP1;2α;βtαβij � tαβij � 13 Trðtαβij � tαβij Þ1⃡h i;ð6Þwhile the parameters formagnetically-induced electronic component of thepolarization have the form:~Pij ¼ � 2e3VΔEP1;2α;β3~r 0αβij t0αβij � Tr ~r αβij � tαβij� �� �;~Pij ¼ � 2eVΔEP1;2α;βt0αβij ~r αβij þ~r 0αβij tαβij� �;Π⃡ij ¼ � 2eVΔEP1;2α;β~r αβij � tαβij þ tαβij �~r αβij � 23 Tr ~r αβij � tαβij� �1⃡� �;ð7Þwhere ΔE =U+ JH, tαβij and~r αβij are the spin dependent matrix elements ofthe hopping and position operator, correspondingly, expanded in Paulimatrices as t̂ij ¼ t̂0ij � σ̂0 þPx;y;zγ îtγij � σ̂γ, ~̂rij ¼ ~̂R 0ij � σ0þPγ0 i~̂Rγ0ij � σ̂γ0 , 1⃡ is a 3 × 3 identity matrix. Jij and ~Pij are the symmetricisotropic (Heisenberg) exchange (scalar) and exchange striction (vector),while the consecutive terms describe antisymmetric (Dzyaloshinskii-Mor-iya) exchange and symmetric anisotropic (Ising-like) interactions, respec-tively. The exchange parameters obtainedwith thismethod are summarizedinTables1 and2.Wefind that the significant exchange interactions inNISOpredominantly originate from the bonds between Ni2 and its Ni neighborsin the layer above (ions 1,2,3, exchange constant J1) and below (ions 4,5,6,exchange constant J2), as shown in Fig. 1c. The obtained parameters obeyJ2 < J1, which can be understood considering geometric Ni-O-Ni angles,∠(Ni-O-Ni, J1) = 129.38∘ and ∠(Ni-O-Ni, J2) = 136.75∘. Typically, the half-filled 2-orbital model predominantly yields AFM interactions, with FMcontributions manifesting as effective suppressions in the AFM iteractionconstants. According to the Goodenough-Kanamori rule33, the bond angle∠(Ni-O-Ni, J1) being close to 90∘ compared to ∠(Ni-O-Ni, J2), leads to J1receiving compensationsby the ferromagnetic contributions.Consequently,this results in a smaller magnitude of J1 compared to J2.We note that the 2-orbital model parameters also provide insight intoimportance of different terms in Eq. (2) and Eq. (3). As for the magneticTable 1 | Values of isotropic exchanges and DMparameters inNISO [meV]Bond α Dist. Jα dkα d?α θα [Deg.]1 3.821 −6.872 1.255 1.046 1382 3.912 −13.095 1.833 −1.053 64Table 2 | Values of isotropic term for polarization and corre-sponding parameters in NISO [μC/m2]Bond α Dist. pkα p?α θα [Deg.]1 3.821 449 −138 752 3.912 613 242 180https://doi.org/10.1038/s44306-024-00020-9 Articlenpj Spintronics |            (2024) 2:17 3energy, isotropic and antisymmetric exchange contributions are found to benon-negligible (see Supplementary Note 1 for the values of the symmetricanisotropy tensor components). Actually, the relativemagnitudes of isotropicand DM interactions indicate that the symmetric anisotropic exchange isinherently small. This fact aligns with Moriya’s paper28 and our analyticalformulas (6). These state that the DM interaction is first-order in SOC,whereas the symmetric anisotropy is second-order in SOC. Among thecontributions to the polarization, only the isotropic term is found non-negligible. Therefore, in the following we focus mainly on these terms. Theactual valuesof theanisotropic termsof themagnetically-inducedpolarizationand their small effects are detailed in SupplementaryNote 3. Consequently, inthe followingwe primarily focus on the isotropic exchange, theDMexchangeand the isotropic term of the magnetically-induced polarization.We note that the antiferromagnetic contribution overestimates theexchange parameters as the 2-orbital model does not take into accountferromagnetic contributions from t2gorbitals andHund’s couplings onnon-magnetic ions accurately. Additionally, the relatively small value of thedenominator in superexchange theory, specifically ΔE = 2.9 eV, furthercontributes to potential overestimations in the overall exchange parameters.Using these parameters in the Monte-Carlo simulations in a 30 × 30 × 1simulation box with periodic boundary conditions, we obtain the transitiontemperature, overestimating the experimental one by approximately afactor of two. The transition temperature was identified by the peak in theheat capacity in Monte-Carlo simulations using the calculated parameters(Supplementary Note 