# Fileset

[PhysRevLett.131.216501.pdf](https://mdr.nims.go.jp/filesets/2f8ba9b4-d38d-4561-abef-f3bbec7a577e/download)

## Creator

Norimasa Sasabe, Masaichiro Mizumaki, Takayuki Uozumi, [Yuichi Yamasaki](https://orcid.org/0000-0002-8560-3462)

## Rights

[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Ferroic order for anisotropic magnetic dipole term in collinear antiferromagnets of (t2g)4 system](https://mdr.nims.go.jp/datasets/3a281e12-3f5a-4825-93a6-9593db6a0f44)

## Fulltext

Ferroic Order for Anisotropic Magnetic Dipole Term in Collinear Antiferromagnets of (t2g)4 SystemFerroic Order for Anisotropic Magnetic Dipole Term inCollinear Antiferromagnets of ðt2gÞ4 SystemNorimasa Sasabe,1,* Masaichiro Mizumaki,1,* Takayuki Uozumi,2 and Yuichi Yamasaki3,41Japan Synchrotron Radiation Research Institute, SPring-8 Kouto, Sayo, Hyogo 679-5198, Japan2Graduate School of Engineering, Osaka Metropolitan University, Sakai, Osaka 599-8531, Japan3Center for Basic Research on Materials, National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0047, Japan4RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan(Received 27 February 2023; accepted 21 August 2023; published 22 November 2023)We present the possibility of x-ray magnetic circular dichroism on RuO2 with collinear antiferromag-netism (AFM). Given that the crystal symmetry breaks the time reversal symmetry when the antiparallelspin aligns along the [100] direction, the expectation vector of the anisotropic magnetic dipole operator htiremains uncanceled along the [010] direction. Our Letter reveals that the magnetic dipole (Tz) term in thex-ray magnetic circular dichroism is induced by the residual hti. Because the features of the magneticmoment can be detected via absorption measurements even in the AFM, this technique will be useful fordetermining the magnetic phase, the Van Vleck–type paramagnet or the excitonic AFM in ðt2gÞ4 system.DOI: 10.1103/PhysRevLett.131.216501Spatial inversion and time reversal symmetry (TRS)breaking are the most fundamental information for macro-scopic properties in condensed matter physics; the emer-gence of electric polarization and magnetization leads tospatial inversion and TRS breaking, respectively [1].Simultaneous spatiotemporal symmetry breaking allowsthe magnetoelectric effect [2], and is sometimes realized inmultiferroic materials [3], where ferroelectric polarizationand ferromagnetic magnetization coexist [4]. Symmetrybreaking can be detected by using the symmetric feature ofthe measurement probe. For example, circularly polarizedlight with breaking the TRS can detect the magnetization,which is known as magnetic circular dichroism (MCD) [5].However, the existence of magnetization is a sufficientcondition, but not a necessary condition for the occurrenceof MCD. In other words, the MCD can be allowed even inthe absence of magnetization, if the TRS is brokenthroughout the crystal [6,7].In recent years, emergent phenomena have beenobserved in the TRS breaking antiferromagnets with nonet magnetization. For example, Mn3Sn is a 120° anti-ferromagnetic structure with negative vector chirality on anetwork of breathing Kagomé lattices, which breaks theTRS. This material has been reported to be a Weylantiferromagnet, and exhibits the anomalous Hall effect(AHE) and magneto-optical Kerr effect [7–9]. AlthoughMn3Sn has a noncollinear magnetic structure of 120°, theAHE derived from TRS breaking has also been theoreti-cally proposed in a collinear antiferromagnet RuO2 [10,11].When the antiferromagnetic Néel vector N ¼ m2 −m1with magnetic moment mi of the ith Ru ion is orientedalong the [100] direction as shown in Fig. 1(a), the