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Igor Solovyev, Sergey Nikolaev, Akihiro Tanaka

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Altermagnetism and weak ferromagnetismnpj | quantummaterials ArticlePublished in partnership with Nanjing Universityhttps://doi.org/10.1038/s41535-026-00900-9Altermagnetism and weakferromagnetismCheck for updatesIgor Solovyev1 , Sergey Nikolaev2 & Akihiro Tanaka1Using a realistic model relevant to La2CuO4 and other altermagnetic perovskites, we studyinterrelations between weak spin ferromagnetism, anomalous Hall conductivity, σxy, and net orbitalmagnetization M. All of them are intertwined with the vector of Dzyaloshinskii-Moriya interactions.Nevertheless, while weak spin ferromagnetism is induced by interactions having the same sign in allequivalent bonds,σxyandMare related to sign-alternating interactions,whichdonot contribute toanycanting of spins. The microscopic model remains invariant under the symmetry operation fSjtg,combining the shift t of antiferromagnetically coupled sublattices to each other with the spin flip S.Thus, the band structure remains spin-degenerate, but the time-reversal symmetry is broken,providing a possibility to realize σxy in antiferromagnetic substances. The altermagnetic splitting ofbands, breaking fSjtg, does not play a major role in the problem. More important is the orthorhombicstrain, responsible for finite σxy and M.Preface Daniel Khomskii had a unique ability of finding the logic behindseemingly complicated phenomena and internal connections between theirdifferent facets. He substantially enriched the conventional theory ofmagnetism by decorating it with intertwined orbital, lattice, and otherdegrees of freedom. While basically an observer in the field of electronicstructure calculations, hemade enormous contributions to it by providing arational explanation for many puzzles and finding simplified toy modelscapturing the essence of complex computer simulations. This article, whichillustrates the potential of the toy model analysis and explains the deepintertwining between new and old concepts of unconventional anti-ferromagnetism, known as, respectively, altermagnetism and weak ferro-magnetism, is a tribute to his memory.Altermagnetism is regarded as a new phase of matter, where a nearlyantiferromagnetic (AFM) alignment of spins coexists with the spin splittingof bands and robust time-reversal symmetry (T ) breaking,which are typicalfor ferromagnetic (FM) systems1–4. Nevertheless, this new classificationraises new questions, especially on how it fits into our previous knowledgeonAFMmaterials. It is certainly true that the lifting of Kramers’ degeneracyof AFM bands, which was predicted largely due to development of density-functional theory (DFT) calculations, is a new aspect of the problem5–12. Onthe other hand, the possibility of breaking T in certain classes of AFMsystems has been known for decades. As was pointed out byDzyaloshinskii13, using phenomenological symmetry arguments, thematerials whose magnetic unit cell coincides with the crystallographic onepresent a special type of antiferromagnetism, giving rise to such phenomenaas weak ferromagnetism14, piezomagnetism15, and magnetoelectricity16. Avery detailed classificationwas given byTurov17,18, who argued that there aretwo major classes of unconventional antiferromagnets, depending onwhether the spatial inversion I enters the magnetic group in combinationwith T or alone. The first scenario corresponds tomagnetoelectricity, whilethe second one, which encompasses weak ferromagnetism and piezo-magnetism, has clear similarity to what is now called “altermagnetism”.Canonically, weak ferromagnetism refers to net spin magnetic moments inotherwise AFM substances, while altermagnetism emerged from the ana-lysis of the anomalous Hall effect (AHE)9,10. However, from the phenom-enological point of view, these two effects are identical to each other: bothmanifest that T is macroscopically broken and the existence of AHEautomatically implies the existence of weak ferromagnetism, nomatter how“weak” it is. Actually, in his monograph17, Turov considered not only weakferromagnetism, but also AHE and many other phenomena expected inweak ferromagnets. Particularly, already in 1962, Turov and Shavrov pre-dicted that AHE can be