5). Nevertheless, we emphasize that themain physicsdiscussed in the following is purely originating from the relative strengthbetween isotropic and anisotropic exchanges.Magnetic structures in NISOWenowdiscuss themagnetic ground state obtained from the single-q spiralAnsatz analysis inNISO. The 2-orbial parameters show that the J1 and J2 arevery strong andAFM.Thus,we can expect spins in the neighboring layers tobe opposite. Since there are four Ni layers in the hexagonal cell, Fig. 1a, theperiod of the AFM order coincides with the length of the hexagonal c-axis,thus, we have ~qC ¼ ð0; 0; 0Þ. DM interactions modify q-vector from acommensurate phase to an incommensurate one by δ~q. The 2-orbitalmodelparameters show that those bonds fulfill a relation dkα > d?α . This stabilizesspin-spiral states with the propagation vector within the layer. Namely, theq-vector is modified as ~qIC ¼~qC þ δ~q ¼ ðδqx; δqy; 0Þ. Then, we canconsider the situation given in Fig. 2a, where the spin spiral propagateswithin a Ni layer.Magnetic energy contributions from isotropic (Eiso) and DM (EDM)interactions areEs ¼ Eiso þ EDM¼ 14 ðJ1 þ J2Þ 12� δq2x � δq2y� �n� 2ffiffiffi3pδqx dk1 sinðθ1 � ϕÞ þ dk2 sinðθ2 � ϕÞ� �� 2ffiffiffi3pδqy dk1 cosðθ1 � ϕÞ þ dk2 cosðθ2 � ϕÞ� �o:ð8ÞThen, we take derivative of the energy with respect to δqx and δqy, andfind the energy minimum at δ~q,δqx ¼ffiffiffi3p �dk1 sinðθ1 � ϕÞ � dk2 sinðθ2 � ϕÞ� �=ðJ1 þ J2Þδqy ¼ffiffiffi3p �dk1 cosðθ1 � ϕÞ � dk2 cosðθ2 � ϕÞ� �=ðJ1 þ J2Þ:(ð9ÞThewavevector components are plotted inFig. 2ausing theparametersfrom the 2-orbital model. We see that, approximately,ðcos ϕ; sin ϕ; 0Þ ? n̂?, thus, a cycloidal spiral state (Fig. 2b), while theexperiment reported a proper-screw spiral state. However, the wave vectorand the spiral period, δqAns ≈ 0.034 (29 unit cells) are in a good agreementwith reported experimental values δq ≈ 0.029 (30 unit cells)9. Additionally,the symmetric anisotropic interactions tilt the spin-spiral plane. The cal-culated parameter actually give a small rotation ≈ π/8 (see SupplementaryNote 1). Another parameter set, calculated fromGreen’s functionmethod34,gives a proper-screw type spiralwith a very longwave length of 142unit cells(δqGF ≈ 0.007; see Supplementary Note 4). Since the spin spiral type is notimportant for the following discussion of magnetic kink generation andmagnetically-induced polarization, we use a cycloidal spiral as the groundstate of NISO.We note that in this mean-field analysis, the energy does notdepend on the rotation of the spiral plane ϕ. This can be straightforwardlyconfirmed by substituting the analytical formula for the spiral wave vectorEq. (9) into the energy Eq. (8). The resulting formula yieldsEs ¼ 3ððdk1Þ2 þ ðdk2Þ2 þ 2dk1dk2 cosðθ1 � θ2Þ þ 4ðJ1 þ J2Þ2Þ=ð4ðJ1 þ J2ÞÞ,thus, independent of ϕ.Fig. 2 | Magnetic ground state as obtained fromthe single-q spiral Ansatz. a Definition of the spin-spiral parameters. b The components of δqmini-mizing the energy as a function of the spin rotationplane orientation (given by a polar angle ϕ, where therotation plane normal is n̂? ¼ ð� sin ϕ; cosϕ; 0Þ).The energy is minimized by δq∼ ðcosϕ; sin ϕ; 0Þwhich is perpendicular to n̂? , giving rise to a cycloidalspiral. The horizontal axis indicates the rotation of thespin-spiral plane. c The cycloidal spiral ground statefor the Hamiltonian, with magnetic exchange con-stants derived from our superexchange theory. Blacklines connecting the sites and spins of neighbors aredrawn as a guide.https://doi.org/10.1038/s44306-024-00020-9 Articlenpj Spintronics |            (2024) 2:17 4In order to understand the magnetic ground state and the response tothe externalmagneticfields inNISO,we also perform classicalMonte-Carlosimulations (MCS) of the spin model Eq. (2) with the 2-orbital parameters.Incorporating the Zeeman term�PiHzezi , simulations are carried out on a30 × 30 × 1 supercell with periodic boundary conditions.OurMCSreveal the emergenceof ananticipated spiral ground state, seeFig. 