TRS isbroken and the Berry curvature Ω emerges along the [010]direction [10].X-ray MCD (XMCD) has been theoretically proposed tobe possible in a noncollinear antiferromagnet [12–15].Considering the orbital state frozen from the crystal fieldand a 120° magnetic structure with negative vector chirality,FIG. 1. (a) Crystal and magnetic structure with the Néel vectoraligned along the ½1̄00� direction for the rutile structure. (b) Defi-nition of local coordinates of RuO6 cluster, and (c) electronicconfiguration for an intermediate spin state ðt2gÞ4 without thespin-orbit interaction (SOI). Two approximation states, i.e., (d) jjcoupling and (e) LS coupling, are shown for the electronic statewith SOI.PHYSICAL REVIEW LETTERS 131, 216501 (2023)Editors' Suggestion0031-9007=23=131(21)=216501(6) 216501-1 © 2023 American Physical Societyhttps://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevLett.131.216501&domain=pdf&date_stamp=2023-11-22https://doi.org/10.1103/PhysRevLett.131.216501https://doi.org/10.1103/PhysRevLett.131.216501https://doi.org/10.1103/PhysRevLett.131.216501https://doi.org/10.1103/PhysRevLett.131.216501the XMCD originating from the anisotropic magneticdipole (AMD) order is expressed [13]. The antiferromag-netic structure across multiple magnetic sites is understoodas augmented (cluster) multipoles; the magnetic structure isclassified as the augmented magnetic octupoles (AMO)[16]. The operator of AMD is not orthogonal to that ofAMO, and thus its projection component is detectable inXMCD as the Tz term [17]. Recently, the role of the AMDhas attracted much attention, and the relationship betweenthe AMD and AHE has been discussed even for collinearmagnetic structures [18]. In this study, we evaluate thepossibility of XMCD originating from the Tz term in thecollinear antiferromagnet rutile RuO2 from symmetryarguments and spectral calculations. In particular, weconsider the effect of spin-orbit interaction (SOI), whichmay act more strongly in the 4d electron system than in the3d electron system.The AMD operator tα (α ¼ x, y, z) is expressed astα ¼Pβ¼x;y;z Qαβsβ, where s and Qαβ denotes the spin andthe electric quadrupole operators, respectively [19,20]. Thetz term depends on the quadrupole moment of the 4dorbitals even though the spin moment is the same [21,22].The quantization axis (QA) of the collinear spin orderedstate is often determined as the parallel direction of spin(skz) and/or incident x-ray wave vector (kkz) [28], whereasa better approach may be to determine the QA of theantiferromagnet by considering the quadrupole momentaccompanied by the symmetry of the crystal field due to theelectron-lattice interaction [14]. If the magnetic symmetryoperation of the antiferromagnetic order does not matchthat of the crystal, a residual AMD may be allowed, andthen the Tz term in XMCD may appear macroscopically. Inorder to qualitatively understand the collinear antiferro-magnets of the 4d4 compound RuO2 [29], we examined(i) the expression of the AMD and the Tz term from theviewpoint of symmetry and (ii) the dependence of the Tzterm on the SOI. Moreover, the XMCD spectra arecalculated with consideration of the full multiplet effect,and then we discuss the relationship between the AMDand XMCD.The Ru ion is located in a distorted octahedron withoxygen coordination of local D2h symmetry, as shown inFig. 