induced by AFM order19. Apparently, this is themost one can say within phenomenological theories, which do not provideany information about themagnitude of the effect or its microscopic origin.Despite the fact that AHE is frequently regarded as one of possiblemanifestations of time-reversal symmetry breaking in altermagneticsystems2–4, the aspect of spin splitting in this phenomenon remains to beobscured, and so does the general relation between weak spin ferro-magnetism and altermagnetism. The microscopic theory of weak spin fer-romagnetism is essentially the theory of Dzyaloshinskii-Moriya (DM)interactions, originally proposed by Moriya for Mott insulators20 and laterextended to a broader class ofmagnetic systems21–23. In this case, the net spin1Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Ibaraki, Japan. 2Department of MaterialsEngineering Science, The University of Osaka, Toyonaka, Japan. e-mail: SOLOVYEV.Igor@nims.go.jpnpj Quantum Materials |           (2026) 11:54 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41535-026-00900-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41535-026-00900-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41535-026-00900-9&domain=pdfmailto:SOLOVYEV.Igor@nims.go.jpwww.nature.com/npjquantmatsmagnetization appears atfirst order in the spin-orbit (SO)coupling,whereasthe associated energy change enters at second order. For comparison, thesingle-ion anisotropy – another source of weak ferromagnetism24 – pro-duces a net magnetization only in second order and an energy change infourth order.Depending on the symmetry, DM interactions can have severalcomponents: those that have the same sign in equivalent bonds supportweak spin ferromagnetism, while the ones with alternating signs do notcontribute to the FMmoment or any canting of spins. A notable example isthe DM interactions between two magnetic sublattices in CrO2, a sistercompound of altermagnetic RuO2: these interactions are relatively strongbut alternate among eight neighbouring bonds23. Regarding the role playedby the sign-alternating DM interactions, we will argue that they do beardirect and crucial implications for AHE and net orbital magnetizationM,thus clarifying the fundamental difference between AHE and weak spinferromagnetism from the microscopic point of view. On the other hand,while the lifting of Kramers spin degeneracy in the non-relativistic limit isregarded tobe the centralmanifestationof breakingT in altermagnets1–3, wewill show that: (i) the bands can remain spin-degenerate even when T isbroken and (ii) the spin splitting does not play an essential role in theemergence of AHE andM, which are driven by relativistic SO interaction.Finally, in 1997, long before the modern era of altermagnetism, one ofus proposed that weak ferromagnets LaMO3 (M = Cr, Mn, and Fe) canexhibit an appreciable magneto-optical effect (the ac analog of AHE)25.Rather than being related to the weak spin ferromagnetism itself, thisphenomenon better correlated with the behavior of orbital magneticmoments25. Using the modern theory of orbital magnetization26,27, we willargue that the conclusion was essentially correct as AHE and M have asimilar microscopic origin.ResultsLattice symmetryTo be specific, we keep in mind the BO2 layer of orthorhombic perovskitesABO3 with the space group Pbnm or layered perovskites A2BO4 with thespace group Bmab. The characteristic examples are LaFeO3 and La2CuO4,forming AFM order in the layer plane (xy). There are two magnetic sub-lattices, centered at the origin (0, 0) and t ¼ ð12 ; 12Þ. ThePbnm group has foursymmetry operations transforming the plane xy to itself: the unity; I aboutthe origin; the twofold rotation about x combinedwith the shift by t, fC2xjtg;and themirror reflection of x, also combinedwith the shift, {mx∣t}. InBmab,these symmetry operations are combined with the mirror reflection my.Note that, in comparison with the standard setting of the Bmab group, herewe additionally swap the orthorhombic axes x and y. fC2xjtg and {mx∣t}transform twomagnetic sublattices to each other, whileI andmy transformeach sublattice to itself.The phenomenological Landau theory for these and other materials,which were recently considered in ref.28, was formulated by Turov17,regardless of altermagnetic band splitting. Particularly, he predicted