Supported data and manuscript " Separating altermagnetic and ferromagnetic effects in X-ray magnetic dichroism of rutile NiF2" in npj Quantum Mater. 10, 49 (2025)

npj | quantum materialsArticle Published in partnership with Nanjing University https://doi.org/10.1038/s41535-025-00753-8 Separating altermagnetic and ferromagnetic effects in X-ray magnetic dichroism of rutile NiF 2 Check for updates A. Hariki1,3 , K. Sakurai1,3, T. Okauchi1&J.Kuneš2 We present numerical simulations of X-ray magnetic circular dichroism (XMCD) at the L 2,3 edge of Ni in the weakly ferromagnetic altermagnet NiF 2 . Our results predict a significant XMCD signal for light propagating perpendicular to the magnetic moments, which are approximately aligned along the [010] easy-axis direction. The analysis shows that the altermagnetic and ferromagnetic contributions to the XMCD signal can be uniquely distinguished by their dependence on an applied magnetic field. By varying the angle of the field relative to the easy axis, the in-plane orientation of both the Néel vector and the net magnetization can be systematically controlled. We further demonstrate that the XMCD signal, even under fields as strong as 40 T and for any in-plane orientation, can be accurately described as a linear combination of two spectral components, with geometrical prefactors determined by the field’s magnitude and direction. This insight enables experimental validation of the distinctive relationship between the Néel vector orientation and the X-ray Hall vector in the rutile structure. Quantitative simulations supporting these findings are provided. The identification of altermagnets1,2marks the emergence of a new class of collinear magnets, which, despite having no net magnetization, enable the existence of spin-polarized bands1–12, anomalous Hall effect6,10,13–18,odd magneto-optical effects15,19–21, and various other phenomena22 characterized by odd Néel vector dependence. Distinguished from conventional collinearmagnetssuchasferromagnets and antiferromagnets by non-relativistic symmetry operations, the characterization of altermagnets is rooted in separation between electron spin and its orbital motion. In real-world materials, however, relativistic effects such as spin-orbit coupling (SOC) invariably come into play. While typically considered a perturbation to non-relativistic symmetry, SOC becomes a critical factor for the observation of altermagnetic phenomena in some experiments23.Indeed,theinfluence of SOC is twofold: on one hand, it is indispensable for manifesting altermagnetic effects in optical and transport properties, enabling phenomena such as anomalous Hall currents or linear magneto-optical effects. On the other hand, the presence of SOC leads to competing effects, such as weak ferromagnetism, which can obscure or even mimic altermagnetic responses. This duality poses a key challenge in disentangling these intertwined behaviors experimentally. To address this challenge, X-ray spectroscopy has emerged as a powerful probe for isolating intrinsic altermagnetic effects from those induced or modified by SOC. While SOC is a prerequisite for observing magnetic dichroism, the X-ray magnetic circular dichroism (XMCD) technique leverages the large spinorbit splitting inherent to core states, which is fundamentally distinct from weak ferromagnetism or other valence band effects. Computational studies indicate that the contribution of valence SOC to XMCD spectra is generally minor in many compounds containing 3delements20,24. In this article, we demonstrate this point in computational study of the XMCD on the Ni L 2,3 edge in NiF 2 .NiF 2 , like other members of the transition metal difluoride series, crystallizes in rutile structure and antiferromagnetically orders below 73.2 K25. Unlike other compounds in the series with the [001] easy axis, the Néel vector L=m 1 −m 2 points along [010] or the other three directions related by the tetragonal symmetry26,27. This orientation of Lin the rutile structure allows for a finite XMCD19 even without an external magnetic field, assuming a single-domain sample. The valence SOC causes a small canting of the Ni moments, which results in a net magnetization along [100] and gives rise to conventional ferromagnetic XMCD signal. The XMCD spectra for light propagating in the direction ^ kis obtained as FXMCDðωÞ¼2hðωÞ ^ k, where the frequency-dependent Hall vector hðωÞ¼=ðσa zyðωÞ;σa xzðωÞ;σa yxðωÞÞ depends on the orientation of the magnetic moments in the sample. Applying magnetic field Bin the (001) plane one can vary the moments canting as well as the orientation of the Néel vector Lwithin the (001) plane. We simulate such an experimental set-up 1Department of Physics and Electronics, Graduate School of Engineering, Osaka Metropolitan University, Sakai, Osaka, Japan. 