1(b). We assume that the local z1 axis (x1y1 plane) isparallel (perpendicular) to the [110] crystalline direction;thus, the electrons occupy the t2g set (dx2−y2=dyz=dxz) andeg set (d3z2−r2=dxy) [30]. Because of competition betweenthe on-site Coulomb exchange of Hund’s coupling and thecrystal-field splitting (CFS), a strong crystal field favors thelow-spin (S ¼ 0) or intermediate-spin (S ¼ 1) state withðt2gÞ4, whereas the Coulomb exchange favors the high-spinstate (S ¼ 2) with ðt2gÞ3ðegÞ. Before discussing RuO2, weintroduce the electronic structure of the ðt2gÞ4 state [31]. Ifthe SOI is not taken into account, the four electrons areoccupied [32], as shown in Fig. 1(c), where the orbitalangular momentum is quenched (Lz ¼ 0). In contrast, inthe limit of a strong SOI in Fig. 1(d), the t2g orbital splitsinto Jeff ¼ 1=2 and 3=2, and then electrons fully occupythe quartet Jeff ¼ 3=2 state, resulting in a nonmagneticJ ¼ 0 state. Even in the weak SOI case in Fig. 1(e), the SOImakes the nonmagnetic J ¼ L − S ¼ 0 state the moststable. Therefore, Ru4þ is estimated to be nonmagnetic,but magnetism may be acceptable because of the inter-atomic interaction, as discussed in detail below.Figures 2(a) and 2(b) show the collinear antiferromag-netic structures for d3z2−r2 orbital with Nk½1̄00� andNk½100�, respectively. Considering the quadrupole anddirection of the spin moment at each site of antiferro-magnetism (AFM) ordered, the AMD moment hti is notcompletely canceled out and has a ferroic component alongthe [010] direction; see Sec. I of Supplemental Material[21]. Therefore, the pure Tz term XMCD can be expressedwhen x rays are irradiated along the [010] direction in thecase of Nk½100�. By the same argument, tk½110� is allowedif Nk½110�.The expected t vectors for the different orbitals areshown in Figs. 2(c)–2(f). For the dx2−y2 and dxy orbitals,there is also a residual t vector along the ½01̄0� direction;however, they are in the opposite direction to that ford3z2−r2 . In contrast, the residual t vector of dyz is parallel tothe [010] direction. The absolute values of these residual tvectors are the same, whereas the residual t vector of the dzxorbital is zero because the t vectors are antiparallel to thespin moment. In order to study AMD of a multielectronsystem, we employ the summation of the t vectors(TðionÞ ≡Pα tα) at each site and the summation amongnonequivalent sites ðT ≡Pi TðionÞi ; i siteÞ. Because T ≠ 0FIG. 2. Anisotropic magnetic dipole operator hti for thecollinear antiferromagnet rutile, (a), (b) d3z2−r2 , (c) dx2−y2 ,(d) dxy, (e) dyz, and (f) dzx orbitals. The Néel vector is set to[100] for (b) and ½1̄00� for the other frames.PHYSICAL REVIEW LETTERS 131, 216501 (2023)216501-2is a sufficient condition for emerging XMCD of antiferro-magnets, it is necessary to lift the degeneracy due to thecrystal field and spin polarization to cause a difference inthe up and down spin densities of states at the same time. Inthe ðt2gÞ4 case, where there are two holes in dyz and dzx forthe t2g set, as shown in Fig. 1(c), the dyz orbital can solelycontribute to T, and the XMCD signal is expected to arise.In order to discuss the possibility of the XMCD responseof RuO2 quantitatively, it is necessary to examine thecompetition among the three competing interactions, i.e.,the CFS, SOI, and interatomic magnetic interactionbetween Ru1 and Ru2. Therefore, we investigated thefeatures of the simple ðt2gÞ4 model within the mean fieldapproximation of interatomic magnetic interaction whichis treated as a molecular field (MF). The effectiveHamiltonian of the simple ðt2gÞ4 model is described asHeff ¼ ΔðLZÞ2 þ λL · S − gJm · hMF; ð1Þwhere Δ, λ, and gJhMF are the parameters of the CFS, SOI,and MF, respectively. The t2g levels are set as shown inFig. 