Detection of multipolar orders in the spin-orbit-entangled 5d Mott insulator Ba2MgReO6

In electronic solids with strong spin-orbit interactions (SOIs), the spin and orbital degrees of freedom of an electron are quantum mechanically entangled, which may result in an exotic multipolar order instead of a conventional dipolar order such as a magnetic order. Such a higher-degree order is called"hidden order"because of difficulties in experimental detection. Moreover, the number of candidate compounds is limited, especially rare in d electron systems, in which an interplay between SOIs and Coulomb interactions is expected to cause rich physics. Here, we employ state-of-the-art synchrotron X-ray diffraction techniques on a high-quality single crystal to probe subtle symmetry breaking induced by a multipolar order. We unequivocally demonstrate that the double-perovskite Ba2MgReO6 exhibits successive transitions to quadrupolar and then dipolar orders upon cooling, which is consistent with a theory considering SOIs. Our findings are a significant step towards understanding the intriguing physics of multipoles realized by spin-orbit-entangled 5d electrons.

Exotic quantum phases such as high-temperature superconductivity 1,2 , colossal magnetoresistance 3,4 , and heavy Fermion states 5,6 are observed in materials with strong electron-electron correlations, and both the spin and orbital degrees of freedom of an electron are thought to play a role. In particular, strong interactions between spin and orbital moments an effect known as spin-orbit interactions (SOIs)are expected to produce a variety of exotic phenomena [7][8][9] .
In this context, over the past ten years, research has focused on heavy transition metal (TM) compounds [7][8][9] , where the combined effect of electron correlations and SOIs is realized. For example, in Sr2IrO4, a spin-orbit-entangled Mott insulating state emerges as a result of electron correlation effectively enhanced by SOIs 10,11 . In addition, when these spin-orbit-entangled electrons interact with each other through specific direction-dependent interactions, a novel quantum phase called the Kitaev spin liquid occurs 9,12 . Moreover, exotic topological phases can emerge when the spin-orbit-entangled electrons begin to delocalize 7,8 .
Despite these intriguing phenomena, the understanding of 5d electron systems remains incomplete. In particular, the most fundamental symmetry-breaking phase, that is, the multipolar order 13,14 , has not yet been experimentally well established. The electrons with strong spin-orbital entanglement, denoted by a total angular momentum J, may experience various symmetry breaking transitions. These transitions result in complex angular distributions of spin and charge densities, which can be described by the quantum mechanically defined multipole moments 15 . The higher-order multipolar order is generally subtle and hard to detect by traditional experimental probes in comparison to conventional dipolar orders; this is why the multipolar order is also known as the "hidden order" 16,17 . In addition, model 5d materials that can demonstrate the characteristics of multipolar orders are lacking thus far.
There are two requirements to create a multipolar order in an actual material. One is a high local symmetry at the TM site, which leads to an unquenched orbital moment. The other is an appropriate distance and charge transfer between TM ions for electrons to be localized; many 5d TM compounds are weakly correlated metals with relatively large bandwidths. In terms of these requirements, double-perovskite (DP) compounds 18 are good candidates, as they comprise octahedrally-coordinated TM ions that are spatially separated from each other (Fig.   1a). Thus far, several DP compounds have been suggested by both theoretically 8,13,19 and experimentally to exhibit multipolar order: Ba2NaOsO6 [20][21][22] , Ba2MOsO6 (M = Zn, Mg, Ca) 23 , Ba2YMoO6 24 , Ba2MgReO6 25,26 , and A2TaCl6 (A = Rb, and Cs) 27 . A recent nuclear magnetic resonance (NMR) study on Ba2NaOsO6 revealed that the compound shows a noncollinear spin order and a small structural distortion, which was suggested to be due to a quadrupolar order 22 . However, the order parameter or ordering structure was not identified. Moreover, ferro-octupolar order was suggested for Ba2MOsO6 (M = Zn, Mg, Ca) based on the observation of a gapped magnetic excitation spectrum without additional magnetic Bragg peaks 23 .
