# Fileset [Ba2CaOsO6_XMCD_MDR.pdf](https://mdr.nims.go.jp/filesets/be148472-0331-4a4a-8547-c27f40e2cbe3/download) ## Creator G. Shibata, N. Kawamura, J. Okamoto, A. Tanaka, [H. Hayashi](https://orcid.org/0000-0001-7787-9082), [K. Yamaura](https://orcid.org/0000-0003-0390-8244), H. Y. Huang, A. Singh, C. T. Chen, D. J. Huang, S. V. Streltsov, A. Fujimori ## Rights ©2026 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/) ## Other metadata [Possibility of ferro-octupolar order in Ba

_{2}

CaOsO

_{6}

assessed by x-ray magnetic dichroism measurements](https://mdr.nims.go.jp/datasets/fac7c47e-3ab3-4d1c-a44f-91e776b77259) ## Fulltext Possibility of ferro-octupolar order in Ba2CaOsO6 assessed by X-ray magneticdichroism measurementsG. Shibata,1, ∗ N. Kawamura,2 J. Okamoto,3 A. Tanaka,4 H. Hayashi,5, 6, † K. Yamaura,5, 6H. Y. Huang,3 A. Singh,3, ‡ C. T. Chen,3 D. J. Huang,3, 7, 8 S. V. Streltsov,9, § and A. Fujimori1, 3, 7, 10, ¶1Materials Sciences Research Center, Japan Atomic Energy Agency, Sayo, Hyogo 679-5148, Japan2Japan Synchrotron Radiation Research Institute, Sayo, Hyogo 679-5198, Japan3National Synchrotron Radiation Research Center, Hsinchu 300092, Taiwan4Department of Quantum Matter, Hiroshima University, Hiroshima 739-8530, Japan5Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science, Tsukuba, Ibaraki 305-0044,Japan6Graduate School of Chemical Sciences and Engineering,Hokkaido University, Sapporo, Hokkaido 060-0810, Japan7Department of Physics, National Tsing Hua University, Hsinchu 300044, Taiwan8Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 300093, Taiwan9Institute of Metal Physics, 620041 Ekaterinburg GSP-170, Russia10Department of Physics, University of Tokyo, Bunkyo-Ku, Tokyo 113-0033, Japan(Dated: March 15, 2026)Localized 5d2 electrons in a cubic crystal field possess multipoles such as electric quadrupoles andmagnetic octupoles. We studied the cubic double perovskite Ba2CaOsO6 containing the Os6+ (5d2)ions, which exhibits a phase transition to a ‘hidden order’ below T ∗ ∼ 50 K, by X-ray absorptionspectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD) at the Os L2,3 edge. Thecubic ligand-field splitting between the t2g and eg levels of Os 5d was deduced by XAS to be ∼4eV. Ligand-field (LF) multiplet calculation under fictitious strong magnetic fields indicated that theexchange interaction between nearest-neighbor octupoles should be as strong as ∼1.5 meV if a ferro-octupolar order is stabilized in the ‘hidden-ordered’ state, consistent with the exchange interactionof ∼1 meV previously predicted theoretically using model and density functional theory calculations.The temperature dependence of the XMCD spectra was consistent with a ∼18 meV residual cubicsplitting of the lowest Jeff = 2 multiplet state into the non-Kramers Eg doublet ground state andthe T2g triplet excited state.I. INTRODUCTIONCorrelated electronic states in strongly spin-orbit cou-pled systems have attracted strong interest in recentyears [1, 2]. Particular attention has been attracted bythe Jeff =1/2 Mott insulators Ir4+ (5d5) oxides [3, 4]and their doped compounds [5] as well as the Kitaevquantum-spin-liquid candidates Ru3+ (4d5) honeycombcompounds [6]. Recently, it was theoretically predictedthat a magnetic octupolar order can occur in localized5d2 electrons is a cubic crystal field [7–10]. In general,a d2 ion coordinated by ligand atoms in the cubic (Oh-symmetry) environment is expected to undergo a Jahn-Teller distortion, but many cubic crystals with 5d2 ionsremain