1(c) [32]. The Landé g factor becomes gJ ¼ 3=2because of the LS coupling approximation with L ¼ S ¼ 1,and hMF is assumed to be applied along the z direction oflocal coordinates in the RuO6 cluster, as shown in Fig. 1(b).Figure 3(a) shows a phase diagram and the expectedvalue of the total AMDmoment hTzi of the ðt2gÞ4 electronicstate, where the horizontal axis indicates the SOI valuenormalized by the crystal field, i.e., λ=Δ. The left verticalaxis indicates the MF intensity which affect the phasetransition of the ðt2gÞ4 model, and the right vertical axisshows the hTzi value. At λ ¼ 0 (the weak SOI limit), theground state of ðt2gÞ4 is the S ¼ 1 triple states, and theSz ¼ 0 state contains the basis J ¼ 0 of the LS couplingscheme shown in Fig. 1(e). The Sz ¼ 1 state contains thebasis Jz ¼ 1 (J ¼ 1) of the LS coupling scheme, andSz ¼ −1 contains Jz ¼ −1. The J ¼ 0 state, paramagnet-ism (PM), is the most stable state under a moderate SOI; asthe SOI becomes stronger, it soon becomes the nomi-nally nonmagnetic phase, J̃ ¼ 0 (Van Vleck–type PM).However, the interatomic magnetic interaction between theexcited J ¼ 1 states induces the magnetic moment; thus, itsplits the jJ;mi multiplet by the Zeeman energyEMF ¼ −gJm · hMF. Hence, because of the interactionbetween the excitonic triplets, which are sometimes calleda triplon condensation [32] or spin-orbit excitonic anti-ferromagnet [33–35], the magnetic state, J̃ ¼ 1 (excitonicAFM), becomes more stable than the Van Vleck–type PM.We focus on the hTzi values of the ðt2gÞ4 model andcompare them with those of the Ru4þ model whichincludes CFS and MF terms of the interatomic spininteraction within full multiplet effect of Ru4þ; seeSupplemental Material, Sec. II [21] and Refs. [23–27].For the ðt2gÞ4 model, the blue and red dotted lines inFig. 3(a) indicate the hTzi values of the excitonic AFM andVan Vleck–type PM, respectively. With an increase in theSOI parameter λ, the hTzi value of excitonic AFMdecreases, while that of the Van Vleck–type PM remainszero. This indicates that measurement of hTzi value candetect the ðt2gÞ4 electronic state, where the competitionbetween the two states allows quantum phase transition[36,37]. In order to discuss the electronic state of actualRuO2, the hTzi values of the Ru4þ model are simulated, asshown in Fig. 3(b). Here, the red, green, and blue symbolsindicate the behavior of the hTzi value depending on theMF intensity. The black dotted line indicates the hTzivalues from Fig. 3(a), where the CFS Δ in Eq. (1) is 1.0 eVand that λ is estimated as approximately 0.1 eV in ζRu4þ4d .Under this condition, the phase transition between the exci-tonic AFM and Van Vleck–type PM occurs underhMF ∼ 0.01 eV, as shown in Fig. 3(b). From the resultsfor the Ru4þ model, as well as the result of the ðt2gÞ4 model,the phase transition is observed under weak interatomicmagnetic interaction; thus, the electronic states of the Ru4þmodel are regarded as the excitonic AFM and Van Vleck–type PM. The limited basis set of the ðt2gÞ4 model causes aFIG. 3. (a) Phase diagram and hTzi values for the ðt2gÞ4 model.The inset shows the direction of hMF on the RuO6 cluster underhMFkQA. (b) hTzi value of the Ru4þ model. hTzi values areindicated by colored symbols, and the black dotted line indicatesthe hTzi behavior of the ðt2gÞ4 model where gJhMF=Δ ¼ 0.0118when λ=Δ ¼ 0.1. ζ4d is the spin-orbit coupling constant for the4d orbital, where the ζ4d value of Ru4þ ion (ζRu4þ4d ) is 0.161 eV.PHYSICAL REVIEW LETTERS 131, 216501 (2023)216501-3mild decrease in the hTzi value of excitonic AFM; however,the decrease in the hTzi value of the Ru4þ model is largerbecause the occupation in the eg set arises from thestrengthened SOI. Moreover, for the Ru4þ model, thehTzi value becomes nonzero because the intra-atomic spininteraction affects the Van Vleck–type PM. Thus, theelectronic state of the