that inthese orthorhombic systems, the Néel order along x gives rise to the weakferromagnetism along z and the anomalous Hall response σxy17. Both arelinear in the Néel field.ModelThemicroscopicmodel for the BO2 layer includes the following ingredients(see Fig. 1): (i) The hoppings between first, second, and third nearestneighbors (t1, t2, and t3, respectively). (ii) The orthorhombic strain of thesecond nearest hoppings, δt2, having the same form in both magneticsublattices and making the directions x and y inequivalent. As we will see,this is a very important parameter, which is responsible for finite values ofAHE. (iii) Deformation of the third nearest hoppings, δt3, alternatingbetween the sublattices 1 and 2, that results in the altermagnetic splitting ofbands. (iv) SO coupling in noncentrosymmetric nearest-neighbor bonds,which has the form of spin-dependent hoppings, bHsoij ¼ itij � bσ, where bσ ¼ðbσx;bσy;bσzÞ is the vector of spin Pauli matrices. The DM interaction, Dij, issimply proportional to tij and this universal property holds in insulating29 aswell as metallic21–23 regimes. In orthorhombic systems, Dij ~ tij have thefollowing form around each magnetic site30: tij = ( ± tx, ty, tz), where y and zcomponents are the same in all the bonds, while the sign of x componentalternates as shown in Fig. 1c. Thus, if AFM spins are aligned along x, ty andtz are responsible for the weak ferromagnetism along, respectively, z and y31,while tx has no effect on the spin texture. The Bmab symmetry imposesadditional constraints: δt3 = 0 and tz=0.Nevertheless, the band splitting canreappear in the multi-orbital case, considering the hoppings betweenorbitals belonging todifferent irreducible representationsof thepoint group.In conventional antiferromagnets, t is the regular translation. There-fore, ifmagnetic site is in the inversion center, therewill be another inversioncenter between the magnetic sites, meaning that Dij ~ tij = 014,20.bHsoij can be eliminated by the SU(2) rotations of the spins,bUS ¼ e�iφn�bσ , in the sublattice 2 with n ¼ t ijjt ij j, φ ¼ 2 arctanjt ij jt1, which leadsto the redefinition t1 ! t1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ tij=t1� �2r29,32. However, this transforma-tion depends on the bond and, therefore, cannot be performed simulta-neously for all the bonds. Nevertheless, we can still use it to eliminate ty andtz, but to keep tx and bHsoij ¼ ± itxbσx . The corresponding transformation isgiven by bUS with tx = 0. Therefore, the parameters ty and tz, which areresponsible for weak spin ferromagnetism, do not play any role in AHE andorbital magnetization. Finally, we add the Néel field ±Bbσx along x, whichyields FM moment along z31. Then, after global rotation of spins, trans-forming bσx to bσz , we will have the following 4 × 4 Hamiltonian 33, which isdiagonal with respect to spins σ = ± :bHk ¼ hk � δh3kbτz þ h1kbτx � Bbτzbσz � hsok bτybσz; ð1Þwhere two AFM sublattices are described in terms of Pauli matricesbτ ¼ ðbτx;bτy;bτzÞ33, hk ¼ h2k þ δh2k þ h3k , h2k ¼ 2t2ðcos kx þ cos kyÞ,δh2k ¼ δt2ðcos kx � cos kyÞ, h3k ¼ 4t3 cos kx cos ky , δh3k ¼ 4δt3 sin kx sinky , h1k ¼ 4t1 coskx2 cosky2 , and hsok ¼ 4tx sinkx2 sinky2 .The model parameters can be derived from DFT34–37, as we will dobelow for La2CuO4.Alternatively, one can take the exchange interactions, Jk,and evaluate the parameters from the superexchange theory as jtk=t1j ¼ffiffiffiffiffiffiffiffiffiffiJk=J1pand 2tij/t1 = Dij/J129. δt2 and δt3 can be exceptionally large inorthorhombic manganites, to facilitate the formation of spin-spiral multi-ferroic phases23.Note that the large δt3 arises fromthe Jahn-Teller distortion,while the SO-induced term tx is associated with the tilting of the BO6octahedra. A reasonable choice is (in units of t1 < 0): t3 ~− t2 ~ 0.1 and δt3 ~tx~− δt2 ~ 0.0523, whichwill be usedunless it is specifiedotherwise. ∣B∣=1 issufficient to open the band gap. Examples of such band structure, εσk;ν ¼hk þ νffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðσBþ δh3kÞ2 þ ðh1kÞ2 þ ðhsok Þ2q(ν = ± being the band index), andFermi surface are shown in Fig. 1(d),(e). As expected, δt3 splits the bands,while δt2 deforms the Fermi surface.Similar model has been considered by Naka et al. for the analysis ofAHE in κ-type organic antiferromagnets8,10, obeying the