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Ferromagnetic ferroelectricity due to orbital orderingJ. Phys.: Condens. Matter 38 (2026) 161501 https://doi.org/10.1088/1361-648X/ae5e14OPEN ACCESSRECEIVED2 February 2026REVISED23 March 2026ACCEPTED FOR PUBLICATION10 April 2026PUBLISHED20 April 2026Original content fromthis work may be usedunder the terms of theCreative CommonsAttribution 4.0 licence.Any further distributionof this work mustmaintain attribution tothe author(s) and the titleof the work, journalcitation and DOI.PERSPECTIVEFerromagnetic ferroelectricity due to orbital orderingI V Solovyev∗Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba,Ibaraki 305-0044, Japan∗ Author to whom any correspondence should be addressed.E-mail: SOLOVYEV.Igor@nims.go.jpKeywords: ferroelectricity, ferromagnetism, orbital ordering, Hund’s rulesAbstractRealization of ferromagnetic (FM) ferroelectricity, combining two ferroic orders in a single phase,is the longstanding problem of great practical importance. One of the difficulties is that ferromag-netism alone cannot break the inversion symmetry I . Therefore, such a phase cannon be obtainedby purely magnetic means, as in other multiferroics with complex magnetic order. Here, we showhow it can be designed by considering orbital degrees of freedom. The idea can be traced back toa basic principle of interatomic exchange coupling, which states that an alternation of occupiedorbitals along a bond (i.e. antiferro orbital order) favors FM interactions between the spins. Thenew aspect of this canonical picture is that the antiferro orbital order breaks I , so that the bondbecomes not only FM but also ferroelectric (FE). Then, we formulate basic principles governingthe realization of such a state in solids, namely: (i) the magnetic atoms should not be located ininversion centers, as in the honeycomb lattice; (ii) the orbitals should be flexible enough to changethey shape and minimize the energy of exchange interactions; (iii) this flexibility can be achievedby intraatomic interactions, which are responsible for Hund’s second rule and compete with thecrystal field splitting; (iv) for octahedrally coordinated transition-metal compounds, the mostpromising candidates are iodides with a d2 configuration and relatively weak d–p hybridization.The situation is illustrated on the van der Walls compound VI3, which we expect to be FM FE.1. IntroductionMultiferroicity means a coexistence of two or more ferroic orders within one phase. Thus, literally, itshould be a combination of ferromagnetism, ferroelectricity, ferroelasticity, or any other property withthe prefix ferro. Such a combination is of great practical importance: Because these ferroic properties areintertwined, multiferroics offer an excellent platform for cross-control of polarization by the magneticfield and magnetization by the electric field [1]. The prefix ferro appears to be very important also inthe practical context: the larger the magnetization (polarization), the weaker the magnetic (electric) fieldneeded to achieve this cross-control.However, today the term multiferroicity is understood in a more general sense, when ferroelectricitycoexists with any type of magnetic order and not necessarily the ferromagnetic (FM) one [2–4], whileFM ferroelectrics (FEs) are very rare [5].FM ferroelectricity is expected in materials with two or more sublattices, where one sublattice hostsferromagnetism and another ferroelectricity [6, 7]. Typically, these ferroic properties have differentmicroscopic origin and only weakly depend on each other. A notable exception is SrMnO3 under epi-taxial strain, which was predicted to be simultaneously FM and FE, and both of these properties stemfrom the Mn sublattice [8, 9]. Another possibility is the synthesis of artificial heterostructures, combin-ing layers of FM and FE materials [10].A new route for realizing FM ferroelectricity has been proposed in [11]. The basic idea is traced backto Goodenough–Kanamori–Anderson (GKA) rules for interatomic exchange interactions [12–15], whichstate basically that population of alike orbitals at two sites of the bond (the ferro orbital order) favorsantiferromagnetic (AFM) coupling, while population of unlike orbitals (the antiferro orbital order) will© 2026 The Author(s). Published by IOP Publishing Ltdhttps://doi.org/10.1088/1361-648X/ae5e14https://crossmark.crossref.org/dialog/?doi=10.1088/1361-648X/ae5e14&domain=pdf&date_stamp=2026-4-20https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://orcid.org/0000-0002-2010-9877mailto:SOLOVYEV.Igor@nims.go.jpJ. Phys.: Condens. Matter 38 (2026) 161501 I V Solovyevmake this coupling FM. Nevertheless, what was overlooked in the canonical GKA picture is that, besidesFM coupling, the antiferro orbital order breaks the inversion symmetry I in the bond so that it becomessimultaneously FM and FE. The idea was further adapted for periodic solids, arguing that in certainmaterials the spontaneous antiferro orbital order can yield the FM-FE state. The van der Walls FM semi-conductor VI3 is one of potential candidates for realizing such a state [11, 16].In this article, we further elaborate the basic conditions and perspectives of realizing the orbitallyinduced FM-FE state in real materials. After reminding in section 2 key results of the modern theory ofelectric polarization, in section 3 we will consider possible mechanisms of inversion symmetry break-ing in solids. The main conclusion is that the FM-FE state cannot be realized by magnetic means alone:the FM order is too simple to break I . Then, in section 4, we will turn to the orbital degrees of free-dom and show how, by populating different combinations of orbitals at two sites of the bond, one cancontrol not only the exchange coupling but also the electric polarization in the bond. Thus, the orbitaldegrees of freedom appear to be that additional ingredient, which can help us in realizing the FM-FEstate. In section 5, we will turn to more practical aspects and consider the conditions, which should bemet in order to realize the FM-FE state in solids: type of the lattice, electronic configuration of magneticions, type of the ligand atoms, details of the electronic structure, etc. Particularly, the antiferro orbitalorder can be obtained by minimizing the energy of exchange interactions [17–19], meaning that occu-pied orbitals should be flexible enough to change their shape depending on the spin coupling. This flex-ibility can be achieved by intraatomic interactions responsible for Hund’s second rule, which forces theatomic ground state to have maximal orbital degeneracy and competes with the Jahn–Teller (JT) dis-tortion, acting in the opposite direction and tending to freeze