# Fileset

[supplemental_RuO2.pdf](https://mdr.nims.go.jp/filesets/1c3ced85-0960-413e-9647-a1b3ce82696d/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

Supplemental Material: Ferroic order for anisotropic magnetic dipole termin collinear antiferromagnets of (t2g)4 systemNorimasa Sasabe1, Masaichiro Mizumaki1, Takayuki Uozumi2, 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, JapanS1. ANISOTROPIC MAGNETIC DIPOLE OPERATORThe anisotropic magnetic dipole (AMD) moment t of aparticle electron state with orbital l and spin s , i.e., |ϕ⟩ =∑⟨lz,sz⟩ alz,sz |l, lz, s, sz⟩, is defined as the expectation value of theintra-atomic magnetic dipole operator ⟨t⟩ = ⟨ϕ|t̂|ϕ⟩ [1]; lz andsz represent the orbital and spin quantum numbers, respec-tively, and alz,sz is a normalization factor. The AMD operatortα (α = x, y, z) is expressed astα = [s − 3(r̂ · s)r̂]α =∑β=x,y,zQαβsβ, (1)where s and r̂ are the operators for spin and the unit vector ofposition, respectively, and Qαβ denotes the electric quadrupoleoperators [2, 3]. The tz term for dxy, dxy, dzx, dx2−y2 , and d3z2−r2are 2/7, -1/7, -1/7 , 2/7, and -2/7, respectively.Next, PT symmetry for the AMD moment is discussed.The magnetic space group of antiferromagnetism ordered isPnn′m′ when N ||[100] of the rutile structure, including themagnetic symmetric operations P, TMxτ1/2, Myτ1/2, andTMz [4]. Here, T , P, Mα (α=x, y, z), and τ1/2 denote thetime reversal, spatial inversion, mirror perpendicular to theα-axis, and translation of the (1/2, 1/2, 1/2) operations, re-spectively. Those operations for the AMD moment t result inT t = −t, Pt = t, and τ1/2t = t. The mirror operation isexpressed asMyt = (−tx, ty,−tz), (2)indicating that all components other than the component per-pendicular to the mirror are inverted. Thus, it can be con-firmed that t||[010] is allowed from the viewpoint of magneticsymmetry. When the vector N is inverted, ⟨t⟩ is inverted.S2. Model HamiltonianFor the Ru4+ model describing an electronic state of RuO2,Hamiltonian is given byHi = Hatom + HCEF + HMF, (3)where index i indicates Ru1 and Ru2 shown in Fig. 1(a) of themain text. The first term Hatom is expressed asHatom = ϵd∑γd†γdγ + ζ4d∑γ1,γ2(l · s)γ1,γ2 d†γ1dγ2+ ϵp∑γp†γpγ + ζ2p∑γ1,γ2(l · s)γ1,γ2 p†γ1pγ2+12∑γ1,γ2,γ3,γ4gdd(γ1, γ2; γ3, γ4) d†γ1dγ2 d†γ4dγ3+∑γ1,γ2,γ3,γ4gdp(γ1, γ2; γ3, γ4) d†γ1p†γ2pγ4 dγ3 ,(4)where d †γ represents the creation operator for a 4d electron, in-cluding a combined index γ with orbital and spin, and p †γ rep-resents the creation operator for a 2p core state [5–8]. Hatomincludes 4d level (ϵd), spin-orbit coupling constant for the 4dorbital (ζ4d), 2p level (ϵp), spin-orbit coupling constant for the2p orbital (ζ2p), Coulomb interaction between the 4d states(gdd), and the Coulomb interaction between the 4d and 2pstates (gdp). These spin-orbit coupling constants and the Slaterintegrals included in gdd and gdp are estimated from the ioniccalculation within the Hartree–Fock–Slater (HFS) method [9].The ζ4d of Ru4+ is 0.161 eV, and more we investigated spin-orbit interaction (SOI) dependence of the electronic state bychanging ζ4d. For the Slater integrals, 60% of the HFS valuesare used [10, 11]. The second term HCEF is determined byconsidering the one-electron potential of D2h symmetry [11],expressed asVcrys = B20C(2)0 + B22(C(2)2 +C(2)−2)+B40C(4)0 + B42(C(4)2 +C(4)−2) + B44(C(4)4 +C(4)−4),(5)whereC(k)q =√4π2k + 1Ykq, (6)and Ykq represents a spherical harmonic [12]. Consideringthe previous study in Ref. 11, the 10Dq value between eg(d3z2−r2 /dxy) and t2g (dx2−y2 /dyz/dzx) is 2.6 eV and the eg split-ting between dxy and d3z2−r2 is 0.6 eV. For the splitting of t2gorbitals in the present study, ∆ for the splitting between dx2−y2and dyz/dzx is 1.0 eV and the splitting between dyz and dzx is0.55 eV. The third term HMF is expressed asHMF =∑γ1,γ2(h(i)MF · s)γ1,γ2 d†γ1dγ2 , (7)where h(i)MF denotes the molecular field for the spin part of theRu1 and Ru2.2[1] J. Stöhr, Journal of Electron Spectroscopy and Related Phenom-ena 75, 253 (1999).[2] J. Stöhr, Journal of Magnetic Materials 200, 470 (1999).[3] T. Oguchi and T. Shishidou, Phys. Rev. B 103, 024412 (2004).[4] L. Šmejkal, R. González-Hernández, T. Jungwirth, and J.Sinova, Science Advances 6, eaaz8809 (2020).[5] K. Okada and A. Kotani, J. Phys. Soc. Jpn. 58, 2578 (1989).[6] A. Tanaka and T. Jo, J. Phys. Soc. Jpn. 61, 2040 (1992).[7] M. Taguchi, T. Uozumi and A. Kotani, J. Phys. Soc. Jpn. 66,247 (1997).[8] M. Matsubara, T. Uozumi, A. Kotani, Y. Harada and S. Shin, J.Phys. Soc. Jpn. 69, 1558 (2000).[9] R. D. Cowan, The Theory o f Atomic S tructure and S pectra(University of California Press, Berkeley, CA, 1981).[10] 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).[11] C. A. Occhialini, V. Bisogni, H. You, A. Barbour, I. Jarrige, J.F.Mitchell, R. Comin, and J. Pelliciari, Phys. Rev. Research. 3,033214 (2021).[12] N. Sasabe, M. Kimata, and T. Nakamura, Phys. Rev. Lett. 126,157402 (2021).