In this article, we focus on Ba2MgReO6 25,26,28 . Using state-of-the-art synchrotron X-ray diffraction (XRD) techniques on high-quality single crystals, we have successfully observed orderings of both magnetic dipoles at the magnetic transition temperature Tm = 18 K and charge quadrupoles at the quadrupolar order temperature Tq = 33 K and unambiguously determined their patterns for the first time. Our observations are in good agreement with the theoretical model that considers strong SOIs 13 . This compound provides us with a new opportunity to study quantum phases appearing among strongly correlated electrons with strong SOIs.
Ba2MgReO6 is a 5d Mott insulator 25,26 that crystallizes in a DP (elpasolite) structure with a non-magnetic MgO6 octahedra arranged in a rock-salt structure. b Energy diagram of a single electron occupying the 5d orbitals in a Re 6+ ion. The crystal field from the surrounding six oxygen ions splits the fivefold-degenerate 5d states by 10Dq ~ 4 eV to a triply degenerate t2g state and a doubly degenerate eg state. A strong spin-orbit interaction λ as large as 0.5 eV further splits the lower-lying t2g state by (3/2)λ into a Jeff = 1/2 (Γ7) doublet and a Jeff = 3/2 (Γ8) quartet by mixing the spin angular momentum (S = 1/2) and effective orbital angular momentum (Leff = 1). The spin-orbit-entangled Jeff = 3/2 quartet is the ground state of Ba2MgReO6 and has a multipolar degree of freedom.
face-centred cubic lattice in the space group Fm-3m, as illustrated in Fig. 1a. The Re 6+ ion possesses a 5d 1 electron configuration and adopts a spin-orbit-entangled Jeff = 3/2 quartet as the ground state 29 (Fig. 1b). This Jeff = 3/2 quartet state has been inferred by the large reduction in the paramagnetic moment 26 (Fig. 2a): 0.68μB instead of the expected value of 1.73μB for S = 1/2. In addition, the electronic entropy released at low temperatures is 11.3 J K −1 mol −1 , which is close to the electronic entropy value of Rln4 = 11.4 J K −1 mol −1 (where R is the gas constant) expected for a quartet state 26 (Fig. 2b).
Upon cooling, Ba2MgReO6 undergoes two phase transitions at Tq = 33 K and Tm = 18 K, as shown by the temperature dependences of the inverse magnetic susceptibility and heat capacity in Figs. 2a and 2b, respectively. In the magnetically ordered phase below Tm, an exotic spin structure is inferred from the small saturation moment of 0.3μB and an unusual easy-axis along the [1 1 0] direction, the latter of which is incompatible with standard Landau theory for a cubic ferromagnet (Fig. 2c). These magnetic behaviours are very similar to those observed in Ba2NaOsO6 22 . Thus, a quadrupolar order is expected to occur at high To reveal the characteristics of the successive phase transitions in Ba2MgReO6, we employed two kinds of XRD techniques using synchrotron radiation. First, the magnetic order below Tm has been investigated in terms of the resonant effect at the X-ray absorption edge, which is generally superior in determining the ordered structures of small dipole moments using small single crystals 11,[30][31][32] . Second, structural changes across the two transitions at Tq and Tm are examined by non-resonant XRD experiments to probe the quadrupolar order and its influence on the magnetic order. The very small structural change accompanied by the quadrupolar order, which is approximately one order of magnitude smaller than that induced by the orbital order in 3d electron systems 33,34 , can only be detected with ultra-high-intensity synchrotron X-ray beam using high-crystallinity samples. Figure 3a shows a typical XRD profile taken below Tm at 6 K, for the (1 0 0)t reflection which is not allowed for the body-centered-tetragonal lattice attained below Tq, as described later; the subscripts 't' and 'c' denote the index based on the tetragonal and cubic cells, respectively. The peak is strongly enhanced at around the Re LIII absorption edge of 10.535 keV, indicative of its magnetic origin. Moreover, polarization analysis (shown in Supplementary Fig. S1) shows that the reflection appears not in the σ-σʹ channel but in the σ-πʹ channel, further confirming it is a magnetic reflection. This (1 0 0)t magnetic reflection appears only below 18 K (Fig. 3b), indicating that the dipole moments of Re arranged below