3a. This result validates the applicability of our single-q spiral Ansatz.Upon the applicationof an externalmagneticfieldperpendicular to the layer,a SF transition is found. This supports the observations reported in refs. 16–18.Remarkably, before theSF transition,weobserve the solitons, favoredbyDMinteraction, shrinking, while the regions between them, with spins along themagnetic field, experiencing a notable increase (Fig. 3b). In comparison tothe experimentally determined transition field of approximately 20 T, ourcalculations yield an overestimated field of approximately 100 T. This dis-crepancy is attributed to the overestimated exchange parameters and finitesize effects in the MCS. Despite the success of the MCS, finite size effectssignificantly influence important spiral deformations by forcing the spiral tobe commensurate with the simulation box. Therefore, in the following wederive and employ a continuous Ginzburg-Landau-type theory.1D continuous modelApplication of the externalmagneticfield canmodify the spiral period. Sucha phenomenon, often elusive inMCSdue tofinite size limitations, can resultin overlooking of interesting physics like spiral period changes. For instance,De Gennes investigated modifications of the solitonic lattice period by anexternal magnetic field in liquid crystals35 and concluded that the spiralperiod follows the analytical expression:LðHz ÞLð0Þ ¼ 2π� �2KðkÞEðkÞ; ð10Þwhere L(0) is the original period of the perfect spiral at zero field, and K(k),E(k) are elliptic integrals of the first and second kind, respectively. Here, weformulate a continuous theory to capture such period changes and resultingmagnetically-inducedpolarizationwithout any restrictionof theperiodicity.As we see from MCS, the obtained magnetic structures are quasi-1D,therefore we can effectively model NISO as a 1D-chain. This is due to thefollowing reasons. First, the spiral lies in the xy-plane. Second, importantexchanges are only J1 and J2 bonds. They involve the group of bondsconnected by the Ĉz3 operation. Those give rise to the energy expressionidentical to that of a 1D chain.Figure 4 shows the spin-spiral ground state and the states it deforms intounder an external magnetic field perpendicular to the layer, as obtained byminimizing 1D AFM continuous chain energy in Eq. (37). First, applicationof an externalmagneticfield generates a kink array state (Fig. 4c, g, k): insteadof rotating in space with the constant wave vector, antiferromagnetic orderparameter now rotates in a non-uniform fashion. Indeed, when~A is pointingperpendicular to the magnetic field ~H, sublattices cant along the field, whichleads to a gain ofZeeman energy, linear in the canting angle (at the expense ofthe quadratic loss in the antiferromagnetic exchange energy between thesublattices). Thus, the regions with ~A ? ~H expand, forming plateaus in Ax(Fig. 4c), which increases the gain of the Zeeman energy on canting. Incontrast, between theplateaus, kinks inAx (or solitons)occurwhere~A quicklyrotates through the field direction. At these points, one sublattice aligns withthefield andanother–opposite to it, and therefore the gain ofZeemanenergyon the canting is not possible. The shape of these kinks is analogous to thesolitons in nonlinear dynamics because they are solutions of the Sine-Gordonequation. The continuousmodel reveals thatwith further increase of the fieldstrength, the kinks are pushed apart. At a higherfield, theflat spiral turns intoa conical one and its plane flops perpendicular to thefield (Fig. 4b, f, j), whichenables a higher gain ofZeemanenergy fromspin canting along thefield. Thekinks, present both in the flat and conical spiral phases, can be viewed asparticles interacting with each other via exchange of magnons. Furtherincrease of the magnetic field strength drives the transition into SF phase.These transitions are very similar to the one found in our MC simulations(Fig. 3b). However, in this continuous theory, the transition is muchsmoother due to unrestricted simulation box size.Fig. 3 | Magnetic structures as obtained from the Monte-Carlo simulations.aMagnetic ground state of NISO as obtained from classical Monte-Carlo simula-tions. The red