occupied orbitals in one particular con-figuration. Then, in section 6, we discuss results of numerical simulations for VI3, using for these pur-poses realistic model derived from first-principle electronic structure calculations. Finally, in section 7,we summarize our material design strategy for FM FEs and discuss possible implications of the Hund’sphysics to the properties of magnetic materials.2. Theoretical backgroundIn quantum mechanics, the electric polarization is given by the expectation value of the position oper-ator r⃗:P⃗=− eV⟨Ψ |⃗r |Ψ⟩,where −e is the electron charge, V is the volume, and Ψ is the ground state wavefunction. Since Itransforms the polar vector r⃗ to −⃗r, the spontaneous polarization can develop only when I is macro-scopically broken. In this sense, the search of new types of FE materials is essentially the search of newpossibilities how to break I . Some of such possibilities will be briefly reviewed in section 3.According to the modern theory of electric polarization in periodic systems [20, 21], P⃗ can be com-puted either in the k-space, via the Berry connection:P⃗=− ie(2π)3M∑n=1ˆBZ⟨n⃗k|∇⃗k⃗|n⃗k⟩d⃗k, (1)or, equivalently, in the r-space, via the Wannier functions wn for the occupied states:P⃗=− eVM∑n=1ˆr⃗ |wn (⃗r) |2d⃗r. (2)The latter expression is especially important for understanding microscopic origin of magnetoelectriccoupling. Since r⃗ does not depend on spin degrees of freedom, all information about magnetic statedependence of P⃗ is accumulated in wn. Equation (2) is also useful for finding the analytical dependenceof P⃗ on the magnetization in the framework of superexchange theory [22–24].3. Mechanisms of inversion symmetry breaking3.1. Hybridization between bonding and antibonding statesThe first canonical example of FE materials is the so-called d0 perovskites, which include such represent-ative compounds as BaTiO3 and KNbO3 [25]. From the viewpoint of electronic structure in paraelectric2J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevFigure 1. Total and partial densities of states of cubic BaTiO3 and KNbO3 in the local density approximation. The Fermi level isin the middle of the band gap (shown by dot-dashed line).cubic phase, the key aspect of these materials is the strong hybridization between the transition-metald and oxygen p states, which splits them around the Fermi level, forming the bonding O 2p band andantibonding Ti 3d or Nb 4d band (figure 1). Because of this hybridization, both materials are insulators.Moreover, the bonding and antibonding states can be viewed as, respectively, symmetric and antisym-metric states relative to some inversion center. Then, the polar distortion, δQ, mixing these symmetricand antisymmetric states, will additionally shift the occupied O 2p band to the lower energy region, giv-ing a possibility to realize the FE phase with spontaneously broken I . This mechanism was proposedlong ago by Bersuker using model considerations [26]. A transparent illustration was give by Cohen onthe basis of first-principles electronic structure calculations [27].The corresponding energy gain is even in δQ. This is different from the conventional JT effect, whichsplits the degenerate states and, therefore, is odd in δQ [28]. To emphasize this difference, the mechan-ism is called pseudo JT effect. Then, the electronic energy gain should be combined with the energy loss,∼(δQ)2, resulting from the harmonic ion core motion [25]. Depending on parameters, the total energycan have a minimum at finite δQ, signaling that the FE phase is stable and I is broken.However, the d0 perovskites are intrinsically nonmagnetic. The magnetism would require a partialpopulation of antibonding transition-metal band. Therefore, it will reduce the energy gain caused bymixing of bonding and antibonding bands, that will act against the instability towards the FE state.Nevertheless, certain populations of transition-metal states may still lead to the FE instability [29] andaccording to the first-principles calculations the FM ferroelectricity can be indeed realized in the d3 com-pound SrMnO3 under epitaxial strain [8, 9].Another important ingredient is the lone pair 6s2 electrons residing on the A sites of such perovskitesas PbTiO3 and BiMnO3 [6]. The off-centrosymmetric displacements of A sites induce the hybridiza-tion of the occupied 6s2 states with unoccupied states of opposite parity, which additionally stabilizesthe polar phase.3.2. Type-II multiferroicityI can be broken by magnetic order. Such materials are called type-II multiferroics [3]. The canonicalexamples are spin-spiral insulators, where the onset of conical or cycloidal magnetic order gives rise tospontaneous polarization [30, 31].Thus, one can have a unique combination of ferroelectricity and magnetism, which can be used tomutually control each other applying either electric or magnetic field [1]. However, literally, the mul-tiferroicity means something different. It implies the combination of several ferroic orders, so that thematerial should be not simply magnetic, but FM. Therefore, the next question is: Can the FM orderalone break the inversion symmetry?To answer this question, one should understand why certain magnetic textures break I . To be spe-cific, let us consider the centrosymmetric bond connecting two noncollinear spins. This bond can beregarded as a part of cycloidal texture (figure 2). Then, the noncollinear configuration of spins in thebonds can be presented as the superposition of collinear FM and AFM counterparts [24]. While the FMconfiguration is transformed to itself by I about the bond center, the AFM configuration is transformed3J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevFigure 2. Spin spirals and inversion symmetry breaking. (a) Conical and (b) cycloidal magnetic order. q is the propagation vec-tor. (c) Illustration of inversion symmetry breaking by noncollinear spins in centrosymmetric bond.