Ru4þ model with hMF ¼ 0 corre-sponds to a pure Van Vleck–type PM.Figures 4(a) and 4(b) show the results of the spectralcalculation by employing the Ru4þ model under TRSbreaking; the incident x-ray directions in Figs. 4(a) and4(b) are parallel to [010] and [100], respectively. The upperand lower results in each figure are the x-ray absorptionspectra (μþ þ μ−) and XMCD (μþ − μ−), respectively,where μþ and μ− represent the helicities of x rays. Thesolid lines indicate the total intensity between the Ru1 andRu2 sites, and the dotted lines in Fig. 4(b) indicate theXMCD intensity on the Ru1 site surrounded by the graydotted square. We determined the XMCD responses ofRuO2 reflecting the AMD, Van Vleck–type PM (red line)and excitonic AFM (blue line), where the Van Vleck–typePM (excitonic AFM) is hMF ¼ 0.003 eV (0.3 eV) withζRu4þ4d . The black line in Fig. 4(a) indicates the XMCDsimulation of the pure Van Vleck–type PM with ζRu4þ4d , andthe green line indicates that of the ideal excitonic AFMwithhMF ¼ 0.3 and ζ4d ¼ 0.0 (in units of eV). In thesecalculations, the MF is applied in the [100] direction; thus,the spin component of Ru1 (Ru2) tends to be parallel along[100] (½1̄00�), where the spin direction of Ru1 is antiparallelto that of Ru2 without the SOI (ζ4d ¼ 0.0 eV).As shown in Fig. 4(a), the XMCD signal is detectable forthe excitonic AFM, whereas it is difficult to detect thesignal of the Van Vleck–type PM. In contrast, in Fig. 4(b),neither the XMCD signal of the Van Vleck–type PM northat of the excitonic AFM is observed. This behavior isconsistent with the expression for tk½010� whenNk½1̄00�, asshown in Fig. 2; therefore, the AMD is the origin of theXMCD of the RuO2 collinear antiferromagnetic state, asindicated by the green lines.The XMCD signal of the excitonic AFM originated fromthe direction of the AMD moment, which is confirmed inXMCD of Mn3Sn [38,39]. Focusing on the red line inFig. 4(a), we discuss the XMCD signal of the Van Vleck–type PM. It is well known that the Van Vleck PM is mixedwith the magnetic states when a magnetic field is applied,and then the XMCD signal is observed in the Van VleckPM under a magnetic field [40]. Therefore, the red line is areasonable result; this weak XMCD signal results in acanted magnetic moment due to hMF ¼ 0.003 eV. Becausethe XMCD signal arises in both states, the differencebetween the Van Vleck–type PM and the excitionicAFM is discussed. One clear difference is that theXMCD intensity of the excitonic AFM significantlyexceeds that of the Van Vleck–type state, which is affectedby the Tz value. In particular, the Tz value of the excitonicAFM is not easily influenced by external magnetic field[38], whereas that of Van Vleck–type PM is sensitive [40].In addition to the intensity, the XMCD features differbetween the excitonic AFM and Van Vleck–type PM;particularly, the structure near 73 eV in LII structure. Thus,investigation of these XMCD spectral features is useful forclarifying the electronic state of RuO2.Finally, we discuss the possibility of detecting theXMCD signal and Tz term in actual RuO2 crystals. Ithas been reported that the magnetic diffraction at (100) isobserved in a bulk crystal RuO2 by neutron scattering, andthe magnetic moment is estimated to be approximately0.05 μB at the ambient temperature [31]. In addition, aresonant x-ray scattering (RXS) study of the Ru L edge inRuO2 has revealed that the Néel vector is nearly alignedwith the c axis [41]. The small magnetic moment suggeststhat it is close to the Van Vleck PM with moderate spin-orbit coupling, in which the pure XMCD Tz term would notbe allowed for the magnetic structure of the bulk RuO2.Moreover, the XMCD signal of RuO2 under AFM alongthe c axis is not observable considering the configuration ofthe AMO hti. Thus, we consider that ferroic Tz is notobserved in bulk RuO2. Instead of ferroic Tz, contributionof the antiferroic Tz term may be included in the RXSspectrum for (100) diffraction, but it is usually difficult toseparate from the strong Sz term. In contrast, for the AHE-allowed thin films, the magnetic structures should differfrom those in the bulk owing to the change in spinanisotropy caused by the epitaxial strain [42]. BecauseFIG. 