2Department of Condensed Matter Physics, Faculty of Science, Masaryk University, Brno, Czechia. 3 These authors contributed equally: A. Hariki, K. Sakurai. e-mail: [email protected]; [email protected] npj Quantum Materials | (2025) 10:49 1 1234567890():,; 1234567890():,; and show that h(ω) can be to a high accuracy described by two spectral functions Δ ALT (ω)andΔ FM (ω) hðωÞ¼ΔALTðωÞMð110Þ ^ LðBÞþΔFMðωÞmðBÞ:ð1Þ Here, ^ Lis the unit vector in the direction of L,mis the net magnetic moment per atom and Mð110Þis the operation of mirror symmetry by the (110) plane. The first term is exact for a collinear antiferromagnet with rutile structure, assuming no valence SOC and only monopole (no exchange) interaction between the core and valence states19,28. The second term is an ad hoc approximation based on the smallness of m. Expressing the conductivity tensor in terms of several fundamental spectral functions follows the spirit of refs. 24,29–32, and we show that it has a particularly simple form (1)incase of rutile antiferromagnets. Results In Fig. 1, we demonstrate that our calculations accurately capture the experimental X-ray absorption spectra (XAS) of NiF 2 , which are dominated by atomic multiplet features. Note that we use a single value for the lifetime broadening, which somewhat exaggerates the sharpness of the L 2 features. The orientation and size of the Néel vector Land magnetization m vectors is determined by the external field B, the inter-atomic exchange J i and the single-ion anisotropy (SIA). We treat the inter-atomic exchange on the mean-field level while SIA appears through solution of the atomic problem with SOC and crystal-field. We begin our presentation with B∥[100]. In this configuration, as reported in ref. 33,a field of 0.7 T is sufficient to select the [010] domain (out of the four possible Lorientations). Increasing the magnetic field leads to a growth of ∣m∣.InFig.2a we present the XMCD spectra for ^ kalong the applied field. The Hall vector points along [100] direction and has an amplitude h(ω)=Δ ALT (ω)+Δ FM (ω)∣m∣. In the absence of an external field, the calculated net magnetization is m=m s +m l = (0.023 +0.027)μ B , which somewhat overestimates the 0.03 μ B reported in literature27,34. Calculations with reduced SOC, which reproduce the experimental value of magnetization can be found in Supplementary Figs. 4 and 5. The XMCD spectra calculated for various applied fields allow us to extract Δ ALT (ω)andΔ FM (ω),asshowninFig.2c. The Δ ALT (ω)closely resembles the corresponding density obtained without valence SOC, a limit in which one can distinguish an altermagnet using non-relativitic symmetry. The validity of Eq. (1)isconfirmed in Fig. 2a, b. Fields up to 10Tresultinmorethanatwofoldincreaseof∣m∣, providing sufficient variation of the XMCD spectra to facilitate a similar analysis of typical experimental data. Next, we rotate the applied field, B¼Bðcos φB;sin φB;0Þ, in the (001) plane, as illustrated in Fig. 3(b). The calculated orientations of Land m, along with the magnitude ∣m∣, are shown in Fig. 4. These results align well with previous theoretical and experimental studies27,34.Themagneticorder arises from a competition between SIA, which favors the [010] and [100] orientations of Land m, respectively, and the external field, which Land m being perpendicular, with m∥B.Atlowfield strengths, the orientations of themagneticmomentsareonlyslightly perturbed. In the highest studied field of 40 T, the net magnetization mfollows the rotation of B,thoughit remains misaligned, while the orthogonality of mand Lis approximately maintained, reflecting the rigidity of Ni moments. In Fig. 5we show the field dependence of the XMCD spectra for the incoming light directions [100] and [010]. A key observation is the comparison of the full calculation, which accounts for specific orientations of the Ni moments, and the spectra derived using Eq. (1) with the previously obtained spectral distributions Δ ALT (ω)andΔ FM (ω). We find that for all studied field strengths, up to 40 T, and across all field angles φ B ,Eq.