× is the inversion center.The noncollinear spin configuration can be decomposed in ferromagnetic and antiferromagnetic counterparts. The former isinvariant under I, while the latter is invariant under IT . Since I cannot coexist with IT , the inversion symmetry is broken.Reproduced from [24]. CC BY 4.0.to itself by IT , where I is combined with the time reversal T , which additionally flips the spins1.However, I cannot coexist with IT because otherwise T ≡ IIT would be another symmetry opera-tion and the system would become nonmagnetic. The only possibility to resolve this conflict betweenFM and AFM counterparts coexisting in the noncollinear texture of spins is to break I [24]. This is thephenomenological reason why spiral magnetic order can induce the electric polarization. The micro-scopic explanation was proposed by Katsura, Nagaosa, and Balatsky [32] and refined later in the series ofpublications [23, 24, 33, 34].The rule is generic and applies not only to spin-spiral compounds but to all type-II multiferroics.In all these materials, the magnetic texture should involve simultaneously FM and AFM counterparts,transforming via, respectively, I and IT [24]. Therefore, the answer to the above question is no: the FMorder is just too simple to break the inversion symmetry.4. Orbital degrees of freedomThus, the conventional type-II multiferroics cannot be FM: in order to break I by spin degrees of free-dom, it is necessary to destroy the ferromagnetism. This means that in order to achieve the FM-FE state,it is essential to consider additional degrees of freedom, which would control the magnetic coupling inthe bonds and simultaneously break I . Below, we explore the potential of orbital degrees of freedom,which play a very important role in magnetism.4.1. GKA rules and local breaking of inversion symmetryThere are five d orbitals and the magnetic properties of solids strongly depend on which orbitals areoccupied and which are empty, and how the occupied orbitals are aligned relative to each other. Thecanonical example is the phenomenological GKA rules, proposed back in the 1950s [12–15].These rules state essentially that if electrons occupy the same orbitals at two sites of the bond (theso-called ferro orbital order), the exchange coupling will be AFM (figure 3).If they occupy different1 Note I does not change the direction of spin, which is the axial vector.4https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevFigure 3. Examples of ferro and antiferro orbital order around the inversion center. Ferro orbital order tends to stabilize AFMcoupling and preserve the inversion symmetry in the bond. Antiferro orbital order stabilizes FM coupling and breaks the inver-sion symmetry.Figure 4. Illustration of superexchange interactions: (a) example of occupied and unoccupied orbitals in the bond; (b) transferintegrals between occupied and empty states in the case of FM and AFM alignment. The splitting is∆↑↑ = U− JH/2≡ Ũ and∆↑↓ = U+ JH/2.orbitals, aligned perpendicular to each other (antiferro orbital order), the exchange coupling will mostlikely be FM.However, there is one important point, which was overlooked in this canonical GKA picture: theantiferro orbital order not only stabilizes the FM coupling but, since the occupied orbitals across theinversion center become inequivalent, also breaks I in the bond. Thus, at least in the single bond, onecan easily realize the FM-FE state.4.2. Superexchange theoryThe superexchange theory provides the microscopic explanation for the GKA rules [35]. It starts withthe atomic limit, where occupied and empty states are splits by the on-site Coulomb repulsion U andintraatomic exchange interaction JH, and treats transfer integrals between them as a perturbation. U insuch a picture enforces the charge neutrality of the atomic configuration with the integer number ofelectrons, while JH is responsible for Hund’s first rule [36].Then, the exchange coupling J ij is obtained in the second-order perturbation theory for the energydifference between FM and AFM configurations in the bond. For instance, considering the superex-change processes involving the occupied 3x2 − r2 (3y2 − r2) and unoccupied y2 − z2 (z2 − x2) eg orbitalsat the site i (j) (see figure 4), J ij will be given byJij =x[(t12ij)2+(t12ji)2]− 2(t11ij)22Ũ(1+ x),where t11ij =−1/2, t12ij =−√3/2, and t12ji = 0 are the transfer integrals between occupied (1) and unoc-cupied (2) orbitals in terms of the two-center Slater–Koster integral ddσ [37], Ũ= U− JH/2, and5J. Phys.: Condens. Matter 38 (2026) 161501 I V Solovyevx= JH/Ũ2. Thus, the coupling is FM when 2JH > U. The tendency towards ferromagnetism is furtherenhanced if there are other d electrons, which tend to form high-spin state (due to JH) and, therefore,favor the FM configuration when electrons transfer to neighboring sites with the same direction of spin.For instance, when beside eg there are inner t2g electrons, which interact via JH, but do not participate inthe hoppings processes, J ij is described by the same equation but with x= NJH/Ũ, where N is the totalnumber of d electrons. Then, the FM coupling is stabilized when 3N+12 JH > U. Such a situation is real-ized in perovskite manganites [38].These are typical considerations for exchange interactions, which were known for decades.Nevertheless, similar arguments can be applied also to P⃗, starting from the expression (2) of the mod-ern theory of electric polarization [20, 21]. Indeed, in the first-order perturbation theory, the occupiedWannier function at the site i will be given by|wi⟩= |α1i ⟩+∑j ̸=i|α1i→j⟩,where α1i is the ‘head’ residing at the central site i and α1i→j are the ‘tails’ induced by the electron hop-pings at the neighboring sites j. The latter can be evaluated using the perturbation theory as|α1i→j⟩=−t12ijŨ|α2j ⟩.Then, assuming ⟨αai |⃗r |αbj ⟩ ≈ R⃗jδijδab [22] and using equation (2), it is straightforward to find the follow-ing expression for electric polarization in the bond ij:P⃗ij =eV(t12ij)2−(t12ji)2Ũ2(R⃗i − R⃗j). (3)Thus, the polarization is finite if t12ij ̸= t12ji , i.e. when the hoppings between occupied and empty orbit-als are nonreciprocal. Such nonreciprocity is produced by the antiferro orbital order, which makes thedirections i→ j and j→ i inequivalent.4.3. Kugel–Khomskii theoryKugel–Khomskii theory is basically a generalization of the superexchange theory, treating spin andorbital degrees of freedom on an equal footing [17–19]. If orbital degeneracy is lifted by the lattice dis-tortion, deciding which orbitals are occupied and which are empty, it is reduced to the conventionalsuperexchange theory, specifying the exchange coupling for the given configuration of occupied orbitals.In this case, the lattice distortion acts as an external constraining field in the spin–orbital (OS) system.However, if the lattice distortion is weak, there is certain element of self-organization, when the systemdecides the type of the orbital alignment by minimizing the energy of exchange interactions with respectto the orbital degrees of freedom for the given configuration of spins. If this orbital alignment appears tobe of the antiferro type, it stabilizes the FM coupling and induces the electric polarization as describedabove.The basic idea can be again illustrated by considering the superexchange processes between two egorbitals in the single bond oriented along x (see figure 4). If the eg levels are degenerate, the occupiedorbitals can be searched in the most general form, as a linear combination of the |x2 − y2⟩ and |3z2 − r2⟩states:|α1i ⟩= cosϑi|x2 − y2⟩+ sinϑi|3z2 − r2⟩.The unoccupied orbital is orthogonal to |α1i ⟩:|α2i ⟩=− sinϑi|x2 − y2⟩+ cosϑi|3z2 − r2⟩.2 The corresponding exchange energy in the bond is defined as−Jijei · ei, where ei is the unit vector in the direction of spin magneticmoment at the site i. Furthermore, following previously adopted notations [24], we use vector symbols, such as P⃗, to denote polarvectors and bold symbols, such as e, to denote pseudovectors.6J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevFigure 5. Example of the antiferro orbital order realized in perovskite (a) and honeycomb (b) lattice. In the perovskite lattice, thedirections of polarization induced in each bond are shown by arrows. The inversion centers are shown by circles (which coincidewith the positions of oxygen atoms in undistorted perovskite structure).Then, using Slater–Koster parametrization in terms of ddσ [37], one can find the following expres-sions for the transfer integral:t12ij =12sin(ϑi −ϑj)− 14sin(ϑi +ϑj)−√34cos(ϑi +ϑj)and the energy gain ∆E↑↑ij =− (t12ij )2+(t12ji )2Ũcaused by the electron hoppings between occupied and unoc-cupied orbitals in the bond for the FM configuration of spins:∆E↑↑ij =−4√3sin4ϑ+ cos4ϑ− 2cos2∆ϑ+ 4Ũ,where ϑ=ϑi+ϑj2 and ∆ϑ= ϑi −ϑj. Minimizing ∆E↑↑ij with respect to ϑ and ∆ϑ, it is straightforward tofind that ϑ= 14 arctan(4√3) and ∆ϑ= π2 , meaning that the orbital order is indeed of the antiferro type.Then, the electric polarization (the x component) in the bond can be found from equation (3) asPxij =− eaVŨ2cos(2ϑ+π3),where a is the bondlength.5. Towards practical realizationIn the previous section we have seen that the FM-FE state can be easily realized in the single bond bythe antiferro orbital order. In this section, we will discuss whether such state can be realized in realmaterials having periodic structure.5.1. Crystal lattice: perovskite versus honeycombThe orbital order has been intensively studied in magnetoresistive manganites and other transition-metal compounds crystallizing in the perovskite structure [39–45]. Particularly, the antiferro orbitalorder is believed to be responsible for the FM character of exchange interactions in LaMnO3 [13, 40],YTiO3 [42], LaVO3 [43], and YVO3 [44]. However, in all these materials the magnetic atoms are loc-ated in the inversion centers. Therefore, although the electric polarization can be induced in each bond,there will always be another bond with the opposite direction of electric polarization, as explainedin figure 5(a). Thus, most likely, these perovskite materials will be antiferroelectric. They can host very7J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevFigure 6. Atomic level splitting for electronic configurations d1 (left) and d2 (right). The octahedral environment of ligandsaround the transition-metal sites will split the 3d levels into threefold degenerate t2g and twofold degenerate eg states, corres-pondingly in the lower and upper part of the spectrum (the so-called 10Dq splitting). The Jahn–Teller distortion tends to lift theorbital degeneracy and further split the t2g manifold into twofold degenerate e ′g and nondegenerate a1g states (∆tr splitting). Thisdistortion works in the opposite directions for d1 and d2, stabilizing, respectively, a1g and e ′g states. The d2 configuration is sub-jected to atomic Hund’s second rule effects, which tend to reenforce the ground state with maximal multiplicity and reverse theorder of a1g and e ′g states.interesting properties, such as weak ferromagnetism, net orbital magnetization, etc [46], but hardly suit-able from the viewpoint of FE applications.The honeycomb lattice seems to be more promising. In this case, I connects two magnetic sublat-tices, while the atoms themselves are located not in the inversion centers, as explained in figure 5(b).Therefore, if we succeed in realizing the antiferro orbital order between the sublattices, our system canbecome simultaneously FM and FE.5.2. Electronic configuration: d1 versus d2The next important question is how to realize the antiferro orbital order on the honeycomb lattice.First, we explore the conventional picture based on the JT distortion, which takes place, for instance,in LaMnO3 [47]. We assume that the transition-metal atoms are in the octahedral environment, whichresults in the 10Dq splitting of 3d levels into threefold degenerate t2g and twofold degenerate eg states, asexplained in figure 6. Then, we consider the situation when there is either one or two 3d electrons resid-ing in the t2g shell, corresponding to the electronic configurations d1 and d2. Thus, the system is subjec-ted to the JT distortion, which further splits degenerate t2g levels into twofold degenerate e ′g and nonde-generate a1g states. The corresponding parameter is denoted as ∆tr. Moreover, the JT theorem states inthis respect that the distorted system should have a nondegenerate ground state [48]. Therefore, the JTdistortion should work in opposite directions for the configurations d1 and d2, stabilizing, respectively,a1g and e ′g states (figure 6). Another important point is that for two sites connected by I the JT distor-tion is expected to be the same. Thus, the JT mechanism can stabilize ferro orbital order but not theantiferro one and alone does not break the inversion symmetry. Furthermore, since the ground state isnondegenerate, the orbital degrees of freedom are frozen. This suppress the Kugel–Khomskii mechanismof the orbital ordering when the latter is driven by superexchange processes.Fortunately, the JT distortion is relatively weak in the t2g systems so that other mechanisms can eas-ily compete with it. One of such prominent mechanisms is Hund’s second rule, driven by intraatomicinteractions. In isolated atoms, the proper interaction parameter (the Racah parameter) is given byB= 9F 2−5F 4441 , in terms of the radial Slater integrals F2 and F4 [49, 50]. For comparison, the intraatomicexchange coupling responsible for Hund’s first rule is JH = F 2+F 414 . Since F 4 ∼ 0.63F 2 [36], the ratio BJHisabout 0.1, which naturally explains the hierarchy of atomic Hund’s rule, where the second rule should beconsidered only after the first one. Since for 3d ions JH ∼ 1 eV [36, 39], B is about 0.1 eV. The importantpoint is that typical crystal-field splitting of t2g levels caused by the JT distortion is also of the order of0.1 eV [45]. Therefore, these two mechanisms can compete with each other.8J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevFigure 7. Crystal structure of VI3 and V2O3. (a) Linked network of VI6 octahedra in VI3. (b) Vanadium atom positions in VI3.Two sublattices are shown by different colors. d∥V–V and d⊥V–V are interatomic distances in and