rectangle indicates the area, for which we show the spin texture inPanel (b). bMagnetic structure inside the area, indicated by the red box in Panel (a)for different values of an external magnetic field, perpendicular to the spiral plane(three-fold rotation axis).Fig. 4 | Spin-spiral evolution by an external mag-netic field as obtained from our continuousspin model. a–d Continuous spin, e–h real spacecontinuous spin textures and (i–l) sphere area cov-ered by each spin array as calculated using thecontinuous model Eq. (37) at several external mag-neticfield strengthsHz. a, e, i SF state; (b, f, j): conicalspiral; (c, g, k): kink state; (d, h, l): flat spiral state.H  [T]z0102025x (sites)aefbcdghijklhttps://doi.org/10.1038/s44306-024-00020-9 Articlenpj Spintronics |            (2024) 2:17 5In the kink array phase, fast change of the angle θ near the kink atposition X results in a delta function-like change of the gradient∂θ∂x / δðx � XÞ. In this case, the interactions between kinks are found bysolving Euler-Lagrange equation for the magnetic structure deformation,inducedby the soliton, and result in the following interactionbetweenkinks,VðrÞ ¼ D4βJ  e�βr; ð11Þwhere β ¼ffiffiffiffiffiffiffiffiffiffiHz=Jp, and r is the distance between the kinks. This Yukawa-like interaction can be understood as a long-range repulsion caused byexchanging virtual massive magnons (similar to how Coulomb interactionis caused by exchanges of virtual massless photons). High magnetic fieldresults in the enlargementof the areawith~A ? ~H becauseof the exponentialdependence on Hz in Eq. (11). This is reminiscent of physics found inmultiferroic TbFeO336.Figure 5 illustrates the relationship between spiral period and thepolarization change. Our continuous theory predicts a period change thatclosely matches De Gennes’s analytical formula Eq. (10). Then, the differ-ence in the polarization (computed as a dipolemoment of the cluster in Fig.1c divided by the volume) between spiral and SF states is given as:ΔPisoz ¼ PisoSF;z � Pisospiral;z ≈ π2δq2ðp?1 þ p?2 Þ: ð12ÞThemodel parameters result in a very small change due to the isotropicexchange, ΔPisoz ¼ 0:923 μC/m2. Such a minor polarization change is likelyto be imperceptible in experiments. In the first step (flat spiral phase), achange in polarization is induced by a change in the spiral period (kinkdistance). In the second step (conical phase, above 16 T, indicated by adotted line in Fig. 5), the curve shows a sharp increase of polarizationresulting from a ferromagnetic component developing in xy-plane due tothe SF contribution.Figure 6 illustrates the phase diagram in the (Hz, Ez) plane. Here weaccount for the electric field perturbatively by introducing the lowest orderenergydensity correction− EzPz.Wefind that a small electricfield along theferroelectric polarization stabilizes SF state while an oppositefield favors theflat spiral. Asymmetry with respect to the sign of Ez is caused by the non-centrosymmetric structure ofNISO,with thepyroelectric polarization set bya specific cation ordering. The Pz contributions from the spiral and FMstates are opposite, therefore they are stabilized by opposite electric fields.We note that the largemagneticfield continuously deforms the SF state intothe FM state.Polarization change during spin flop-to-FM transitionOurMCS results indicate the transition from SF phase to the FM state as weapply an external magnetic field (Fig. 7a). Notably, the experimental datareveals a much greater polarization change during this transition whencompared to the change between the spiral and the SF phases. During thisprocess, we can express the isotropic exchange contribution to the polar-ization as follows:Pisoz ¼ 3 cosð2θÞðp?1 þ p?2 Þ; ð13Þwhere θ is the angle between the spin and the xy-plane. In our MCS thisangle exhibits a linear dependence on the externalmagneticfield strength, asdepicted in Fig. 7b. This linear dependence is also found in the experiment16.Consequently, we anticipate that themagnetically-induced polarizationwillresemble the behavior shown in Fig. 7c.Importantly, themagnitude