Tm.
We observe four magnetic reflections exhibiting resonance enhancement in our resonant XRD experiments (shown in Supplementary Fig. S2 Next, we investigated the nature of the phase transition at Tq = 33 K using the non-resonant synchrotron XRD technique. As summarized in Fig. 4 and Supplementary D, a clear cubic-to-tetragonal structural transition is observed at Tq, which is evidenced by a split of the (0 0 24)c Bragg peak (inset of Fig. 4b). The relation between the high-temperature cubic and low-temperature tetragonal unit cells is depicted in Figure 3c. As shown in Fig. 4b, the tetragonal distortion (ct − √2at) develops gradually below Tq, leading to a very small distortion of ~0.09% at 25 K (Supplementary Table S2). The distortion in Ba2MgReO6 is much smaller than in other DPs, for example, ~0.47% in Ba2CaWO6 36 .
In addition to the split of fundamental reflections, 141 additional (superlattice) reflections, which are extremely weak (less than 0.005% of the strongest fundamental reflection), are  Supplementary Fig. S3). The intensity of the superlattice peaks is approximately proportional to the square of δx. A comparison between the simulated and observed intensities provides an approximate estimate for δx at 6 K of 0.022 Å.
The distortion of an octahedron, shown in Fig. 4d, is described by a combination of two normal modes of an octahedron, εu and εv, which are depicted in Fig. 4e and Supplementary   Fig. S5. Among the dipole, quadrupole, and octupole moments originating from the Jeff = 3/2 quartet 13 , only a quadrupole moment, which is an anisotropic distribution of electronic charge, can induce lattice distortion by a linear coupling through electron-phonon interactions. The quadrupole moments in an octahedral field are classified into two-dimensional Γ3 (Qx2-y2, Q3z2-r2) and three-dimensional Γ5 (Qxy, Qyz, Qzx) representations, which are analogous to the eg and t2g orbitals of the 3d electron, respectively. Among these moments, only the former quadrupolar moments are compatible with the observed lattice distortions: in other words, only Qx2-y2 and Q3z2-r2 can linearly couple with εv and εu, respectively (Fig. 4e). The fact that the two distortions take place simultaneously in one octahedron means that the quadrupole moment is a linear combination of the Qx2-y2 and Q3z2-r2 moments. The Qx2-y2 component may be predominant because the amplitude of the εv mode is four times larger than that of the εu mode at 6 K: 0.044(5) Å and 0.01(1) Å, respectively ( Supplementary Fig. S5). Note that the εv distortion is uniform in a layer and stacks in a staggered manner along the c axis, while the εu distortion is common for all the ReO6 octahedra (Fig. 4d). This result indicates an antiferroic alignment of the Qx2-y2 moment and a ferroic alignment of the Q3z2-r2 moment, as depicted in  Q3z2-r2 moments are increased with developing the antiferromagnetic order in Ba2MgReO6.
There is one notable discrepancy between our experimental results for Ba2MgReO6 and the theoretical prediction of the quadrupolar order 13 . We find a quadrupolar order with a linear combination of the Qx2-y2 and Q3z2-r2 moments below Tq in Ba2MgReO6, whereas the theory predicts a simple antiferroic quadrupolar order of purely Qx2-y2 moments, as depicted in the top left of Fig. 5. Notably, a similar type of quadrupolar (orbital) order with two components occurs in LaMnO3, where electron-phonon couplings induce ferroic quadrupolar moments in the antiferroic quadrupolar order [39][40][41] . It is plausible that electron-phonon couplings, which are not considered in the theoretical calculation, and other effects such as quantum fluctuations play a role in stabilizing the Qx2-y2-Q3z2-r2 quadrupolar order in Ba2MgReO6.
In conclusion, our study establishes the existence of successive phase transitions to multipolar orders for the correlated spin-orbit-entangled 5d electrons in the DP Ba2MgReO6.
The quadrupolar order below Tq = 33 K is composed of antiferroically arranged Qx2-y2 and ferroically arranged Q3z2-r2 moments. A noncollinear antiferroic arrangement of dipolar order appears at Tm = 18 K, in which the quadrupolar order is significantly modulated. Our observation is consistent with the mean-field theory for spin-orbit-entangled electrons.
Therefore, Ba2MgReO6 provides an opportunity to experimentally investigate the symmetry breaking of the multipolar degree of freedom in 5d electron systems for the first time.
In future studies, the detection of elementary excitation from these multipolar orders must be of great interest and importance in capturing the dynamics of multipoles. There may be nontrivial excitations that are distinct from conventional magnetic excitations, as predicted by theories 42,43 . Also, it is highly desirable to construct a theory considering electron-phonon couplings or quantum fluctuations, which may confirm our experimental observations. Understanding the nature of interactions between multipolar moments would help us in exploring an exotic superconductivity mediated by multipolar fluctuations 44

Method
Sample preparation High-quality single crystals of Ba2MgReO6 were grown by the flux method in an enclosed space for controlling the oxidation state of rhenium ions. BaO, MgO, and ReO3 powders with a ratio of 2:1:1 were mixed with a flux composed of BaCl2 and MgCl2 in an argon-filled glove box, and the mixture was sealed in a platinum tube. The tube was heated at 1300 °C and then slowly cooled to 900 °C, followed by furnace cooling to room temperature. Black shiny crystals with the octahedral morphology were obtained after the residual flux was washed away with distilled water.