same orthor-hombic symmetry, who arrived to similar conclusions: crucial importanceof sign-alternating part of SO interaction and orthorhombic strain, andrelative unimportance of altermagnetic band splitting. These conclusionshave been further supported by the analysis of more sophisticated multi-orbital model for orthorhombic perovskites38.Hidden symmetriesA very interesting situation occurs when δt3 = 0 (i.e., without alter-magnetic splitting of bands). Since T bσz ¼ �bσz , the Néel field Bbτzbσzbreaks T , but remains invariant under fT jtg, where T is combined withthe lattice shift of site 1 to site 2 (and vice versa), which additionallychangesbτz to�bτz . Then, since Bbτzbσz is real and T ¼ SK , whereK is thecomplex conjugation and S ¼ ibσy flips σ to− σ, T can be replaced by S,so that the Néel field is also invariant under fSjtg � ibσybτx. On the otherhttps://doi.org/10.1038/s41535-026-00900-9 Articlenpj Quantum Materials |           (2026) 11:54 2www.nature.com/npjquantmatshand, the spin-orbit interaction, hsok bτybσz , is invariant under T , butchanges its sign when it is combined with the lattice shift, fT jtg.However, sincebτy is complex, this sign change does not occur if one usesfSjtg instead of fT jtg. Therefore, the spin-orbit interaction is invariantunder fSjtg, so as the full Hamiltonian (1). This means that eigenstateswith σ = ± differ only by a phase factor. The latter guarantees that (i) theσ = ± bands are degenerate and (ii) the contributions of these bands toσxy are equal to each other and, instead of the partial cancellation, whichwould occur in ferromagnets, we have an addition of suchcontributions.Importantly, the fSjtg symmetry is a consequence of inversionalinvariance (the “centrosymmetric antiferromagnetism” in the definition ofTurov17)39. This is an interesting example of intertwined fundamentalproperties ofmatter, when I controls the properties related to time-reversalsymmetry breaking.Berry curvature and AHEσxy is givenby theBrillouin zone (BZ) integral of theBerry curvatures,Ωσk40,41(in atomic units):σxy ¼ �ZBZdkð2πÞ2Xσ¼±f σkΩσk ; ð2Þwhere f σk is Fermi-Dirac distribution function andΩσk ¼ �2Imh∂kxuσk j∂kyuσki. Without loss of generality, one can considerthe casewhere ν=−bands are partially occupiedbynel electronswhile ν=+bands are empty. Searching eigenvectors for ν = − in the form:uσk�� � ¼ cos θσkeiϕσksin θσk !; ð3Þit is straightforward to find (Supplementary, Sec. A):θσk ¼ � 12arctanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðh1kÞ2 þ ðhsok Þ2qσBþ δh3kð4Þ(0≤ θσk < π) and ϕσk ¼ σ arctan hsok =h1k� �(0≤ ϕσk < 2π), which are, respec-tively, even and odd in tx (SO coupling). Therefore, the Berry curvature,which can be expressed as the cross product,Ωσk ¼ sin 2θσk ∂kθσk × ∂kϕσk  z ,is odd in tx. Since sin 2θσk ∂kθσk ¼ � 12 ∂k cos 2θσk� �, σxy can be reformulatedin terms of group velocities, ∂kεσk , at the Fermi surface (Supplementary, Sec.B):σxy ¼ � 12ZBZdkð2πÞ2Xσ¼±∂f σk∂εσkcos 2θσk ∂kεσk × ∂kϕσk  z; ð5Þas was generally pointed out by Haldane 42.If δt3 is finite, Ωσk contains both odd and even components in B, thatimmediately follows from the form of θσk . Thus, one can writeΩ±k ¼ Ωok ±Ωek , whereΩok andΩek are, respectively, oddandeven inB.Here,Fig. 1 | Parameters and basic electronic structure.Hopping parameters between (a) first nearestneighbors and (b) second and third nearest neigh-bors. c Parameters of spin-orbit interaction aroundtwo magnetic sites (denotes as 1 and 2). The direc-tions of the bonds are shown by dotted arrows.d Example of band structure. The inset shows high-symmetry points of the Brillouin zone.e Corresponding Fermi surface for nel = 1 with andwithout orthorhombic strain δt2 = 0.15.https://doi.org/10.1038/s41535-026-00900-9 Articlenpj Quantum Materials |           (2026) 11:54 3www.nature.com/npjquantmatswe used thatΩ�k can be obtained fromΩþk by changing the signs of both Band tx, which follows from the formof bHk . This yields σxy ¼ σIxy þ σIIxy withσIxy ¼ �ZBZdkð2πÞ2 f þk þ f �k� �Ωok ð6ÞandσIIxy ¼ �ZBZdkð2πÞ2 f þk � f �k� �Ωek; ð7Þwhere f þk þ f �k� �and f þk � f �k� �are, respectively, even and odd inB. Thus,both σIxy and σIIxy are odd in B. σIIxy is caused by the altermagnetic splitting ofbands and is finite only when the band with one spin is occupied, while thebandwith opposite spin is empty. This is somewhat similar to ferromagnets,where AHE is caused by imperfect compensation between contributionswith different spins. More intriguing is the existence of finite σIxy in case ofspin-degenerate bands due to special symmetry of the Hamiltonian (1), aswas discussed above.Large-B limitMore insight can be gained by consideringΩok andΩek in the limit of large B(Supplementary, Sec. B3):Ωok �2t1txBjBjt21 � t2xtan2 kx2t21 þ t2xtan2 kx2 tan2 ky2sin2ky2� kx $ ky !