of this polarization change far exceeds thatobserved during the spiral→ SF phase transition. The polarization changeacross the sequence of phase transitions, spiral→ conical spiral→ SF→FM, reproduces well the experimental data16.DiscussionOur study of the complex interplay of magnetic and electric fields,magnetic exchanges and DM interactions in corundum nickelatederivative NISO has uncovered intriguing properties of multiferroickinks and their implications for the field of MFs. The methodologicaladvancements allowed a quantitative modelling of key magnetoelectricphenomena in NISO.We have derived analytical formulas for magnetic exchanges andmagnetically-induced polarization, starting from a minimal 2-orbitalmodel, suitable for computing these parameters from first principles.These formulas can be applied to other S = 1 MFs (e.g. Ni3TeO6,Ni2ScSbO6). Additionally, we have developed a continuous theory toaddress complex changes in the spin-spiral structures, which extendsour understanding of spiral MFs. This theory should be applicable to awide class of materials where spiral state is stabilized byDzyaloshinskii-Moriya interactions, for example BiFeO3. Our analysisstarts from the ground state with spiral ordering within the layers. Wefind that a spiral structure deforms into an array of multiferroic kinksupon the application of an external magnetic field in the plane of thespiral. These kinks play a crucial role in determining the period of thespiral, leading to a small polarization that is opposite to the FMFlat Spiral ConicalFig. 5 | The response to an externalmagneticfieldHz, applied along the three-foldaxis in NISO. (Upper panel) Spin spiral period calculated from the continuousmodel (red line) and using the analytical formula in ref. 35 (blue line). (Lower panel)magnetically-induced polarization resulting from the isotropic exchange striction.The dotted line indicates phase transition between the flat spiral state and the conicalstate shown in Fig. 6.Fig. 6 | Phase diagram of NISO as obtained by minimization of the continuousmodel, Eq. (37), with a small electric field Ez. Yellow, green, and red areas areindicating flat spiral phase, conical phase, and SF phase, respectively. The tendencytowards FM state is indicated with white in color gradients.https://doi.org/10.1038/s44306-024-00020-9 Articlenpj Spintronics |            (2024) 2:17 6contribution. Importantly, externalmagnetic fields can directly controlthe distances between these kinks. When a greater magnetic field isapplied parallel to the three-fold axis, it induces the SF transition,consistent with experimental observations.Using analytical and numerical models with parameters calculatedfrom first principles, we have identified a three-step phase transition(spiral→ conical spiral→ SF→ FM) under an applied magnetic field.Our ab-initio model shows that the most important mechanism of themagnetically-induced polarization in NISO is the isotropic exchangestriction. During the first phase transition, the polarization undergoes asmall change, closely related to the magnetic kink distance (spiralperiod). The kink density produces a contribution to the polarizationthat opposes the ferromagnetic contribution, resulting in polarizationchange similar to the spiral period change. In the second step, thepolarization curve shows a sharp increase related to the FM componentdevelopment in the xy-plane due to SF contribution. In the third step,the polarization changes significantly, simultaneously with the linearchange of the canting angle θwith themagnetic field. These polarizationchanges imply the possibility of switching between these magneticstructures by an external electric field, enabling the electric control ofmagnetism. The proposed scenario is supported by the excellentagreement with the experimental data on the field dependence of thepolarization.In summary, the work advances the understanding of magnetoelectriceffect in spiral multiferroics. NISO shows a rich phase diagram with flatspiral ground state, magnetic kink arrays, conical kink phase and SF phase.We reveal their