(1) describes the calculated spectra with a relative accuracy better than 1%. Finally, we present the data in a form that is closely aligned with a potential experimental setup. In this configuration, the external field Bis parallel to the light beam ^ k,andthesampleisrotatedalongthec-axis, which is perpendicular to the beam. For B= 0 the Néel and magnetization vectors rotate with the sample, resulting in a cos φBdependence of the spectra. For a finite field the XMCD spectrum is described by Eq. (1)as FXMCDðωÞ¼2ΔALTðωÞsinðφLþφBÞþ2ΔFMðωÞjmjcosðφmφBÞ 2ΔALTðωÞcosð1þαÞφBþ2ΔFMðωÞm0cosð1αÞφB; ð2Þ where φ L ,φ m and ∣m∣depend on φ B and the field amplitude, as illustrated in Fig. 4. Using the leading-order approximation φ L ≃φ m +π/2, ∣m∣≃m 0 and φ m ≃αφ B ,whereαand m 0 depend on the field amplitude, we arrives at the bottom line. For sufficiently large fields (α≈0.34 at 10 T) this dependence allows for experimental verification of the relationship ^ hALT ¼Mð110Þ ^ LðBÞ between the in-plane orientation of the Néel vector Land the direction of the altermagnetic part of the Hall vector ^ hALT in the X-ray range. The results of a simulation for a field of 10 T are shown in Fig. 6. Finally, we comment on the X-ray sum rules for orbital and spin moments35 and refer the reader to Supplementary Fig. 6 for quantitative analysis. There is no fundamental reason why the sum rules35 should not apply to altermagnets the same way they apply to ferromagnets. However, given the small magnitude of the net moments and the oscillatory nature of the spectra, their application requires careful consideration. The present Full calculations fulfill the sum rules by design. The approximate formula of Eq. (1)fulfills the spin sum rule with 15% deviation over the studied parameter range, while deviation for orbital sum rules are as large as 30%. That Eq. (1) cannot capture the sum rules accurately is apparent from its form, implying that the effective moments obtained from sum rules scale with m.Theactual spin m s and orbital m l are not collinear and their relative contribution to the net mstrongly varies from approximately 1:1 without field to 5:1 at 40 T, see Supplementary Fig. 2. Discussion We have conducted numerical simulations of XMCD at the L 2,3 edge of Ni in the rutile altermagnet NiF 2 under an external magnetic field. Our results demonstrate that the XMCD signal across a wide range of field amplitudes and ab-plane orientations can be expressed as a linear combination of two distinct spectral functions. These functions represent the altermagnetic and ferromagnetic contributions. The coefficients of this linear combination depend on the orientations of the Néel Land the magnetization mvectors. The ferromagnetic contribution scales with the amplitude of the magnetization. Moreover, the two contributions exhibit different angular 0 1 850 860 870 x3 Exp. Ni L-edge XMCD Intensity Energy (eV) F＋ Fー XMCD Fig. 1 | X-ray absorption at Ni L 2,3 edges. The XAS calculated for the two circular polarizations (red and blue) at the Ni L 2,3 edge together with the XMCD intensities (shaded). The calculated spectral intensities are broadened by a Lorentzian of 0.30 eV (HWHM). The experimental Ni L 2,3 -edge XAS spectrum taken from ref. 41 is shown for comparison. The experimental baseline was offset for the sake of clarity. https://doi.org/10.1038/s41535-025-00753-8 Article npj Quantum Materials | (2025) 10:49 2 dependencies as the external field rotates away from the easy-axis direction. This distinction provides an opportunity for experimental verification of the peculiar relationship28 between the X-ray Hall vector and the Néel vector in the rutile structure. The utility of Eq. (1) lies in the observation that the shape of the XMCD spectra in rutile altermagnets is roughly independent of the Néel vector Linplane orientation and its dependence on the photon incidence vector k allows to determine Luniquely. This contrasts the XMCD behavior in MnTe where the XMCD amplitude strongly depends on Land even vanishes for specific orientations20. X-ray magneto-optics offers a distinct approach to isolating altermagnetic effects, leveraging the specificinfluence of valence spin-orbit coupling (SOC) in lighter elements, such as 3dtransition metals. The valence SOC modifies the magnetic ground state by inducing canted Fig. 2 | X-ray circular dichroism for magnetic field along (100) direction. a Ni L 2,3 -edge XMCD intensities in NiF 2 calculated independently for each magnetic field Band φ B =0 ∘with no approximations to the method (“Full”). bXMCD intensities computed as a linear combination (“Approx.”)of(c)Δ ALT (ω) (blue, left axis) and Δ FM (ω) (red, right axis) following Eq. (1). Δ ALT (ω) in the non-relativistic limit, calculated without the Ni 3dvalence SOC, is also shown (thin black, left axis). The inset in (b) shows the difference in the XMCD intensities at the Ni L 3 -edge between the full calculations in (a) and the approximations in (b). Fig. 3 | Orientation of magnetic moments in the (001) plane of NiF 2 .aTop view of the rutile structure with the Néel vector Lin the [010] direction. The red arrows mark the local moments m 1,2 in the Ni sites, mis the net magnetization. bThe definitions of angles φ B ,φ L and φ m measured from [100] direction where Bin the magnetic field applied in the (001) plane. Note that the competition between the magneto-crystalline anisotropy and the external field leads to the misalignment of net magnetization mand the external magnetic field B. Fig. 4 | Magnetic moments for the magnetic field in general orientation within the (001) plane. Calculated relation between the angles of the external field φ B and athe Néel vector φ L ,bthe FM moment φ m , and cthe amplitude of the FM moment ∣m(B)∣for selected amplitudes of the external field B. The results over a wider range for the angles and amplitudes are provided in Supplementary Fig. 4. https://doi.org/10.1038/s41535-025-00753-8 Article npj Quantum Materials | (2025) 10:49 3 moments and non-collinearity, while also affecting the excitation energies and the transition matrix elements. These intertwined effects are often difficult to disentangle in transport measurements or visiblerange magneto-optics. In X-ray magneto-optics, however, the dominance of core-level SOC in transition metal elements naturally separates these influences. As a result, the role of valence SOC is largely confined to determining the orientation of magnetic moments, which can also be externally controlled. This separation enables the systematic identification of the valence SOC effects, distinguishing them from the altermagnetic contribution. Methods We perform a density functional theory (DFT) calculation for the experimental structure of NiF 2 36 using the Wien2K package37.The crystal field within the Ni 3dshell is derived from the self-consistent DFT band structure using the Wannier90 and wien2wannier packages38,39,see Supplementary Note 1 for the computational details. Since NiF 2 is a large-gap Mott insulator, the Ni2+atomic model adequately accounts for the Ni L 2,3 -edge XAS spectrum dominated by the intra-atomic multiplet effects as shown by the early studies by de Groot et al.20,40. The atomic Hamiltonian spanning the space of 2pand 3dshells consisting of the 3d crystal field, 2pand 3dSOC, 3d−3dand 2p−3dCoulomb interaction, Weiss mean-field and the external magnetic field is diagonalized. Full optical conductivity tensor in the Ni L 2,3 range is calculated in dipolar approximation using the Fermi golden rule. The total conductivity is the sum of the contributions from the two Ni sublattices in the rutile structure. The magnetic ground state is obtained with Weiss mean-field theory. The Weiss field acting on local spin (Zeeman field) is calculated with the Heisenberg exchange parameters J= 1.47 meV derived from the experiment34. 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This work was supported by JSPS KAK-ENHI Grant Numbers 21K13884, 21H01003, 23K03324, 23H03817 (A.H.), and by the project Quantum Materials for Applications in Sustainable Technologies (QM4ST), funded as project No. CZ.02.01.01/00/22 008/ 0004572 by Programme Johannes Amos Commenius, call Excel-lent Research and by the Ministry of Education, Youth and Sports of the Czech Republic through the e-INFRACZ (ID:90254). Author contributions A.H., K.S., and T.O. performed the calculations. A.H. and J.K. conceived the research program, analyzed the data and written the manuscript. All authors reviewed the manuscript. Competing interests The authors declare no competing interests. Additional information Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41535-025-00753-8. Correspondence and requests for materials should be addressed to A. Hariki or J. Kuneš. 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