4. X-ray absorption spectroscopy (upper result) andXMCD (lower result) spectra around absorption Ru-L edgeswhere (a) kk½010� and (b) kk½100�: the right (left) structuresdenote LII (LIII).PHYSICAL REVIEW LETTERS 131, 216501 (2023)216501-4the compressive strain strengthens the crystal field, itreduces λ=Δ, resulting in an increase in the excitonicmagnetic moment. Conversely, it is expected that tensilestrain from the substrate stabilizes the Van Vleck–type PM.This strain control causes a phase transition between theVan Vleck–type PM and excitonic AFM of the ðt2gÞ4system; therefore, our theoretical prediction can be verifiedby paying attention to the hti behavior or XMCD Tz. Inorder to estimate the Tz term, we consider that the XMCDsum rule must be carefully applied because of the QAtreatment between Ru1 and Ru2. For acquiring the Tzvalues, theoretical XMCD studies with full multiplet theoryare helpful, along with band calculations.In summary, we theoretically studied how the SOIsuppresses the Tz value and XMCD spectra by focusingon the AMD behavior of the collinear antiferromagneticstate of RuO2. Using the ðt2gÞ4 model, we investigated thephase diagram and Tz value of the ðt2gÞ4 system, where thephase transition and its quantum states could be controlledby changing the tensile strength. Moreover, XMCD cal-culations were performed using the Ru4þ model based onfull multiplet theorem. It was shown that the Tz term isacceptable for lifting orbital degeneracy owing to thecrystal field and exchange interaction when the Néelvectors are oriented along the [100] and [110] directions.Because XMCD is detectable in the hti behavior or Tz valueof the antiferromagnet, the technique is suitable forevaluating the excitonic (triplon) magnetic moment inthe ðt2gÞ4 system. The AMD is related to the Berrycurvature [18]; thus, spectroscopic and imaging analysisof the AMD through XMCD measurement is a usefulmethod for investigating the AHE and related phenomena[43,44]. Considering the investigations of noncollinearMn3Sn and the present XMCD study on collinear RuO2,we expect that studying the hti behavior is key to under-standing new AHE materials and developing functionaldevices.We thank T-h. Arima, M. Hireshberger, H. Nakao, M.Kimata, T. Nakamura, and H. Kusunose for stimulatingdiscussion. This work was supported in part by PRESTOGrant No. JPMJPR177A, Grant-in-Aid for ScientificResearch No. JP16H05990 and No. JP19H04399 fromthe Japan Society for the Promotion of Science (JSPS),MEXT Quantum Leap Flagship Program (MEXTQ-LEAP) Grant No. JPMXS0120184122, and JSTCREST Grant No. JPMJCR1861.*Present address: Faculty of Science, Kumamoto University,Kumamoto 860-8555, Japan.[1] W. Eerenstein, N. Mathur, and J. F. Scott, Nature (London)442, 759 (2006).[2] M. Fiebig, J. Phys. D 38, R123 (2005).[3] M. Fiebig, T. Lottermoser, D. Meier, and M. Trassin,Nat. Rev. Mater. 1, 1 (2016).[4] Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T. Arima,and Y. Tokura, Phys. Rev. Lett. 96, 207204 (2006).[5] P. Stephens, Annu. Rev. Phys. Chem. 25, 201 (1974).[6] T. Sato, N. Abe, S. Kimura, Y. Tokunaga, and T.