Ferro-Octupolar Order and Low-Energy Excitations in D $^ 2$ Double
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Ferro-octupolar order and low-energy excitations in d2 double perovskites of Osmium Leonid V. Pourovskii,1, 2 Dario Fiore Mosca,3 and Cesare Franchini3, 4 1Centre de Physique Th´eorique,Ecole Polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France 2Coll`egede France, 11 place Marcelin Berthelot, 75005 Paris, France 3University of Vienna, Faculty of Physics and Center for Computational Materials Science, Vienna, Austria 4Department of Physics and Astronomy ”Augusto Righi”, Alma Mater Studiorum - Universit`adi Bologna, Bologna, 40127 Italy Conflicting interpretations of experimental data preclude the understanding of the quantum mag- netic state of spin-orbit coupled d2 double perovskites. Whether the ground state is a Janh-Teller- distorted order of quadrupoles or the hitherto elusive octupolar order remains debated. We resolve this uncertainty through direct calculations of all-rank inter-site exchange interactions and inelastic 2 neutron scattering (INS) cross-section for the d double perovskite series Ba2MOsO6 (M= Ca, Mg, Zn). Using advanced many-body first principles methods we show that the ground state is formed by ferro-ordered octupoles coupled within the ground-stated Eg doublet. Computed ordering temper- ature of the single second-order phase-transition and gapped excitation spectra are fully consistent with observations. Minuscule distortions of the parent cubic structure are shown to qualitatively modify the structure of magnetic excitations. Identification of complex magnetic orders in spin- 23], its origin remains unclear, in particular regarding orbital entangled and electronically correlated transition the rank of the multipolar interactions and the degree metal oxides has emerged as a fascinating field of study, of JT distortions. Considering that the non-Kramers Eg enabling the discovery of new quantum magnetic states doublet does not carry dipole moments it would be le- originating from exchange interaction between effective gitimate to expect that conventional quadrupolar cou- pseudospins carrying high-rank multipoles [1, 2]. Of par- plings in a JT-broken symmetry would promote an anti- ticular interest is the spin-orbit magnetic physics realized ferro (AF) quadrupolar order [5]. This transparent pic- in rock-salt ordered double perovskites (DP) A2BB0O6, ture does not seem to be consistent with recent experi- with B0 being a heavy ion like Os, Mo and Re [3–5]. The ments: X-ray diffraction (XRD) does not find structural strong spin-orbit coupling (SOC) strength splits the ef- distortions (larger than 0.1%) and, whereas no conven- fective L = 1 t2g levels on the magnetic B0 ions into a tional magnetic order is detected by neutron diffraction 3 lower j = 2 quadruplet ground state (GS) and a dou- (upper limit 0.1 µB), muon spin relaxation still in- 1 ≈ blet j = 2 excited state. Depending on filling of the t2g dicates time-reversal (TR) symmetry breaking thereby levels, the entanglement between the spin S and orbital ruling out quadrupolar order [17]. To account for the L degrees of freedom gives rise to different total angu- experimental measurements a ferro-octupolar (FO) or- lar momentum J = S + L states with distinct magnetic dered GS is proposed, involving a coupling between the properties. lower Eg and excited T2g state mediated by quadrupo- The Jahn-Teller (JT) active 5d1 DP are described by lar operators [17–19]. This model reproduces a spin-gap 3 observed in the excitation spectra [17, 18] and is overall an effective Jeff = 2 model [3]. The unusual canted antiferromagnetic GS with small ordered moment in d1 reasonably compatible with the experimental scenario, but it makes use of some problematic assumptions. Only Ba2NaOsO6 (and related DP) attracted a lot of atten- tion [3, 6–14] and has been shown to arise through a com- a subset of inter-site exchange interactions (IEI) allowed plex interplay of dipolar and quadrupolar interactions within Jeff =2 are assumed to be non-zero. Moreover, between Kramers ions coupled with JT distortions [14– the included quadrupole IEI, which cannot be directly 16]. Conversely, the nature of ordered phases in the com- inferred from experiment, are tuned to obtain the desired 2 properties of the FO phase. paratively less studied 5d version Ba2MOsO6 (M=Ca, arXiv:2107.04493v1 [cond-mat.str-el] 9 Jul 2021 Mg, Zn) is still debated [1, 5, 17–20]. These cubic DPs Inspired by the apparent adequacy of the proposed FO have two electrons with S=1 resulting in a total angu- 2 state for the d Ba2MOsO6 DP series (in short: BCOO, lar momentum Jeff =2, where the levels