between the honeycomb planes. Theinversion centers are denoted by×. (c) Linked network of VO6 octahedra in V2O3. (d) Vanadium atom positions in V2O3, wherefour sublattices are shown by different colors.While the JT distortion tends to quench the orbital degrees in one particular configuration, Hund’ssecond rule acts in the opposite direction and tends to realize the ground state with maximal orbitaldegeneracy. For the d2 configuration, it can revert the order of the a1g and e ′g levels. Therefore, in theatomic limit, one electron will reside on the nondegenerate a1g orbital, while another one will be in thedegenerate subspace spanned by two e ′g orbitals (see figure 6). A more rigorous picture can be obtainedby considering exact two-electron states [11], following the arguments of Tanabe and Sugano [51]. Thiswill reactivate the Kugel–Khomskii mechanism of the orbital ordering so that the occupied orbitals canadjust their shape to minimize the energy of exchange interactions and, hopefully, realize the antiferroorder, breaking the inversion symmetry. Simple toy-model considerations supporting this idea can befound in [11].The Hund’s rules are essentially many-electron effects. They do not operate in the one-electron con-figuration d1, where the JT distortion is the only mechanism that splits the t2g levels. Therefore, in orderto realize the antiferro orbital order on the honeycomb lattice, it is important to consider two-electron(or other) systems, in which Hund’s rule effects are operative.5.3. Details of electronic structure: VI3 versus V2O3In this section we will consider two potential d2 candidates, VI3 and V2O3, and argue that the antiferroorbital order stabilizing the FM-FE state can be probably realized in VI3 but not in V2O3. The reasonlies in details of the electronic structure, which is also related to the differences in the crystal structureand types of the ligand atoms.VI3 is the van der Waals material. At room temperature, it has R3 structure [52], while V2O3 crystal-lizes in corundum R3c structure [53] (figure 7). The main structural motif of both VI3 and V2O3 is thehoneycomb planes formed by V3+ ions belonging to two different sublattices, which are connected byI . The ions in the same sublattices are connected by threefold rotations and translations. In V2O3, thesituation is additionally complicated by the fact that there are two pairs of the V3+ sublattices formingalternating honeycomb planes as shown in figure 7(d). In any case, as long as we consider only the V3+ions, the stacking of honeycomb planes looks very similar in VI3 and V2O3. The main difference lies inthe packing of the VI6 and VO6 octahedra in the van der Waals and corundum structure. In the formercase, the octahedra form edge-sharing network in the honeycomb planes, but remains disconnected inthe c direction. However, in the corundum structure of V2O3, some of the VO6 octahedra belonging toadjacent honeycomb planes share their faces [54]. It has strong impact on the V–V distances in (d∥V–V)and between (d⊥V–V) the planes. In the van der Waals structure of VI3, d⊥V–V = 6.55Å is substantially lar-ger than d∥V–V = 3.95Å, whereas in V2O3, d⊥V–V = 2.71Å is even shorter than d∥V–V = 2.88Å. Thus, in9J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevFigure 8. Total and partial densities of states of VI3 and V2O3 in the local density approximation. The Fermi level is at zero energy(shown by dot-dashed line). The 10Dq splitting leads to the formation of two distinct V 3d bands denoted as t2g and eg .Table 1. Parameters of model Hamiltonian for V2O3 and VI3 (in eV): trigonal splittingbetween e ′g and a1g states (∆tr), 10Dq splitting between t2g and eg states, on-site Coulombrepulsion U, intraatomic exchange coupling JH responsible for Hund’s first rule, and Racahparameter B responsible for Hund’s second rule. Reproduced from [11]. CC BY 4.0.compound ∆tr 10Dq U JH BV2O3 0.15 2.31 3.05 0.85 0.09VI3 0.01 1.49 1.21 0.75 0.07V2O3, d⊥V–V is the additional parameter of the crystal structure, which strongly influences the orbitalorder and can even result in the formation of molecular states shared by two V3+ [55, 56]. Since the I5p states are rather diffuse, the d–p hybridization is weaker in VI3, which is also reflected in longer V–Vdistances.Furthermore, the electronic structure appears to be rather different in VI3 and V2O3 (see figure 8).Using the electronic structure in the local density approximation (LDA), one can construct the effect-ive model for the V 3d bands in VI3 and V2O3, which are primarily responsible for the magnetism.Namely, the one-electron parameters can be related to the matrix elements of LDA Hamiltonian in theWannier basis for the V 3d bands located in the interval [−0.5,2] eV for VI3 and [−1,4] eV for V2O3.The effective Coulomb interactions can be evaluated using constrained random-phase approximation(cRPA) [57]. Technical details can be found in [45] and [11].The model parameters depend on the electronic structure, which explains the sharp differencebetween VI3 and V2O3 (see figure 8 and table 1). Namely:• The interaction parameters for V 3d bands in cRPA depend on how well they are screened by otherbands, particularly by O 2p and I 5p bands. The closer the bands are, the stronger the screening.Thus, the parameters U, JH, and B are generally weaker in VI3 (table 1). Moreover, JH and B are typ-ically less screened than U. Therefore, the ratio JHU is substantially larger in VI3, which is importantfor stabilizing FM interactions [19];• The trigonal splitting between e ′g and a1g states (∆tr) is substantially smaller in VI3, so that ∆trB < 1in VI3 but > 1 in V2O3 (table 1). This will reactivate the Kugel–Khomskii mechanism and drive theformation of antiferro orbital order in VI3 but not in V2O3;• 10Dq splitting is also smaller in VI3. This facilitates the mixing of the e ′g and eg states belonging tothe same representation by Hund’s rule interactions and further minimizes the energy of these inter-actions. This further support the Kugel–Khomskii mechanism and formation of antiferro orbital orderin VI3.Thus, VI3 appears to be a good candidate for spontaneous breaking of inversion symmetry by theantiferro orbital order and realizing the FM-FE state. The d2 configuration itself does not necessarilyguarantee the emergence of antiferro orbital order, where many things depend on details of the elec-tronic structure.10https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevFigure 9. Total and partial densities of states as obtained in the Hartree–Fock approximation for the ferromagnetic state, wherethe Racah parameter B responsible for Hund’s second rule was set to either 0 (left) or 0.07 eV (right). The Fermi level is at zeroenergy (the middle of the band gap). Reproduced from [11]. CC BY 4.0.The lattice distortion seems to be also different in VI3 and V2O3. Below 150 K, the corundumstructure of V2O3 undergoes a monoclinic distortion [58], which further stabilize the e ′g levels, followingthe JT theorem [48]. The experimentally observed AFM order can be successfully explained by detailsof this monoclinic structure [56]. On the other hand, the lattice distortion in VI3 is a matter of contro-versy, which will be briefly discussed in section 7. It can be the sign of orbital degeneracy, which canlead to the realization of the FE-FM state as we propose [11].6. What can one expect from VI3?Finally, we turn to numerical mean-field Hartree–Fock (HF) calculations for the model, which wasobtained for the V 3d bands of VI3 as described above [11]. Some of the model parameters are sum-marized in table 1. The total energy in the HF approximation is expressed in terms of the one-electrondensity matrix at each site of the lattice, n̂= [nσσ′ab ], which is calculated self-consistently, where a and bare the orbital indices (two eg , one a1g, and two e ′g), while σ and σ ′ are the spin indices (↑ or ↓). Thenondiagonal matrix elements with respect to ↑ and ↓ are typically induced by relativistic spin-orbit (SO)interaction. Otherwise, n̂ is diagonal. The details can be found in [45].The corresponding densities of states for the FM state with and without the on-site interactionsresponsible for Hund’s second rule are shown in figure 9. The strength of these interactions is controlledby the Racah parameter B. As expected, when B= 0, the type of occupied states is solely determinedby weak trigonal splitting ∆tr, so that two 3d electrons tend to occupied twofold degenerate e ′g states.Therefore, the ground state appears to be nondegenerate. The a1g states are located mainly in the unoc-cupied part and mixed with e ′g ones by intersite hoppings. However, the situation changes dramatic-ally when B is finite, as schematically explained in figure 6. In this case, two 3d electrons reside on thea1g orbital and one of the e ′g orbital. Thus, the ground state is degenerate. The shape of the occupiede ′g states is controlled by superexchange interactions, which further lower the energy via the antiferroorbital ordering [17–19].Similar picture can be obtained using more sophisticated dynamical mean-field theory [11].6.1. Orbital ordering and inversion symmetry breakingLet us start with restricted HF calculations without SO interaction, where we additionally constrainthe form of n̂ to satisfy the R3 symmetry of VI3 lattice. Technically, n̂ is averaged by the matrices ofthreefold rotations in each sublattices to guarantee the R3 symmetry. Then, n̂ is averaged over thesublattices to guarantee the inversional symmetry. This R3 solution is regarded as the reference point(figure 10).Then, we relax the constraints. First, we turn off the rotational constraint and average n̂ onlyover the sublattices, enforcing the inversional invariance. The atoms within the sublattices areconnected by translations, but no longer by the threefold rotations. The corresponding symmetry11https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevFigure 10. Orbital ordering in the ferromagnetic state of VI3 as obtained in model Hartree–Fock calculations by enforcing theoriginal trigonal R3 symmetry, the triclinic P1 symmetry, and fully relaxing the symmetry (P1). The crystallographic inversioncenters are denoted by×. To sublattices of honeycomb lattice are displayed by different colors.∆E is the corresponding energychange relative to the state with the R3 symmetry. Reproduced from [11]. CC BY 4.0.Figure 11. Spin-wave stiffness tensors (in meVÅ2) for the orbital states of the R3, P1, and P1 symmetry, and corresponding spin-wave dispersions near the Γ-point of Brillouin zone. Reproduced from [11]. CC BY 4.0.of the orbital order is P1. It changes the shape of the occupied orbitals, but only slightly lowers theenergy.Finally, we relax the inversional constraint and treat n̂ in two sublattices as independent variables.The corresponding symmetry is P1. In comparison with the P1 state, the occupied orbitals in two sub-lattices are additionally rotated relative to each other, so that I becomes broken, resulting in significantenergy gain (figure 10).The next important question is whether the FM spin order is stable or not. The answer depends onthe spin-wave stiffness D̂, which can be evaluated using linear response theory for interatomic exchangeinteraction [59]. In the present context, it is basically a perturbation theory with respect to infinitesimalrotations of spins. If D̂ is positive definite, the FM order is stable. If not, it is unstable. The results aresummarized in figure 11. One can clearly see that the FM state is unstable if the orbital configurationrespects the R3 symmetry of the lattice. Nevertheless, lowering the orbital symmetry will stabilize the FMstate.12https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevFigure 12. Results of Hartree–Fock simulations with spin–orbit interaction and magnetic field. (a) Directions of spin (MS) andorbital (ML) magnetic moments without magnetic field. Two sublattices are denoted by red and blue colors. The numerical valuesofMS andML are given in parentheses. The crystallographic inversion center is denoted by×. (b) Magnetic-field dependenceof magnetization (top) and electric polarization (bottom). The field is applied parallel to z.MzS,MzS +MzL, and Pz are the val-ues of, respectively, spin magnetic moment, total magnetic moment, and electric polarization along z. Reproduced from [11].CC BY 4.0.6.2. Magnetic field control of electric polarizationIn the previous section we have seen that, in the FM state of VI3, I can be broken by the orbital order.The next important questions are: (i) how large is the electric polarization? (ii) How can it be controlledby a magnetic field?The key point here is that the orbital order breaks not only inversional but also the threefold rota-tional symmetry. Therefore, the magnetic moments are not necessarily aligned perpendicular to the hon-eycomb plane or lie in this plane (figure 12). Since the rotational symmetry is broken by the orbitalorder, the a1g and e ′g states do not longer belong to different representations and are allowed to mix.The angle between magnetic moments and the hexagonal z-axis is induced by the SO interaction anddepends on the degree of this mixing. This angle can be also controlled by the external magnetic fieldH= (0,0,H) along z. Therefore, H can be also used to control the electronic degrees of freedom, suchas the a1g–e ′g mixing and the polarization P⃗. This is the main mechanism of magneto-electric coupling