-h. Arima,Phys. Rev. Lett. 124, 217402 (2020).[7] T. Higo, H. Man, D. B. Gopman, L. Wu, T. Koretsune,O. M. van’t Erve, Y. P. Kabanov, D. Rees, Y. Li, M.-T.Suzuki et al., Nat. Photonics 12, 73 (2018).[8] S. Nakatsuji, N. Kiyohara, and T. Higo, Nature (London)527, 212 (2015).[9] M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki, D. Nishio-Hamane, R. Arita, Y. Otani, and S. Nakatsuji, Nat. Phys. 13,1085 (2017).[10] L. Šmejkal, R. González-Hernández, T. Jungwirth, and J.Sinova, Sci. Adv. 6, eaaz8809 (2020).[11] L. Šmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12,040501 (2022).[12] S. Wimmer, S. Mankovsky, J. Minár, A. N. Yaresko, and H.Ebert, Phys. Rev. B 100, 214429 (2019).[13] Y. Yamasaki, H. Nakao, and T. hisa Arima, J. Phys. Soc.Jpn. 89, 083703 (2020).[14] N. Sasabe, M. Kimata, and T. Nakamura, Phys. Rev. Lett.126, 157402 (2021).[15] G. van der Laan, Phys. Rev. B 104, 094414 (2021).[16] M.-T. Suzuki, T. Koretsune, M. Ochi, and R. Arita, Phys.Rev. B 95, 094406 (2017).[17] H. Kusunose, R. Oiwa, and S. Hayami, J. Phys. Soc. Jpn.89, 104704 (2020).[18] S. Hayami and H. Kusunose, Phys. Rev. B 103, L180407(2021).[19] J. Stöhr, J. Magn. Magn. Mater. 200, 470 (1999).[20] T. Oguchi and T. Shishidou, Phys. Rev. B 70, 024412 (2004).[21] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.131.216501 for detailsof the anisotropic magnetic dipole operator and the em-ployed Hamiltonian, which includes Refs. [22–27].[22] J. Stöhr, J. Electron Spectrosc. Relat. Phenom. 75, 253(1995).[23] R. D. Cowan, The Theory of Atomic Structure and Spectra(University of California Press, Berkeley, 1981), Vol. 3.[24] K. Okada and A. Kotani, J. Phys. Soc. Jpn. 58, 2578(1989).[25] A. Tanaka and T. Jo, J. Phys. Soc. Jpn. 61, 2040 (1992).[26] M. Taguchi, T. Uozumi, and A. Kotani, J. Phys. Soc. Jpn.66, 247 (1997).[27] M. Matsubara, T. Uozumi, A. Kotani, Y. Harada, and S.Shin, J. Phys. Soc. Jpn. 69, 1558 (2000).[28] J. Crocombette, B. Thole, and F. Jollet, J. Phys. Condens.Matter 8, 4095 (1996).[29] Z. Hu, H. von Lips, M. S. Golden, J. Fink, G. Kaindl,F. M. F. de Groot, S. Ebbinghaus, and A. Reller, Phys. Rev.B 61, 5262 (2000).[30] C. A. Occhialini, V. Bisogni, H. You, A. Barbour, I. Jarrige,J. F. Mitchell, R. Comin, and J. Pelliciari, Phys. Rev. Res. 3,033214 (2021).[31] T. Berlijn, P. C. Snijders, O. Delaire, H.-D. Zhou, T. A.Maier, H.-B. Cao, S.-X. Chi, M. Matsuda, Y. Wang, M. R.Koehler et al., Phys. Rev. Lett. 118, 077201 (2017).PHYSICAL REVIEW LETTERS 131, 216501 (2023)216501-5https://doi.org/10.1038/nature05023https://doi.org/10.1038/nature05023https://doi.org/10.1088/0022-3727/38/8/R01https://doi.org/10.1038/natrevmats.2016.46https://doi.org/10.1103/PhysRevLett.96.207204https://doi.org/10.1146/annurev.pc.25.100174.001221https://doi.org/10.1103/PhysRevLett.124.217402https://doi.org/10.1038/s41566-017-0086-zhttps://doi.org/10.1038/nature15723https://doi.org/10.1038/nature15723https://doi.org/10.1038/nphys4181https://doi.org/10.1038/nphys4181https://doi.org/10.1126/sciadv.aaz8809https://doi.org/10.1103/PhysRevX.12.040501https://doi.org/10.1103/PhysRevX.12.040501https://doi.org/10.1103/PhysRevB.100.214429https://doi.org/10.7566/JPSJ.89.083703https://doi.org/10.7566/JPSJ.89.083703https://doi.org/10.1103/PhysRevLett.126.157402https://doi.org/10.1103/PhysRevLett.126.157402https://doi.org/10.1103/PhysRevB.104.094414https://doi.org/10.1103/PhysRevB.95.094406https://doi.org/10.1103/PhysRevB.95.094406https://doi.org/10.7566/JPSJ.89.104704https://doi.org/10.7566/JPSJ.89.104704https://doi.org/10.1103/PhysRevB.103.L180407https://doi.org/10.1103/PhysRevB.103.L180407https://doi.org/10.1016/S0304-8853(99)00407-2https://doi.org/10.1103/PhysRevB.