undistorted owing to the strong spin-orbit cou-pling (SOC) of the 5d electrons [11]. This drew theattention of many researchers to B-site-ordered double-perovskite oxides containing 5d2 ions (Os6+, Re5+) such∗ email: shibata.goro@jaea.go.jp† present address: Institute for Solid State Physics, The Univer-sity of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581,Japan‡ present address: Department of Physics and Astrophysics, Uni-versity of Delhi, Delhi 110007, India§ email: streltsov@imp.uran.ru¶ email: fujimori@phys.s.u-tokyo.ac.jpas Ba2BOsO6 and Ba2B′ReO6, where B is an alkaliearth and B′ is a rare earth. Figure 1 shows the crystalstructure of the B-site-ordered double-perovskite oxidesBa2CaOsO6; the OsO6 octahedra exhibit Oh symmetryand are isolated from one another, which enables studythe intrinsic electronic structures of 5d2 systems.The magnetic susceptibility of Ba2CaOsO6 containingOs6+ ions exhibits a cusp-like anomaly at T ∗ ∼ 50 K,but neutron diffraction has shown no magnetic Braggpeaks below T ∗ while muon-spin rotation (µ-SR) has re-vealed a small local magnetic moment of ∼ 0.2 µB [12]and a staggered magnetic moment of ∼ 0.05 µB per Osion [13]. According to X-ray diffraction, the crystal re-mains cubic down to the lowest temperatures, preclud-ing an electric quadrupolar order that should lead to aJahn-Teller distortion [14]. Therefore, the origin of the‘hidden order’ below T ∗ is consistent with an orderingof the magnetic octuploes. Theoretically, exchange cou-pling between neighboring Os ions favors ferro-octupolarorder in the fcc sublattice of the double perovskite [9].The formation of the magnetic octupole from the 5d2configuration in a cubic crystal field is illustrated inFig. 2(a) based on the two-electron energy diagram thathas been obtained by recent resonant inelastic X-ray scat-tering (RIXS) studies [15, 16]. Under the Oh symmetry,the lowest state of the t22g multiplet, which has the to-tal effective angular momentum of Jeff = 2, should split2FIG. 1. Double-perovskite crystal structure of Ba2CaOsO6drawn using the VESTA software [19]. The Os-Os and Os-Odistances are estimated to be 5.9012 Å and 1.9109 Å, respec-tively [17].into the non-Kramers Eg doublet ground state and thetriply degenerate T2g excited state, separated by a resid-ual cubic splitting ∆c [17, 18]. Depending on the na-ture of additional perturbation on the Os6+ ion, the non-Kramers doublet may split into either eigenstates of theelectric quadrupole operators such as |ψg,↑⟩ ≡ |Jzeff = 0⟩and |ψg,↓⟩ ≡ 1√2(|Jzeff = 2⟩ + |Jzeff = −2⟩), or eigen-states of the magnetic octupole operator T ∝ JxJyJz(the overline denotes symmetrization), such as |ψg,±⟩ ≡1√2(|ψg,↑⟩ ± i|ψg,↓⟩). If all the Os ions are in one ofthe two eigenstates |ψg,±⟩ of the T operator, the ferro-octupole-ordered state is realized. In order to investigatethe ground state and low-energy excited states of theOs6+ ion, which are more directly related to the multi-pole orders, linear or circular dichroism of X-ray absorp-tion spectroscopy (XAS) and its temperature dependenceare expected to provide us with valuable information.In this work, we measured XAS and X-ray magneticcircular dichroism (XMCD) at the Os L2,3 edges ofBa2CaOsO6 and analyzed the spectra using ligand-field(LF) multiplet theory. In particular, we addressed thequestion of why the non-Kramers Eg doublet prefersthe magnetic octupole to the electric quadrupole as theground state on the basis of the experimental and calcu-lated low-energy states of the Os6+ ion under magneticfields. In particular, multiplet calculation under ficti-tious strong magnetic fields suggested that the exchangeinteraction between nearest-neighbor Os ions should bec~20 meVNon-Kramersdoublet EgJeff = 2E(a)Jeff = 1~0.4 eV Jeff = 2Jeff = 0~0.4 eV Jeff = 0 T2gB = 0 B ||<111> 0ET2gEEgSOCLF ~4 eV ’~0.7 eV 32(b)jeff = 1/2jeff = 3/2t2gegFIG. 