connection to the magnetically-induced polarization, andthe possibility of electric switching betweenmagnetic phases via an externalelectric field. Given the importance of themanipulation of the spiralMFs byexternal fields, we expect that our findings will motivate new experimentsand facilitate the applications of spiral MFs in the next-generationmemoryand spintronic devices.MethodsFirst principles calculationsDensity functional theory calculations have been performed for theexperimental crystal structure10 (see Fig. 1a for the hexagonal cell) using arhombohedral cell. These calculations employed norm-conserving pseu-dopotentials within the Quantum ESPRESSO package37. The plane wavecutoff was set to 1088 eV, the Brillouin zone was sampled by a 7 × 7 × 7Monkhorst-Pack k-point mesh.The hopping parameters of Eq. (1) is calculated using the MaximallyLocalized Wannier Fucntion (MLWF) technique38 as implemented in theWannier90 code39–41.The screened Coulomb and exchange interactions of Eq. (1) are cal-culated using constrained RPA technique42, which yields U = 2.3 eV andJH = 0.6 eV.The crystal structures were visualised with VESTA43.Analytical formula for the parameters of spin modelsThe second order perturbation energy with respect to electron hopping inthe 2-orbital model is formulated as:Eð2Þ ¼ � 1ΔEP1;2α;βj α�; ih ∣̂tij∣βþ; j�j2 þ j α�; j�∣̂tji∣βþ; i�j2� �; ð14Þwhere ∣αþ ð�Þ; ii indicates occupied (unoccupied) spin-orbital at site i andΔE =U+ JH. Now, we assumeHamiltonianmatrix elements t̂ij are orderedas ∣1i∣2i (pairs of Kramers’ states). Consequently, the corresponding orbitalket vectors are straightforwardly defined as: ∣1i ¼ 10� �and ∣2i ¼ 01� �.Furthermore, when considering the occupied and unoccupied spin states,we assume them to have general spinor functions:∣�i ¼ � sin θ2 e�iϕcos θ2 !ð15Þand∣þi ¼ cos θ2sin θ2 eiϕ !: ð16ÞThese states have the maximal spin projection in the direction of anassociated classical spin~e ¼ þh ∣~̂σ∣þi ¼ ðsin θ cos ϕ; sin θ sin ϕ; cos θÞ.Next, we decompose general 4 × 4 hopping matrices (with the properphase) into their orbital and spin components, as described by the followingequations:t̂ij ¼ T̂0ij � σ̂0 þXx;y;zγiT̂γij � σ̂γ; ð17Þwith the orbital partT̂ηij ¼tη11ij tη12ijtη21ij tη22ij !; ðη ¼ 0; x; y; zÞ ð18Þwhere σ̂0 is the 2 × 2 identity matrix and σ̂γ; γ ¼ x; y; z are the spin Paulimatrices. Here, we choose the phases such that non-SOC related terms arerepresented solely bypure real coefficients, while SOC terms are representedsolely by pure imaginary coefficients. Then, taking energy difference fordifferent spin orientations and mapping, it is straightforward to findanalytical formual Eq. (6).Fig. 7 | Schematics for the SF transition and theresulting magnetically-induced polarization.a Schematic representation of the SF phase. Thearrows indicate spins in each layer. b Evolution ofthe angle θ after SF transition under the externalmagnetic field as obtained from the MCS.c Polarization change during the transition from SFphase to FM state.L1L2L1L2L1L2Hza b c6 140https://doi.org/10.1038/s44306-024-00020-9 Articlenpj Spintronics |            (2024) 2:17 7The spin model can also be derived with respect to spin operatorsacting on S = 1 multiplets. The procedure is simply finding the correspon-dence between a general spin Hamiltonian:Hs ¼ Ŝi �J⃡ij � Ŝj; ð19Þand the second order perturbation energy:Heff ¼ �XjM1ΔEjMT̂ ij ∣jM�jM�∣T̂ ji: ð20ÞHere,J⃡ij is a 3 × 3 tensor, T̂ ij is a hopping integral defined as: T̂ ij ¼PαβPσσ 0 tαβσσ 0ij ĉyiασ ĉjβσ 0 and ∣jM�represents the intermediate states∣1=2;m1�� ∣1=2;m2�� ∣1=2;m3� 2 S ¼ 1=2 at site j. For instance,0; 1h ∣Heff ∣1; 1i ¼ 1ffiffi2p ðJxzij þ iJyzij Þ. This approach results in the followingexpressions:Jxxij ¼ þ 121ΔEX1;2α;βþtαβ"#ij tαβ#"ij� ��þ tαβ""ij tαβ##ij� ��þ tαβ##ij tαβ""ij� ��þ tαβ#"ij tαβ"#ij� ��� �ð21ÞJxyij ¼ � i21ΔEX1;2α;β�tαβ"#ij tαβ#"ij� ��þ tαβ""ij tαβ##ij� ��� tαβ##ij tαβ""ij� ��þ tαβ#"ij tαβ"#ij� ��� �ð22ÞJxzij ¼ � 121ΔEX1;2α;βþtαβ##ij tαβ"#ij� ��� tαβ#"ij tαβ""ij� ��þ tαβ"#ij tαβ##ij� ��� tαβ""ij tαβ#"ij� ��� �ð23ÞJyxij ¼ � i21ΔEX1;2α;β�tαβ"#ij