ð8ÞandΩek � �2δh3kBΩok þ4t1txδt3jBj3 sin kx sin 2ky � kx $ ky� �: ð9Þ(i) The altermagnetic contribution Ωek , which is proportional to δt3, isexpected to be smaller than Ωok by the factorjδt3jB . (ii) Both terms are anti-symmetricwith respect to the permutation of kx and ky. Therefore,σxy, givenby the BZ integral, will vanish unless there is some “anisotropy”, dis-criminating between kx and ky. The role of this anisotropy is played by theorthorhombic strain δt2. The Berry curvature itself does not depend on δt2.However, δt2 deforms the Fermi surface [Fig. 1e], and thus yields finite σxy.Although the term “orthorhombic strain” is relevant to one particular typeof lattice, the idea itself ismuchmore general. For instance, in RuO2we dealwith the σzx component instead of σxy9. Therefore, this is the tetragonaldistortion, c/a, which plays the same role and is responsible for finite σzx inRuO239.The behavior of Berry curvature and σxy is summarized in Fig. 2. Ωokand Ωek have nodes along Γ�M�Γ0. Moreover, Ωek has additional nodesalong X�M�Y0, as can be also clearly seen from the above expressions inthe large-B limit. The magnitude of Ωek is generally smaller, which is againconsistent with the above estimate for largeB. The actual contribution ofΩekto σxy is even smaller because the largest band splitting is expected alongΓ�M�Γ0 [Fig. 1(d),(e)], which is the nodeline ofΩek . Therefore, for realisticvalues ∣δt3∣ ≲ ∣t3∣, the contribution of Ωek to σxy is practically negligible.Furthermore, jΩokj is the largest along X�M�Y0, where the bands are spin-degenerate. On the other hand, σxy rises rapidly with the increase of δt2. Thekink of σxy aroundnel = 1.9 is related to the depopulation of states near theXFig. 2 | AnomalousHall effect. (a) odd (Ωok) and (b)even (Ωek) components of the Berry curvature. Band-filling dependence of σxy for different values of (c) δt3and (d) δt2.https://doi.org/10.1038/s41535-026-00900-9 Articlenpj Quantum Materials |           (2026) 11:54 4www.nature.com/npjquantmatspoint, which are lower in energy than the ones in the Y (Y0) point due to theorthorhombic strain.We would like to emphasize that although Ωok and Ωek can be con-sidered independently, both contributions are proportional to tx and drivenby relativistic SO interaction. Therefore, there is no AHE without SOinteraction. Conversely, a finite AHE can exist even in the absence of bandsplitting δt3. In the latter case, it is given solely by Ωok .Orbital magnetizationM is given by the BZ integral ofMσk ¼ Imh∂kx uσk jbHσk þ εσk � 2μj∂ky uσki ð10Þ(per two sites), where μ is the chemical potential26,27. IntroducingM±k ¼ Mok ±Mek , one can identify two contributions,M ¼ MI þMII,similar to AHE (Supplementary, Sec. C). In the insulating state for nel = 2, itis sufficient to consider only the orthorhombic strain part of bHσk and εσk .Then,M is given by the BZ integral ofMink ¼ �2δh2kΩok .The case of La2CuO4The simplest realistic model for La2CuO4 can be constructed for the x2-y2 band near the Fermi level, as suggested by DFT calculations, andusing for these purposesWannier functions technique34,35 (Fig. 3). 2B ≈U (the on-site Coulomb repulsion) can be evaluated from constrainedrandom phase approximation36,37. It yields the following parameters(see Methods): U = 2.196 eV, t1 =− 439.02 meV, t2 = 34.07 meV, δt2 =5.06 meV, t3 =− 30.27 meV, tx =− 1.27 meV, and ty =− 3.70 meV.Mreplicates the shape of σxy, including the kink position [Fig. 3d]. This isto be expected: for the narrow-band compounds, the k-dispersion ofbHσk and εσk is relatively weak and, therefore,Mσk � Ωσk43. Nevertheless,the k-dispersion of δh2k additionally modulates the sign-alternatingΩokalong X�M�Y0, thus making Mink ≤ 0 throughout the BZ [Fig. 3(e)]and causing M to be finite. This can be viewed as a piezomagnetisminduced by the orthorhombic strain.The numerical