2. Energy levels of the Os6+ (5d2) ion in the cubic (Oh)crystal field. (a) The t22g part of the 5d2 multiplet levels.The t2geg part is located at higher energies separated by thet2g-eg splitting of ∆LF ∼ 4 eV, see panel (b). The lowestmultiplet state Jeff = 2 is split by a residual cubic splitting,∆c, into the non-Kramers Eg doublet ground state and theT2g triplet excited states. Under a finite magnetic field B ∥⟨111⟩, the non-Kramers doublet is split into the two magneticoctupolar eigenstates |ψg,±⟩ (defined in the text) separatedby ∆Eg (∝ B3) and the triplet is split by the Zeeman energy∆ET2g (∝ B). (b) One-electron energy levels of the Os 5dorbitals. ∆LF is the t2g-eg ligand-field splitting. Spin-orbitcoupling (SOC) splits the t2g level further into the jeff = 12and jeff = 32levels separated by 32ζ′, and the jeff = 32level isoccupied by two electrons.as strong as ∼1.5 meV if the hidden order is causedby a ferro-octupolar order. This is consistent with therecent density-functional theory (DFT) calculation thatpredicted the exchange interaction of ∼1 meV [20].II. RESULTSFigure 3 shows XAS and XMCD spectra taken atT = 10 K and 60 K under the magnetic fields of B = ±7 T. (For experimental details, see Appendix A.) TheXAS spectra show a double-peak structure both at theL3 and L2 edges [Figs. 3(a) and (b), respectively], re-flecting the t2g-eg ligand-field splitting of ∆LF ∼ 4 eV ofthe Os 5d level [defined in Fig. 2(b)]. This ∆LF valueis nearly identical to the value deduced from the O K-edge XAS measurement [15]. Only the XAS peaks arisingfrom transition to the t2g states show finite XMCD sig-nals. The XMCD intensity at the Os L2 edge [Fig. 3(d)]is high (∼ 1% of XAS) while that at the Os L3 edge[Fig. 3(c)] is extremely low (<∼ 0.1%). Such highly L3-L2 asymmetric XMCD intensities have also been ob-served in some ferrimagnetic double-perovskite oxidescontaining Os [21, 22]. The XMCD sum rules yield theinduced orbital and ‘effective’ spin magnetic moments(see Appendix A) of Morb = −0.016±0.002µB/Os and3 !"#$%&'()*+,-.-/&-&#&-&$&-&%&0&-&%'123*+,-.-/%&-4$%&-4&%&-55%&-5 67898:*;:;<=>*+?;@/%$-"$%$-"&%$-#5%$-# 67898:*;:;<=>*+?;@/&-&#!&-&#&&-&$!'123%$-#4!%$-#4&6-*A-*+?;@/BC* #'()BC* $'()BC* #'123BC* $'123*!*D*%&*E*!*D* &*E*!*D*%&*E*!*D* &*E"*D*F*G+,/ +H/+I/ +J/FIG. 3. X-ray absorption spectroscopy (XAS) and x-ray mag-netic circular dichroism (XMCD) spectra of Ba2CaOsO6 atthe Os L2,3 edges measured at T = 10 K and 60 K under themagnetic fields of ±7 T before background subtraction. (a),(b) XAS spectra at the Os L3 and L2 edges, respectively. (c),(d) XMCD spectra at the Os L3 and L2 edges, respectively.Inset shows the expanded XMCD spectra at the Os L2 edge.M effspin = 0.058±0.007µB/Os at T = 10 K, and M effspin de-creases by ∼0.004 µB at T = 60 K. The relatively largeunquenched orbital magnetic moment suggests that thestrong SOC of Os 5d orbitals indeed plays an importantrole in the electronic structure of Ba2CaOsO6.Figure 4 shows comparison of the observed Os L2,3-edge XAS spectra after background subtraction and theXMCD spectra with LF multiplet calculation. (De-tails of the background subtraction are described in Ap-pendix A.) We used the