70.024412http://link.aps.org/supplemental/10.1103/PhysRevLett.131.216501http://link.aps.org/supplemental/10.1103/PhysRevLett.131.216501http://link.aps.org/supplemental/10.1103/PhysRevLett.131.216501http://link.aps.org/supplemental/10.1103/PhysRevLett.131.216501http://link.aps.org/supplemental/10.1103/PhysRevLett.131.216501http://link.aps.org/supplemental/10.1103/PhysRevLett.131.216501http://link.aps.org/supplemental/10.1103/PhysRevLett.131.216501https://doi.org/10.1016/0368-2048(95)02537-5https://doi.org/10.1016/0368-2048(95)02537-5https://doi.org/10.1143/JPSJ.58.2578https://doi.org/10.1143/JPSJ.58.2578https://doi.org/10.1143/JPSJ.61.2040https://doi.org/10.1143/JPSJ.66.247https://doi.org/10.1143/JPSJ.66.247https://doi.org/10.1143/JPSJ.69.1558https://doi.org/10.1088/0953-8984/8/22/013https://doi.org/10.1088/0953-8984/8/22/013https://doi.org/10.1103/PhysRevB.61.5262https://doi.org/10.1103/PhysRevB.61.5262https://doi.org/10.1103/PhysRevResearch.3.033214https://doi.org/10.1103/PhysRevResearch.3.033214https://doi.org/10.1103/PhysRevLett.118.077201[32] C. Svoboda, M. Randeria, and N. Trivedi, Phys. Rev. B 95,014409 (2017).[33] G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).[34] N. Kaushal, R. Soni, A. Nocera, G. Alvarez, and E. Dagotto,Phys. Rev. B 101, 245147 (2020).[35] T. Feldmaier, P. Strobel, M. Schmid, P. Hansmann, and M.Daghofer, Phys. Rev. Res. 2, 033201 (2020).[36] A. Jain, M. Krautloher, J. Porras, G. Ryu, D. Chen, D.Abernathy, J. Park, A. Ivanov, J. Chaloupka, G. Khaliullin,B. Keimer, and B. Kim, Nat. Phys. 13, 633 (2017).[37] S. Kunkemöller, D. Khomskii, P. Steffens, A. Piovano,A. A. Nugroho, and M. Braden, Phys. Rev. Lett. 115,247201 (2015).[38] M. Kimata, N. Sasabe, K. Kurita, Y. Yamasaki, C. Tabata, Y.Yokoyama, Y. Kotani, M. Ikhlas, T. Tomita, K. Amemiya,H. Nojiri, S. Nakatsuji, T. Koretsune, H. Nakao, T.-h.Arima, and T. Nakamura, Nat. Commun. 12, 5582(2021).[39] S. Sakamoto, T. Higo, M. Shiga, K. Amemiya, S. Nakatsuji,and S. Miwa, Phys. Rev. B 104, 134431 (2021).[40] S. Agrestini, C.-Y. Kuo, K. Chen, Y. Utsumi, D. Mikhailova,A. Rogalev, F. Wilhelm, T. Förster, A. Matsumoto, T.Takayama, H. Takagi, M.W. Haverkort, Z. Hu, and L. H.Tjeng, Phys. Rev. B 97, 214436 (2018).[41] Z. H. Zhu, J. Strempfer, R. R. Rao, C. A. Occhialini, J.Pelliciari, Y. Choi, T. Kawaguchi, H. You, J. F. Mitchell, Y.Shao-Horn, and R. Comin, Phys. Rev. Lett. 122, 017202(2019).[42] Y. Yokoyama, Y. Yamasaki, M. Taguchi, Y. Hirata, K.Takubo, J. Miyawaki, Y. Harada, D. Asakura, J. Fujioka, M.Nakamura, H. Daimon, M. Kawasaki, Y. Tokura, and H.Wadati, Phys. Rev. Lett. 120, 206402 (2018).[43] S. Nakatsuji and R. Arita, Annu. Rev. Condens. MatterPhys. 13, 119 (2022).[44] L. Šmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12,031042 (2022).PHYSICAL REVIEW LETTERS 131, 216501 (2023)216501-6https://doi.org/10.1103/PhysRevB.95.014409https://doi.org/10.1103/PhysRevB.95.014409https://doi.org/10.1103/PhysRevLett.111.197201https://doi.org/10.1103/PhysRevB.101.245147https://doi.org/10.1103/PhysRevResearch.2.033201https://doi.org/10.1038/nphys4077https://doi.org/10.1103/PhysRevLett.115.247201https://doi.org/10.1103/PhysRevLett.115.247201https://doi.org/10.1038/s41467-021-25834-7https://doi.org/10.1038/s41467-021-25834-7https://doi.org/10.1103/PhysRevB.104.134431https://doi.org/10.1103/PhysRevB.97.214436https://doi.org/10.1103/PhysRevLett.122.017202https://doi.org/10.1103/PhysRevLett.122.017202https://doi.org/10.1103/PhysRevLett.120.206402https://doi.org/10.1146/annurev-conmatphys-031620-103859https://doi.org/10.1146/annurev-conmatphys-031620-103859https://doi.org/10.1103/PhysRevX.12.031042https://doi.org/10.1103/PhysRevX.12.031042