value of M is small ( ~ − 5 × 10−7μB), but finite. Itshould be noted that the orbital magnetism in the one-orbital model ispurely itinerant, as it is caused by intersite processes. The additional, single-site contribution to M can emerge in the more general multi-orbitalframework25. However, for the eg states in La2CuO4, this contribution is alsosmall, as it is induced by interactions with occupied t2g states, which arelargely separated from the eg ones. Thus,M is expected to be smaller thanspin net magnetic moment 8Sty/t1 ~ 0.03 μB, where S ¼ 12, meaning thatLa2CuO4 is the canonical weak spin ferromagnet. Nevertheless, this situa-tion is not generic and there are indeed cases where the spin net magneticFig. 3 | Construction of realistic model for La2CuO4. a Crystal structure.b Electronic structure near the Fermi level. The red line shows the tight-bindingdispersion of the x2-y2 band. c Corresponding Wannier functions. d Band-fillingdependence of σxy and orbital magnetization. e The integrand,Mink , specifying theorbital magnetization in the insulating state for nel = 2 electrons.https://doi.org/10.1038/s41535-026-00900-9 Articlenpj Quantum Materials |           (2026) 11:54 5www.nature.com/npjquantmatsmoment can vanish, as expected from the form of DM interactions inRuO223. Then, what is the proper order parameter classifying such uncon-ventional AFM state44,45? We believe that the legitimate choice is M25.Similar toAHE, it is ultimately related to theBerry curvature and inducedbysign-alternating part of the SO coupling. Furthermore, although beingtypically small, it remains finite (unlike spin net magnetic moment) and inmany respects replicates the behavior of AHE. In RuO2 and other AFMrutile compounds, the net spin magnetization can be induced by single-ionanisotropy24.However, this contribution is second order in the SO coupling,whereasM appears already at first order.DiscussionAltermagnetism presents a new turn in the development of weak ferro-magnetism, bringing the analysis to the microscopic level and, thus,revealing new aspects in old-standing problems. Although from a phe-nomenological point of view the phenomena of weak ferromagnetism andAHE are basically identical, the microscopic pictures behind them are dif-ferent and can be linked to, respectively, nonalternating and alternating insign DM interactions. Nevertheless, these components typically coexist asboth of them are induced by the same oxygen displacements, tending toalign the DM vectors perpendicular to magnetic bonds46. This is the reasonwhy weak spin ferromagnetism and AHE frequently coexist and cannot beeasily separated47. In this respect, we show that, as these phenomena havedifferent microscopic origin, they should be separable.The altermagnetic band splitting does not play a key role in AHE. ThefSjtg symmetry of the microscopic Hamiltonian supports the spin degen-eracy of the bands, but does not exclude breaking of T . The lack of the bandsplitting,whichwas recently observed in somepotential altermagnets48, doesnot necessarilymean the absence of AHE. Although the altermagnetic bandsplitting is allowed by symmetry, the symmetry itself does not say anythingabout whether it is large or small. It can be accidentally zero or even for-bidden for certain symmetries, as in La2CuO4. On the other hand, therelativistic SO interaction, which is mainly responsible for AHE andM insuch unconventional antiferromagnets is always finite owing to a lack ofbond inversion symmetry.Furthermore, the fSjtg symmetry imposes a constraint on the form ofSO interactions, which allows us to perform a transformation to some localcoordinate frame, where unconventional antiferromagnet with the SOinteraction can be described as a ferromagnet with only one magnetic siteper cell39. This solves the puzzle of why some nearly collinear antiferro-magnets can exhibit simultaneously AHE andM – two quantities, havingthe same microscopic origin and displaying similar behavior, when theexistence of AHE automatically means the existence ofM and vice versa.The orthorhombic strain, which is typically ignored in models ofaltermagnetism33,49, is another key ingredient responsible forfiniteAHEandM in analogy with piezomagnetism.MethodsDetails of electronic-structure calculations for La2CuO4Weadopt theorthorhombicBmab crystal structureofLa2CuO4, as shown inFig. 