inthe symmetry-broken state of VI3. Furthermore, besides spin magnetic moment, |MS| ∼ 2 µB, there is anappreciable orbital magnetic moment, |ML| ∼ 0.5 µB, induced by the SO coupling and further enhancedby electron–electron interactions on V sites [60, 61]. The inversion symmetry breaking gives rise to aneffective Dzyaloshinskii–Moriya interaction, which is responsible for small canting of magnetic momentsbetween different sublattices (figure 12). Contrary to regular Dzyaloshinskii–Moriya interactions [62,63], caused by off-centrosymmetric displacements of intermediate ligand sites [64], these interactions areinduced solely by antiferro orbital ordering.Results of numerical simulations in the magnetic field are summarized in figure 12. The electricpolarization was evaluated using Berry-phase theory, which was applied to the model Hamiltonian inthe HF approximation [65]. Since threefold rotational symmetry is broken, P⃗ is allowed to have all threecomponents. Here, we focus on the behavior of Pz, which dominates over Px and Py [11]. The mag-netic field dependence of spin and orbital magnetic moments is characterized by the hysteresis loop,where the field applied in the positive direction of z leads to the saturation of magnetization, while inthe opposite direction it leads to the reorientation of magnetization at around H∼−7T. CorrespondingPz has a characteristic butterflylike shape and undergoes the jump ∆Pz ∼ 2.4µCcm−2 at the point ofreorientation of the magnetization. This polarization jump is comparable with the one observed inCaBaCo4O7 (∆P∼ 1.7µCcm−2), which is believed to be the largest polarization change induced bymagnetic field [66].13https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 38 (2026) 161501 I V Solovyev7. Summary and outlookFirst, let us summarize main principles for realizing orbitally induced FM-FE state, which we propose.(i) The magnetic atoms should not be in the inversion centers. Different atoms in the lattice can beconnected by I , and this I can be broken by antiferro orbital order. One promising example of suchlattice is the honeycomb plane, considered in the present work. Of course, there are other possibilit-ies including corundum, some hexagonal and monoclinic lattices realized, for instance, in the P63/mmcphase of BaCrO3 [67] and P2/c phase of CoWO4 [68].(ii) The d2 configuration appears to be rather unique, at least for the octahedral environment. Twoelectrons are required to activate Hund’s second rule and enforce the orbital degeneracy, so that theoccupied orbitals could freely adjust their shape and minimize the energy of exchange interactions viathe orbital ordering. From this point of view, other configurations are less promising. The d1 config-uration can be excluded because single electron is not subjected to the Hund’s rule physics. The levelsplitting in this case is controlled solely by the JT distortion, which lifts the orbital degeneracy, prevent-ing the system from formation of the antiferro orbital order. A similar situation occurs for the d4 con-figuration in the high spin-state, where there is only one hole in the majority-spin shell and, therefore,no room for realizing Hund’s second rule. Furthermore, the d4 configuration is subjected to strong JTdistortion [47]. The d3 configuration is not orbitally active because of large 10Dq splitting. Thus, consid-ering less than half filled shell in octahedral environment, the only promising configuration seems to bed2, which can be realized in Ti2+, V3+, and Cr4+.(iii) The main parameters controlling the ground state of the d2 ions are ∆tr, B, and 10Dq. Thecrystal-field splitting ∆tr is caused by the JT distortion, which tends to lift the orbital degeneracy. TheRacah parameter B is responsible for Hund’s second rule, which acts in the opposite direction, yieldingthe ground state with maximal orbital degeneracy. The octahedral splitting 10Dq does not alter the orderof t2g levels directly. However, since Hund’s interactions mix the e ′g and eg states, which are separated by10Dq, the strength of the Hund’s rule coupling effectively depends on 10Dq. Hund’s second rule doesnot apply to isolated t2g manifold in the limit of large 10Dq: the situation is similar to p electron system,where for the high-spin state there is always only one possible orbital state. Therefore, mixing of the e ′gand eg states is crucial for realizing Hund’s second rule. While B is not sensitive to the crystal environ-ment, ∆tr and 10Dq strongly depend on details of the crystal structure and type of the ligand atoms.For our purposes, one would like to have smaller ∆tr and 10Dq. Since they depend on the hybridiza-tion between transition-metal d and ligand p states, the I− ions seem to be more promising than O2−:the I 5p states are more diffuse, the V–I distance is significantly larger and, therefore, the hybridizationbetween V 3d and I 5p states is substantially weaker.(iv) Another interesting option is the d7 configuration of Co2+ ions in the honeycomb lattice. Suchmaterials are well know and widely discussed as potential candidates for realizing the Kitaev quantumspin liquid state [69, 70]. The canonical examples are BaCo2(AsO4)2 [71] and Na2Co2TeO6 [72]. Thecorresponding Kitaev model is formulated for the Co2+ ions having pseudospin-1/2 doublet in theground state, which can be realized in the large 10Dq limit. However, in order to realize the antiferroorbital order on the honeycomb lattice via Hund’s second rule, we need the opposite situation with rel-atively small 10Dq, which will mix the pseudospin-1/2 doublet with other states. Thus, materials suitablefor the FM-FE state are generally unsuitable for the Kitaev physics (and vice versa).To date, VI3 remains the only potential candidate for realizing the orbitally induced FM-FEstate [11]. Nevertheless, the experimental situation is rather controversial. VI3 is indeed a ferromagnetwith relatively high Curie temperature, TC ≈ 50K [73]. Besides the FM transition, VI3 exhibits at leasttwo structural phase transitions, at around 78 and 32K [74, 75]. Details of these transitions is a mat-ter of dispute: there exist different proposals regarding crystal structure changes, the direction of thesechanges, as well as their interconnection with the magnetic properties of VI3 [11]. There are several sug-gestions that the vanadium atoms across inversion centers in honeycomb layers become inequivalent,either structurally [76] or magnetically [77], meaning that these inversion centers are broken and thematerial can be potentially FE. In any case, the experimental data seems to suggest that the crystal struc-ture of VI3 is rather fragile, which is consistent with our main idea that the orbital degrees of freedomin VI3 