Resonant XRD
Resonant XRD experiments were performed on BL19LXU at SPring-8, Japan 47 . A Ba2MgReO6 single crystal with dimensions of 2 × 2 × 1 mm 3 was fixed to a cold finger of a 4 He closed-cycle refrigerator mounted on a four-circle diffractometer. The linearly polarized incident beam was referred to as the σ polarization in a vertical scattering plane geometry and its intensity was monitored by an ionization chamber. The diffracted beam intensity was measured using a silicon drift detector (SDD). The azimuthal angle φ was defined as φ = 0° when the tetragonal a axis was parallel to the X-ray polarization.
Off-resonant XRD Off-resonant XRD experiments were performed on the beamline BL-8A, Photon Factory (PF) and on the beamline AR-NE1A, Photon Factory Advanced Ring (PF-AR), in High Energy Accelerator Research Organization (KEK), Japan. Incident X-ray beams monochromatized by using Si (111) double crystals with wavelength = 0.68987 Å were used at BL-8A and high-flux and short-wavelength X-ray beams with  0.41827 Å were used at AR-NE1A in order to detect weak superlattice reflections; these wavelengths were calibrated by measuring the diffraction profiles of a standard polycrystalline CeO2 sample. A Ba2MgReO6 single crystal was attached to the top of a sapphire needle with Apiezon-N grease and mounted in a closed-cycle 4 He refrigerator. XRD intensities were collected by the oscillation photograph technique using a large cylindrical image plate to index Bragg reflections and integrating their intensities. The RAPID-AUTO program developed by RIGAKU corp. was used on both the beamlines. Structural parameters were refined with the SHELXL program 48 . Real and imaginary parts of anomalous scattering factors calculated with D. T. Cromer and D. Liberman's method were used 49 .

C. Possible space group for the quadrupolar order phase below Tq
We investigated the crystal structure of the quadrupolar order phase below Tq by the non-resonant synchrotron X-ray experiments performed at the beamline AR-NE1A at Photon Factory Advanced Ring. Above Tq, a structural refinement at 40 K confirms an undistorted cubic structure of the space group Fm-3m, as reported previously (Supplementary Table S1).
At Tq, we observe a clear structural change: several Bragg peaks split and 141 superlattice reflections appear. When we ignore the extremely weak superlattice reflections, the crystal structure below Tq is reasonably refined by assuming a tetragonal space group I4/mmm, as shown in Supplementary Tables S2 and S3

D. Structural parameters for Ba2MgReO6
Table S1 | Synchrotron X-ray single-crystal structure analysis of Ba2MgReO6 at 40 K in the paramagnetic phase. (x, y, z), Uiso, and Unm are the atomic coordinates, isotropic thermal parameter, and anisotropic thermal parameter, respectively. The temperature factor is expressed as exp(-2 2 a *2 (U11h 2 + U22k 2 + U33l 2 ). U11 = U22 = U33 = Uiso for Re, Ba, and Mg atoms. The occupancy at each site is fixed to 1. The values in parenthesis represent standard deviations. The dataset for this analysis was collected at the beamline BL-8A at Photon Factory. The refinement was performed assuming space group of Fm-3m. 232 unique reflections were used for the refinement.

E. Estimation of the magnitude of oxygen displacements in the quadrupolar order phase
The magnitude of the displacements of oxygen atoms accompanied by the quadrupolar order is estimated based on the structural analysis using the off-resonant single-crystal X-ray diffraction data. The high-temperature Fm-3m structure includes a single type of oxygen site, 24e at (x 0 0), and a single type of Re site, 4a at (0 0 0), while the P42/mnm structure includes three types of oxygen site, as illustrated in Fig. 4d: an apical site O1, 4e at (0 0 z) and in-plane sites O2, 4f at (x x 0) and O3, 4g at (x −x 0). This site splitting causes a slight elongation of the ReO6 octahedron in the c direction and a rhomboid εv-type distortion of the square formed by four oxygens surrounding the Re ion in the (0 0 1) plane. The displacements of oxygen atoms in the tetragonal P42/mnm structure for temperatures below Tq is depicted in Fig. 4d.
Firstly, the elongation of the ReO6 octahedron is estimated by performing structural refinements assuming the tetragonal space group I4/mmm, as the average structure. The result of structural refinements, as listed in Supplementary Table S3,