3a 50. Electronic structure calculations were performed within DFTusing generalized gradient approximation (GGA) for the exchange-correlation potential51 with spin-orbit coupling as implemented in theQuantum-ESPRESSOpackage, usingultrasoft pseudopotentials52. Energycutoff for wavefunctions and charge density was set to 80 Ry and 80 Ry,respectively. The Brillouin zone was sampled by a 12 × 12 × 16Monkhorst-Packmesh53, and the total energy convergence criterion was put to 10−9 Ry.The results are also verified using local density approximation as imple-mented in the VASP package54,55.The one-orbital model for the Cu states is constructed in the basis ofWannier functions using the procedure of maximal localization as imple-mented in the wannier90 package34,35. The Wannier functions werecalculated byprojecting the states near theFermi level in the rangeof [−2.5,2.5] eV onto the atomic x2-y2 orbitals. The states in the range of [− 0.2,1.0] eVwith respect to the Fermi level were keptfixed. The on-site Coulombinteraction was calculated within the constrained random phaseapproximation36, as implemented in the RESPACK package37 using norm-conserving pseudopotentials without spin-orbit coupling. Energy cutoff forwavefunctions andchargedensitywas set to 100Ryand400Ry, respectively,and the Brillouin zone was sampled by a 8 × 8 × 10Monkhorst-Packmesh.ty appears to be sensitive to the choice of the local coordinate frame,which was necessary for specifying the x2-y2Wannier functions.We believeit is more appropriate to use the same coordinate frame at both Cu sites,which yields ty =− 3.70 meV. Alternatively, one can align the z axes alongthe apical oxygen sites,which are slightly rotated relative to eachother at twoCu sites. This yields ty = − 6.57meV. Such ambiguity is related to the factthat SO coupling parameters depend on the unitary transformatione�iφ0n0 �bσ 29,32, where φ0 and n0 are specified by the choice of the local coor-dinate frame. The interlayer hoppings are about 5 meV and have beenneglected.Data AvailabilityThe data supporting the findings of this study are available within thisarticle. 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B54, 11169 (1996).AcknowledgementsWe are grateful to M. Naka, H. Seo, and A. Lichtenstein for valuablecomments and discussions, M. Katsnelson for drawing our attention to thebook [16], and A. Katanin and S. Streltsov for providing a copy of this book.MANA is supported by World Premier International Research CenterInitiative (WPI), MEXT, Japan. This research did not receive funding.Author contributionsI.S. conceptualized the work, performed the calculations (except specifiedbelow), and wrote the manuscript. S.N. performed electronic structurecalculations and constructedmodel Hamiltonian for La2CuO4. S.N. andA.T.discussed the results and commented on the manuscript. All authorsreviewed the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41535-026-00900-9.Correspondence and requests for materials should be addressed toIgor Solovyev.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.https://doi.org/10.1038/s41535-026-00900-9 Articlenpj Quantum Materials |           (2026) 11:54 7https://doi.org/10.1038/s41535-026-00900-9http://www.nature.com/reprintswww.nature.com/npjquantmatsOpen Access This article is licensed under a Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License,which permits any non-commercial use, sharing, distribution andreproduction in any medium or format, as long as you give appropriatecredit to the original author(s) and the source, provide a link to the CreativeCommons licence, and indicate if you modified the licensed material. 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Toview a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2026https://doi.org/10.1038/s41535-026-00900-9 Articlenpj Quantum Materials |           (2026) 11:54 8http://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/npjquantmats Altermagnetism and weak ferromagnetism Results Lattice symmetry Model Hidden symmetries Berry curvature and AHE Large-B limit Orbital magnetization The case of La2CuO4 Discussion Methods Details of electronic-structure calculations for La2CuO4 Data Availability References Acknowledgements Author contributions Competing interests Additional information