remain active and the decision which orbitals become occupied and which remain empty maydepend on tiny balance between several factors, including the experimental conditions.It remains an open question whether the above principles (i)–(iv) are sufficient for realizing the FM-FE ground state in VI3 and related materials. The orbital degeneracy is the very challenging theoreticalproblem and there are various scenarios of how this degeneracy is resolved and what will be the con-sequences for the properties of real materials [78]. Nevertheless, we believe that the FM ferroelectricity isone the plausible scenarios, that should be carefully considered alongside with others.14J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevBesides FM ferroelectricity, there are other interesting fundamental issues related to the Hund’srule physics. The importance of Hund’s first rule in the physics of strongly correlated materials is wellrecognized today [79]. These effects are driven by interatomic exchange coupling JH. However, the con-sequences of Hund’s second rule, driven by the Racah parameter B, remain largely unexplored. Dueto the hierarchy of atomic Hund’s rules, reflected in the condition B∼ 0.1JH, the Hund’s second ruleeffects should emerge on much smaller energy scale, which posses a challenge for numerical simulations.Nevertheless, the implication of these effects to the physics of strongly correlated materials is ratherinteresting, as they provide a possibility for realizing new phenomena, such as FE ferroelectricity. Theyalso provide a new look on other canonical problems.Particularly, orbital fluctuations were and continue to be the hot topic in the physics of t2g perovskiteoxides, like LaVO3 and YVO3 [78]. One of the key questions is how well these fluctuations are quenchedby the lattice distortions, which are typically present in these systems. The analysis of the crystal-fieldsplitting, derived from the electronic structure calculations, suggests that it can be quite strong [45].However, this analysis was typically based on the minimal three-orbital model constructed for the t2gbands, which excludes the effects of Hund’s second rule. On the other hand, Hund’s second rule isexpected to play an important role in these two-electron materials and compete with the crystal field.Thus, the problem of orbital fluctuations in LaVO3 and YVO3 can be reconsidered on a new level, takinginto consideration the Hund’s second rule effects in a more general model constructed simultaneouslyfor the t2g and eg bands. One of the key parameters of such model controlling the strength of effectiveinteractions responsible for Hund’s second rule will be 10Dq.Another seminal problem is how to improve the exchange-correlation functional in first-principleselectronic structure calculations in order to deal with the orbital magnetism. The density functionaltheory, providing general foundation for modern electronic structure calculations, is formally exact.However, to be practically applicable, it is supplemented with additional approximations, such as localspin density approximation (LSDA) or generalized gradient approximation (GGA), which largely relyon the model of electron gas and fail to account properly for the physics of intraatomic interactions,including those responsible for Hund’s second rule. Very often, the orbital magnetization calculatedwithin LSDA or GGA is severely underestimated. In order to circumvent this problem, one popularstrategy was to mimic the effects of Hund’s second rule in electronic structure calculations, by addinga phenomenological term, which would (i) be proportional to the Racah parameter B and (ii) tend toenhance ML [80–82]. The typical choice is −BM2L [80, 81], similar to the Stoner theory of spin magnet-ism, though there were also other suggestions [82]. Another contribution to the orbital magnetizationis driven by the on-site Coulomb repulsion U [60, 61, 83]. It does not contribute to the energy of isol-ated atoms with the given integer number of electrons [36], but emerges in solids, where the atoms canfreely exchange electrons. The key question in this context is how to properly evaluate the screening ofU, especially in metallic systems [84].By treating U as an adjustable parameter, as is frequently done in electronic structure calculations,one can easily reproduce the experimental magnetization. However, it does not bring us closer to micro-scopic understanding of this problem because there can be other interactions controlling the valueof ML. Particularly, what is the relative importance of on-site Coulomb repulsion U and interactionsresponsible for Hund’s second rule? In this respect, one should clearly understand that the proposedcorrection −BM2L is totally empirical and does not deal with the true physics of atomic Hund’s rules.For instance, it will enhance ML even when there is only one electron, which does not obey any Hund’srules. However, it does not mean that Hund’s rule interactions cannot contribute to ML in principle.A much better solution of the problem can be obtained by considering intraatomic exchange interac-tion energy in the HF approximation, which treats on an equal footing all three parameters: U, JH, andB [60, 61]. The applications can be also rather interesting. For instance, one can expect that the orbitalmagnetization in the d1 and d2 materials should behave differently: in the former case, it is controlledsolely by Coulomb U, while in the latter case there will be an additional contribution associated withHund’s second rule.AcknowledgmentsI am indebted to Ryota Ono and Sergey Nikolaev for collaboration on earlier stages of this project [11].MANA is supported by World Premier International Research Center Initiative (WPI), MEXT, Japan.15J. Phys.: Condens. Matter 38 (2026) 161501 I V SolovyevData availability statementThe data cannot be made publicly available upon publication because they are not available in a formatthat is sufficiently accessible or reusable by other researchers. 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Introduction 2. Theoretical background 3. Mechanisms of inversion symmetry breaking 3.1. Hybridization between bonding and antibonding states 3.2. Type-II multiferroicity 4. Orbital degrees of freedom 4.1. GKA rules and local breaking of inversion symmetry 4.2. Superexchange theory 4.3. Kugel–Khomskii theory 5. Towards practical realization 5.1. Crystal lattice: perovskite versus honeycomb 5.2. Electronic configuration: d1 versus d2 5.3. Details of electronic structure: VI3 versus V2O3 6. What can one expect from VI3? 6.1. Orbital ordering and inversion symmetry breaking 6.2. Magnetic field control of electric polarization 7. Summary and outlook References