ligand-field splitting of ∆LF =4.3 eV and the SOC parameter of ζ = 0.33 eV to achievethe best fit. Here, ζ (≡ −ζ ′) is defined by the SOC energyζ(l ·s) = ζ ′(leff ·s), where l is the orbital angular momen-tum of an Os 5d electron and leff ≡ −l is the effectiveangular momentum of a t2g electron [24]. To reproducethe measured XMCD intensities, we had to assume aneffective molecular field of B = 12±2 T on top of the ex-ternal magnetic field of 7 T in the calculation [Figs. 4(c)and (d)]. The experimental value of |Morb/Meffspin| =0.28±0.04 deduced using the XMCD sum rules under theexternal magnetic field of B = 7 T was reproduced bythe theoretical value of |Morb/Meffspin| = 0.30±0.04 valuecalculated for 10 K under B = 7 T plus the molecularfield of 12± 2 T. This supports our initial assumptionthat the Os 5d electrons in Ba2CaOsO6 are basically lo-calized in spite of the strong hybridization with the O 2porbitals. Using the multiplet calculation, we could sep-arate M effspin into the spin magnetic moment Mspin andthe magnetic dipole MT : Mspin/Meffspin = 0.45±0.05 andMT ≡ 27 (Meffspin −Mspin) = 0.16±0.02 (Appendix A). AsMT/Meffspin is a measure of anisotropic spin distribution !"#$%%&%%!%&%%#%'%&%%#'%&%%!$%&(%$%&))$%&)*+,-.-/01/1234056178#$%%&%"%&%#%&%$%$#&!#$#&!%$#&")+,-.-/01/12340561789:3/1.;<=>?@A0=BC&00D:E<&0 F$%G0=BC&0D:E<&0=BC&0D:E<&HI0!">?AHI0!#>?AHI0!">9DJHI0!#>9DJ5:8 5K85<8 5L8 F$%G F*%GFIG. 4. XAS and XMCD spectra of Ba2CaOsO6 at theOs L2,3 edges after background subtraction compared withligand-field (LF) multiplet calculation. (a), (b) Experimental(blue curves with dots) and calculated (green curves) XASspectra. The white-line background and the extended X-rayabsorption fine structure (EXAFS) oscillations have been sub-tracted (see Appendix A). (c), (d) Experimental (purple andred curves with dots) and calculated (green and lilac curves)XMCD spectra. The structures around the energies of 12.40and 12.41 keV in the experimental XMCD spectra are mag-netic EXAFS oscillations, the period of which coincides withthat of EXAFS in Ba2Na1−xCaxOsO6 [23]. The molecularfield in the multiplet calculation was adjusted to B = 12 ±2 Tso that the measured XMCD intensity at the L2 edge was re-produced.on the Os ion [25–27], the largeMT reflects the distortionof the 5d orbitals induced by spin polarization via strongSOC.Under the external magnetic field of B = 7 T employedfor the XMCD measurements plus the effective molecu-lar field of 12 T, the multiplet calculation predicts thatthe non-Kramers Eg doublet is split by ∆EEg∼1 µeV,and the triplet excited states T2g should show a Zeemansplitting ∆ET2g= 0.74 meV, where ∆ET2gis defined inFig. 2(a). The temperature dependence of the XMCDspectra is compared with the calculation in Figs. 4(c)and (d). Here, the calculated temperature dependencearises from the thermal excitation of the Os6+ ion fromthe Eg doublet ground state to the T2g triplet excitedstate across the residual cubic splitting ∆c, which wascalculated to be 18 meV using the above parameter set,and is consistent with the previous neutron scatteringexperiment [17].The XMCD spectra and their temperature dependenceare consistent with the non-Kramers Eg doublet groundstate and the T2g triplet excited state separated by ∆c ∼18 meV. Because the previous work show no evidence forJahn-Teller distortion [15], the non-Kramers Eg doubletground state, which can host either electric quadrupoleor magnetic octupole, most likely chooses the magneticoctupolar state [7–10]. Unfortunately, the Os