are split due to BMOO and BZOO), we propose in this letter an alter- the crystal field (CF) into a lower Eg doublet and T2g native mechanism based on a direct numerical calcula- triplet [4]. In contrast to the assumptions of the pioneer- tions of all possible interaction channels by means of ing theoretical study of Ref. 4, the intersite exchange in- 2 many-body first principles schemes. Without forcing any teractions in these d DP are inferred to be much smaller pre-assumption on the form of the effective Hamiltonian than the CF [17, 21]. we find a ferro order of xyz octupoles determined by a Though there is clear experimental evidence for a sin- competition between time-even and octupolar IEI within gle phase transition involving the Eg manifold [17, 22, solely the GS Eg doublet. Our data correctly predict the 2 observed second-order phase transition, with computed ordering temperature compatible with the experimental one, and a gapped excitation spectra. Effective Hamiltonian and methods. The effective Hamiltonian for the low-energy degrees of freedom on the Os sublattice is a sum of the IEI (HIEI ) and rem- nant crystal-field (rcf) terms: H = V QQ0 (∆R )OQ (R )OQ0 (R )+ Hi , eff KK0 ij K i K0 j rcf ij KQK Q i Xh i X0 0 X (1) Q where the first sum is over all ij Os-Os bonds, OK (Ri) is the Hermitian spherical tensorh i [24] for J=2 of the rank K = 1...4 and projection Q acting on Os site at the position R , the IEI V QQ0 (∆R ) acts between i KK0 ij the multipoles KQ and K0Q0 on two Os sites con- FIG. 1: Color map of the inter-site exchange interac- nected by the lattice vector ∆Rij = Rj Ri. Finally, tions (IEI) V QQ0 , eq. 1, in BZOO for the [1/2,1/2,0] i 0 4 − KK0 Hrcf = Vrcf 4(Ri) + 5 4(Ri) is the remnant oc- Os-Os pair. The IEI involving hexadecapoles (K=4) are − O OQ QQ0 tahedral CF [17], where K are the standard Stevens negligible and not included. The complete list of V O KK0 operators. for the three compounds is given in the SM [34]. To derive the above Hamiltonian we use density func- tional theory (DFT) [25] + dynamical mean-field theory (DMFT) [26–29] and treat local correlations acting in Eg space can be encoded by spin-1/2 operators, with the 2 the Os 5d ions within the quasi-atomic Hubbard-I (HI) Eg states corresponding to the projectons of pseudo-spin- approximation [30]. All IEI V QQ0 (∆R) are computed 1/2: KK0 within the HI-based force-theorem approach of Ref. 31 = 2, 0 ; = ( 2, 2 + 2, 2 ) /√2, (2) (FT-HI), previously shown to capture high-rank multi- | ↑i | i | ↓i | − i | i polar IEI in spin-orbit oxides [32, 33]. Our DFT+HI written in the Jeff = 2,M basis. The resulting Eg calculations correctly predict the expected Jeff =2 GS Hamiltonian is then| related toi (1) by the projection multiplet, which is split by Hrcf into the ground state Eg doublet and excited T2g triplet. More details can be ˆ ˆT HEg = PHIEI P = Jαβ(∆Rij)τα(Ri)τβ(Rj), found in the Supplementary Materials (SM) [34]. ij NN αβ h Xi∈ X CF splitting and superexchange interactions. The cal- (3) culated CF splitting ∆rcf = 120Vrcf listed in Table I is where the rows of projection matrix P are the Eg states about 20 meV for all members, in agreement with specific in Jeff = 2 basis (2), τα is the spin-1/2 operator for heat measurements and excitation gap inelastic neutron α = x,y, or z. Up to a normalization factor, τy is the scattering (INS) [17, 18, 21]). 2 2¯2¯ octupole O3− Oxyz; the corresponding IEI V33 di- QQ0 ≡ The computed IEI VKK are displayed in Fig. 1 (for rectly maps into Jyy. τx and τz are mixtures of the eg 0 00 BZOO, similar data are obtained for the other members, quarupoles and hexadecapoles. Therefore, the IEI V22 22 see SM). The largest values, 3 meV are significantly and V contribute to Jzz and Jxx, respectively, together ≈ 22 smaller than ∆rcf , in agreement with experiment [17, 18], with the hexadecapole IEI of the same symmetry. Since implying that the ordered phase will be determined by those hexadecapole IEI are negligible (Supp. Table I the IEI acting within the ground-state Eg doublet. The [34]), their admixture into τx and τz reduces the mag- nitude of time-even Jxx and Jzz (Sec. III in SM [34]). Overall, the order in Eg space is determined by a com- Compound ∆rcf Jyy Jzz Jxx petition of the time-even (τx and τz) combinations of quadrupoles and hexadecapoles with the time-odd xyz Ba2CaOsO6 17.1 -2.98 1.48 -0.61 octupole. There are, correspondingly, no IEI coupling τy Ba2MgOsO6 19.2 -2.93 1.67 -0.69 with τx or τz due to their different symmetry under the Ba2ZnOsO6 20.5 -1.71 1.35 -0.50 time reversal. Our calculated Eg IEI for the [1/2,1/2,0] lattice vector TABLE I: Remnant CF splitting ∆rcf and superex- are listed in Table I.