tαβ#"ij� ��� tαβ""ij tαβ##ij� ��þ tαβ##ij tαβ""ij� ��þ tαβ#"ij tαβ"#ij� ��� �ð24ÞJyyij ¼ þ 121ΔEX1;2α;β�tαβ"#ij tαβ#"ij� ��þ tαβ""ij tαβ##ij� ��þ tαβ##ij tαβ""ij� ��� tαβ#"ij tαβ"#ij� ��� �ð25ÞJyzij ¼ þ i21ΔEX1;2α;βþtαβ##ij tαβ"#ij� ��� tαβ#"ij tαβ""ij� ��� tαβ"#ij tαβ##ij� ��þ tαβ""ij tαβ#"ij� ��� �ð26ÞJzxij ¼ � 121ΔEX1;2α;β�tαβ""ij tαβ"#ij� ��þ tαβ#"ij tαβ##ij� ��� tαβ"#ij tαβ""ij� ��þ tαβ##ij tαβ#"ij� ��� �ð27ÞJzyij ¼ þ i21ΔEX1;2α;β�tαβ""ij tαβ"#ij� ��þ tαβ#"ij tαβ##ij� ��þ tαβ"#ij tαβ""ij� ��� tαβ##ij tαβ#"ij� ��� �ð28ÞJzzij ¼ þ 121ΔEX1;2α;β�jtαβ"#ij j2 � jtαβ#"ij j2 þ jtαβ""ij j2 þ jtαβ##ij j2� �; ð29Þwhere the difference in signs between the second-order energiesarises from particle exchanges. This tensor can generally bedecomposed into isotropic interaction: Jij ¼ � 13 TrJ⃡ij;Dzhaloshinskii-Moriya vectors:~Dij ¼ 12Jyzij � JzyijJzxij � JxzijJxyij � Jyxij0B@1CA, and symmetricanisotropy: ⃡Γij ¼ 12 J⃡ij þJ⃡tij� �� Jij1⃡. However, this approach results inthe exactly same result as the classical vector approach Eq. (6). Theequivalence of the two approaches can be easily demonstrated byconsidering the relationship between the hopping integrals Forexample,tαβ""ij ¼ t0αβij þ itzαβij :Substituting this definition into Eq. (9–17) yields the same formula asEq. (6).Here, we provide an explicit notation for the diagonal elements as anexample:Jxxij ¼ þ 121ΔEP1;2α;β�þ tαβ"#ij�tαβ#"ij�� þ tαβ""ij�tαβ##ij�� þ tαβ##ij�tαβ""ij�� þ tαβ#"ij�tαβ"#ij���¼ 1ΔEP1;2α;β�ðt0αβij Þ2 þ ðtxαβij Þ2 � ðtyαβij Þ2 � ðtzαβij Þ2�;Jyyij ¼ þ 121ΔEP1;2α;β�� tαβ"#ij�tαβ#"ij�� þ tαβ""ij�tαβ##ij�� þ tαβ##ij�tαβ""ij�� � tαβ#"ij�tαβ"#ij���¼ 1ΔEP1;2α;β�ðt0αβij Þ2 þ ðtyαβij Þ2 � ðtxαβij Þ2 � ðtzαβij Þ2�;Jzzij ¼ þ 121ΔEP1;2α;β�� 

tαβ"#ij

2 � 

tαβ#"ij

2 þ 

tαβ""ij

2 þ 

tαβ##ij

2�¼ 1ΔEP1;2α;β�ðt0αβij Þ2 þ ðtzαβij Þ2 � ðtxαβij Þ2 � ðtyαβij Þ2�:Therefore, the isotropic interaction is given asJij ¼ � 13 TrJ⃡ij ¼ 13ΔEP1;2αβ�3 t0αβij� �2þ Trðtαβij � tαβij Þ� �:All the other expression can be restored following the same procedure.A similar theory can be developed for magnetically induced electronicpolarization. Themain idea is to expand theWannier functionswith respectto first-order of the hopping and apply the general theory of electronicpolarization in solids. This expansion corresponds to spin-dependentmodulations of the Wannier density, resulting in changes in the Wanniercenters. Starting point of the theory is the general theory for polarization insolids23:~P ¼ � eVXocciwi ĵrjwi� �; ð30Þwherehwi ĵr∣wii represents the position operator’s diagonal elements inthe Wannier basis, and V is the cell volume. Next, the first-orderperturbation expansion of the Wannier function with respect toelectron hopping is:∣wi�≈∣αþ; ii � 1ΔEXjX1;2β∣β�; j�β�; j�∣̂tji∣αþ; ii: ð31ÞBy substituting Eq. (31) into Eq. (30), we obtain pair and single-ionterms as~P ¼Pi~Pi þP<ij >~Pij. The pair interaction terms are as follows:~Pij ¼ eVΔEP1;2αβαþ; ih ∣~̂rij∣β�; j�β�; j�∣̂tji∣αþ; ii þ αþ; ih ∣̂tij∣β�; j�β�; j�∣~̂rji∣αþ; iihþ αþ; j�∣~̂rji∣β�; i�β�; i�∣̂tji∣αþ; j�þ αþ; j�∣̂tji∣β�; i�β�; i�∣~̂rij∣αþ; j�i:ð32ÞSimilar to thehoppingmatrix, thepositionmatrix canbedecomposed into alinear combination of the Pauli matrices as:~̂rij ¼ ~̂R 0ij � σ0 þXγ0i~̂Rγ0ij � σ̂γ0 : ð33Þhttps://doi.org/10.1038/s44306-024-00020-9 Articlenpj Spintronics |            (2024) 2:17 8Then, it is straightforward to find the pair interaction formula Eq. (3)and the corresponding analytical formula Eq. (7).Single-q spiral ansatzIn the simplest approximation, a spin-spiral state can be characterized by asingle q-vector,~ei ¼ n̂1 cosð~q �~RiÞ þ n̂2 sinð~q �~RiÞ; ð34Þwhere n̂1 and n̂2 are orthogonal vectorswithin the plane of the spiral,~q is thepropagation vector and ~Ri is the vector pointing to the site i. This is anAnsatz of a single-q spiral state. The spiral plane can be characterized by itsnormal vector,n̂? ¼ ð� sin ϕ; cos ϕ; 0Þ; ð35Þwhere ϕ indicates the rotation of the spin-spiral plane. In this case, therelative orientation between δ~q and n̂? defines the spiral type. For instance,δ~q k n̂? is a proper-screw spiral state.Layer AFM