L3-edge4XMCD spectra predicted by the multiplet calculation arevery similar between the magnetic octupolar and electricquadrupolar Eg states. Well below the transition temper-ature T ∗ ∼ 50 K, therefore, one may consider that thenon-Kramers doublet is split into two octupolar states|ψg,±⟩ separated by ∆EEg∼ kBT∗ ∼ 4 meV. Such asplitting would be caused by an effective molecular fielddue to interaction with neighboring Os ions. Our multi-plet calculation showed that, magnetic fields along ⟨111⟩directions create purely octupolar states out of the non-Kramers doublet, which is consistent with the scenariothat exchange interaction between the Os ions stabilizesthe octupolar order. Because our multiplet calculationshowed ∆EEg ∼ 1 µeV at B = 7 T plus the effectivemolecular field of 12±2 T during the XMCD measure-ments and because ∆EEg∝ B3, ∆EEg∼ 4 meV wouldbe realized if the internal exchange field were as strongas B ∼ 300 T.III. DISCUSSIONThe magnetic octupole moment, i.e., the expectationvalue of the T operator is calculated to be 1.26, whichis nearly independent of B as long as the sign of B doesnot change. If we assume that ferro-octupolar order isrealized in Ba2AOsO6 as theoretically predicted [9], ex-change interaction J between nearest-neighbor Os ionsshould be as strong as B ∼ 300 T divided by the numberof the nearest-neighbor Os ions in the fcc lattice, 12, asindicated in Fig. 5, that is, J should be as large as ∼ 1.5meV.The above J value is in fair agreement with previ-ous theoretical calculations. In particular, direct calcula-tion of exchange tensors for different multipoles by com-bination of DFT and the dynamical mean-field theory(DMFT) shows that J is expected to be ∼ 0.75 meV [28],but this result depends on chosen Hubbard U parame-ter. On another hand, the Schrieffer-Wolff transforma-tion applied to the two-site Hubbard-like model includ-ing the spin-orbit coupling with parameters estimated byDFT calculations gives J ∼ 1 meV [20]. Finally, it hasto be mention that the same octupolar exchange fieldmay induce a small magnetic moment through Eg-T2ghybridization, as detected by zero-field µSR [12, 13].More direct proof of the octupolar order may be givenby impurity doping that breaks the local symmetry andinduces detectable phenomena such as the formation ofmagnetic dipoles localized near the impurities [20]. Largeangle neutron scattering, as has been applied to the Cepyrochlores, may be used for the direct observation ofmagnetic octupoles [29, 30]. Recently, non-linear Halleffect at high frequencies has been proposed as a probeof magnetic ferro-octupolar order [31, 32]. The latter twomethods would become applicable to Ba2CaOsO6 whensingle crystals become available.FIG. 5. Schematic drawing of the ferro-octupolar order of theOs6+ ions in Ba2CaOsO6. Red and blue colors indicate thedistribution of spin-up and spin-down electrons, respectively.Nearest-neighbor Os atoms are connected by blue lines.IV. CONCLUSIONAlthough XMCD is not a direct probe of magnetic oc-tupoles by principle, the present temperature-dependentXMCD study at the Os L2,3 edge combined with ligand-field multiplet calculation revealed the splitting of thelowest Jeff = 2 state of the t22g multiplet of the Os6+ ioninto the non-Kramers Eg doublet ground state and thethe T2g triplet excited states separated by the residualcubic splitting ∆c ∼ 18 meV. From our ligand-field mul-tiplet calculation of the XMCD spectra, we concludedthat, if the exchange field on the Os6+ ion were as strongas ∼300 T in the ferro-octupolar ordered state, the non-Kramers double ground state would be split by ∼4 meVand may give the transition temperature of T ∗ ∼ 50 Kto the ‘hidden ordered’ state.ACKNOWLEDGEMENTSWe are grateful to M. Haverkort, C. Franchini, L.Pourovskii, and A. Paramekanti for useful discussions.The experiment at BL39XU of SPring-8 was performedwith the approval of the Japan Synchrotron Radia-tion Research Institute (JASRI) under Proposal No.2022A1610. This work was partly supported by theNational Science and Technology Council of Taiwanunder Grant Nos. 103-2112-M-213-008-MY3, 108-2923-M-548-213-001, and 113-2112-M-007-033 and by theJapan Society for the Promotion of Science under GrantNos. JP20K14416, JP22K03535, JP23K11709, andJP25K01657. S.V.S. was supported by the Ministry ofScience and Higher Education of the Russian Federation.A.F. acknowledges the support of the Yushan Fellow Pro-gram and the Center for Quantum Science and Technol-ogy within the framework of the Higher Education SproutProject under the Ministry of Education of Taiwan.5DATA AVAILABILITYAll the data that support the findings of this articleare available upon reasonable request.Appendix A: X-ray magnetic circular dichroismmeasurements and sum rule analysisHigh-quality polycrystalline samples of Ba2CaOsO6were synthesized and characterized as described inRef. [15]. The obtained samples were gray sintered pelletswith a diameter of ∼ 5 mm and a thickness of ∼ 2 mm.Os L2,3-edge XAS and XMCD measurements were per-formed at the hard X-ray beamline BL39XU of SPring-8[33]. The magnetic field B up to 7 T was applied parallelto the X-ray beam using a superconducting magnet. Themeasurements were done in the grazing incident geome-try (∼10◦ incidence angle). We note that the polycrys-talline grain sizes were much smaller than the beam size(∼ 0.4 mm×0.4 mm). The circular polarization of the Xrays was switched at each photon energy at a repetitionrate of ∼ 30 Hz using a diamond phase shifter. The de-gree of the circular polarization of the X rays was betterthan 90%. In order to eliminate spurious XMCD signalsoriginating from differences in the optical paths of thetwo polarizations, the XAS and XMCD spectra taken atB = ±7 T were averaged. The sample was cooled usinga closed-cycle refrigerator. The absorption signals werecollected in the partial fluorescence-yield (PFY) modeusing a silicon drift detector (SDD) located nearly alongthe sample-normal direction (perpendicular to the inci-dent X-ray direction). The Os Lα (5d → 2p 32) and Lβ(5d → 2p 12) fluorescence intensities were monitored forthe Os L3 (2p 32→ 5d) and L2 (2p 12→ 5d) edges, respec-tively. No surface treatments prior to the measurementswere made because the probing depth of the PFY modewas sufficiently large (∼ a few µm). The dead time of theSDD was corrected based on the nonparalyzable model[34, 35].The XAS spectra were normalized so that the edgejump heights at the L3 and L2 edges, which were deducedfrom the center line of the extended X-ray absorption finestructure (EXAFS) oscillation, were equal to the ratio of2:1. To compare the experimental XAS spectra [Fig. 3(a)and (b)] with the calculated ones, one needs to subtractthe white-line (step-like) background and the EXAFS os-cillation in the post-edge region. This has been done byfitting the experimental XAS spectra by the sum of twoLorentz functions and their integrals (arctangent func-tions), and then extracting the Lorentz function part.The results are shown in Fig. 3. We note that, the un-certainties of