chain continuous modelWe express magnetization and AFM order parameter at position x as~MðxÞ ¼~S1ðxÞ þ~S2ðxÞ and~AðxÞ ¼~S1ðxÞ �~S2ðxÞ, respectively. In this for-mulas,~S1ðxÞ and~S2ðxÞ are spins at neighboring layer L1 and L2 and thesesatisfy ~AðxÞ � ~MðxÞ ¼ 0. Then, energy density of the chain at position x isexpressed asδEðxÞ ¼ � J 04 ð∇~AðxÞÞ2 þ ~D � ~AðxÞ× ∇~AðxÞ� �� �� ~H � ~MðxÞ þ J:ð36ÞIn this equation, the first term is the effective isotropic ferromagneticexchange, J 0 < 0, the second one is the antisymmetric DM interaction, thethird termcorresponds to theZeemanenergy due to an externalfield~H, andthe last term is the energy of a ferromagnetic state. The alternating inter-layer AFM exchange coupling in NISO can be considered as intra-layer FMexchanges.Considering inter-layerAFMexchange J, the effective intra-layerFM exchange would be J 0 ¼ �2J . By employing the Fourier expansion ofthe spin, ~AðxÞ ¼ 1ffiffiffiNpPNn¼�N~An eiQnx , we obtain a simplified energyexpression that has to be minimized:Erec ¼PNn¼�N� J 04 Q2n2~An �~A�n þ i~DQn � ~A�n ×~An� �� �� ~H �~A0� �2þ λPl0m0n0p0eiQðl0þm0þn0þp0 Þx~Al0~Am0~An0~Ap0 � 2~An~A�n !;ð37Þwhere the last term is the energy penalty with coefficient λ≫ 1 enforcing aconstraint on the spin length. Similarly, the change of the spin-isotropic partof the polarization per spin can be evaluated as:Pisoz / �1þXNn¼�NQ2n2~An �~A�n: ð38ÞReceived: 17 November 2023; Accepted: 7 March 2024;References1. Kimura, T. et al. Magnetic control of ferroelectric polarization. Nature426, 55–58 (2003).2. Tokura, Y. & Seki, S. Multiferroics with spiral spin orders. Adv. Mater.22, 1554–1565 (2010).3. Tokura, Y., Seki, S. & Nagaosa, N. Multiferroics of spin origin. Rep.Progr. Phys. 77, 076501 (2014).4. Khomskii, D. Classifying multiferroics: mechanisms and effects.Physics 2, 20 (2009).5. Cheong, S.-W. & Mostovoy, M. Multiferroics: a magnetic twist forferroelectricity. Nat. Mater. 6, 13–20 (2007).6. Fiebig, M., Lottermoser, T., Meier, D. & Trassin, M. The evolution ofmultiferroics. Nat. Rev. Mater. 1, 16046 (2016).7. Seki, S., Ishiwata, S.&Tokura,Y.Magnetoelectric natureof skyrmionsin a chiral magnetic insulator Cu2OSeO3. Phys. Rev. B 86,060403 (2012).8. Ruff, E. et al. 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The computations in this study were performed on theNumerical Materials Simulator at the NIMS and the Supercomputer Center,the Institute for Solid State Physics, the University of Tokyo.Author contributionsR.O. performed first-principles calculations, analytical formula derivations,MCS simulations, and 1D continuous model simulations. R.O. and I.S.performed derivation of the multiplet energy in the superexchange theory.R.O. and S.A. performed derivation of the 1D continuous model and theYukawa-like potential. All authors participated in analyzing the data anddiscussions. R.O. wrote themainmanuscript text with contributions from allauthors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s44306-024-00020-9.Correspondence and requests for materials should be addressed toRyota Ono.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in anymedium or format, as longas you give appropriate credit to the original author(s) and the source,provide a link to the Creative Commons licence, and indicate if changeswere made. 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To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024https://doi.org/10.1038/s44306-024-00020-9 Articlenpj Spintronics |            (2024) 2:17 10https://doi.org/10.1038/s44306-024-00020-9http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/ Multiferroic kinks and spin-flop transition in Ni2InSbO6 from first principles Results Basic electronic structure, electronic and spin model for�NISO Magnetic structures in�NISO 1D continuous�model Polarization change during spin flop-to-FM transition Discussion Methods First principles calculations Analytical formula for the parameters of spin�models Single-q spiral�ansatz Layer AFM chain continuous�model References Acknowledgements Author contributions Competing interests Additional information