the dead-time correction, spectral normal-ization by the edge jump heights, and the background-removal procedure described above, may lead to a totalsystematic error of ∼ 10% in the magnetic moments de-duced from the XMCD sum rules [Eqs. (A1), (A2)].The orbital magnetic moment Morb and the ‘effective’spin magnetic momentM effspin have been deduced from theXAS and XMCD spectra using XMCD sum rules [36, 37]:Morb = −4∫L3+L2(µ+ − µ−)dν3∫L3+L2(µ+ + µ−)dν(10−Nd), (A1)M effspin ≡Mspin +72MT= −2∫L3(µ+ − µ−)dν − 4∫L2(µ+ − µ−)dν∫L3+L2(µ+ + µ−)dν(10−Nd).(A2)Here, µ+ (µ−) is the XAS intensity for the positive (neg-ative) helicity as a function of photon energy hν, L3 (L2)is the 2p3/2 → 5d (2p1/2 → 5d) absorption edge, and Ndis the number of electrons in the 5d band. In the presentwork, Nd is assumed to be the formal occupation num-ber of 2 of the Os6+ (5d2) ion. Although Nd is deviatedfrom the formal occupation number due to hybridizationwith ligand (O 2p) orbitals, we have ignored the devia-tion here because the absolute values of Morb and Mspinare not important for the present purposes. As shown inEq. (A2), the magnetic moment deduced from the spinsum rule is the sum of the spin magnetic moment Mspinand an additional term (7/2)MT called ‘magnetic dipole’defined as MT ≡ −2∑i⟨si − 3(si · ri)ri/r2i ⟩, where riand si are, respectively, the position and the spin angu-lar momentum operators of the i-th electron [25–27, 37].The magnetic dipole MT represents the anisotropy ofspin distribution and can be large in systems with a lowsymmetry or with strong SOC [25–27, 37]. Thus, thedifference between M effspin and Mspin gives the degree ofanisotropic spin-density distribution induced by the spin-orbit-entangled electronic structure.Appendix B: Ligand-field multiplet calculationLigand-field multiplet calculations were performed byusing the XTLS 8.5 package [38]. In general, the Slaterintegrals F ’s andG’s (anisotropy of Coulomb interaction)and the SOC coupling constant ζ in solids are smallerthan those of isolated atoms because the wavefunctionsare spatially more extended due to hybridization. Inorder to model this effect, the atomic Slater integralsand ζ, deduced from Hartree-Fock calculations [39, 40],were multiplied by constant factors RSlater and RSOC(0 ≤ RSlater < 1, 0 ≤ RSOC < 1), respectively. RSlaterand RSOC and the cubic ligand-field splitting ∆LF andwere treated as adjustable parameters. We used ∆LF =4.3 eV, ζ = 0.33 eV, and the reduction factor of 40 %for the Slater integrals between the Os 5d orbitals. Notethat the spin-orbit coupling for the 5d shell ζ and thatfor the t2g shell ζ ′ are related via ζ = −ζ ′ [24]. Hund’scoupling JH between two d electrons is related to Slaterintegrals through JH = 349F2 + 20441F4 = 0.27 eV [41].6TABLE I. Parameter values for the Os 5d electrons hybridizedwith O 2p orbitals in Ba2CaOsO6 used in the present work.Parameter Symbol Value (eV)ligand-field splitting ∆LF 4.3Spin-orbit coupling for the 5d shell ζ 0.33Hund’s coupling JH 0.27These parameter values are tabulated in Table I. Weassumed that the incident X ray is parallel to the cubic[001] direction. We confirmed that the spectral line shapedoes not change in the case of cubic symmetry, as longas the incident X ray and the magnetic field are parallel.In the calculation of the Os L2,3-edge XMCD spectra,the Zeeman energy due to the magnetic field −µBB ·(L + 2S), where L and S are the orbital and spin an-gular momenta, respectively, was included. 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