NMR spectra

In a lattice with cubic symmetry only one Na NMR site is present, which results in a narrow single peak spectrum in the insulating paramagnetic (PM) state. Our main discovery is that, on reducing temperature, the Na NMR line undergoes a complex modification, as evident in Fig. 1c, reflecting changes in local electron spin susceptibility and electric field gradient (EFG). That is, we deduce that this modification reflects a splitting of Na into two sites sensing different hyperfine fields due to electronic spins and breaking of the local cubic point symmetry with both sites sensing the same quadrupole frequency, related to the EFG.

To establish the sensitivity of our measurements to putative lattice distortions, orbital order, and magnetism, we first inspect the temperature (T) dependence of 23Na NMR spectra in Ba 2 NaOsO 6 . These 23Na spectra reveal the distribution of the hyperfine fields and the electronic charge and are thus a sensitive probe of both the electronic spin polarization (local magnetism) and charge distribution (orbital order and lattice symmetry). Temperature evolution of 23Na NMR spectra is plotted in Fig. 1c. For T>12 K, the spectrum consists of a single narrow NMR line, evidencing the PM state. Below 13 K, the NMR line broadens and splits into multiple peaks indicating onset of significant changes in the local symmetry, thereby producing EFG, that is, asymmetric (non-cubic) charge distribution. Below 10 K, the 23Na spectra clearly split into six peaks, that is, two sets of triplet lines, labelled as I and II in Fig. 1c, that are well separated in frequency.

Two magnetic sites

Broadly speaking, the emergence of these two sets of triplets indicates appearance of two distinct magnetic sites, that is, two nuclear sites that sense two different local fields, in the lattice. In a field of 9 T, the transition to the long range ordered (LRO) state onsets in the vicinity of 10 K, which is significantly higher than the transition temperature observed in low-field thermodynamic measurements in ref. 12. Our data measured at the different applied magnetic fields indicate that the transition temperature increases with the increasing field, confirming the magnetic nature of the transition (see Supplementary Note 2). The line shape and the fact that the two sets of triplet lines, I and II, are well separated in frequency implies that the low-temperature LRO magnetic state is commensurate. Furthermore, both sets of lines are shifted to frequencies below that of spectra in the PM state. This demonstrates that the net local magnetic fields on both Na sites are of the same sign indicating that the LRO order is likely ferromagnetic.

Point symmetry breaking and EFG

Besides the splitting into sets I and II, that reflects the appearance of two distinct magnetic sites in the low T phase, we observe additional splitting of each set of the spectral lines into three peaks. Moreover, as visible in Fig. 1c, the additional splitting is discernible at temperatures higher than that for the onset of LRO state, apparent in emergence of sets I and II. This splitting, labelled as δ q in Fig. 1c, originates from quadrupole interaction, implying changes in local charge distribution induced by modifications of electronic orbitals and/or local lattice symmetry.

For nuclear sites with spin I>1/2, such as 23Na with I=3/2, and non-zero EFG, quadrupole interaction between nuclear spin and EFG splits otherwise single NMR line to 2I lines, as illustrated in Fig. 2d. Thus, 23Na spectral line splits in three in the presence of non-zero EFG. Nonetheless, for small finite values of the EFG three peaks are not necessarily discernible, in which case significant line broadening can only be observed, as depicted in Fig. 2d. At sites with cubic point symmetry the EFG is zero (see Supplementary Note 5), as is the case for Na nuclei in the high temperature PM phase. Therefore, the observed line broadening and subsequent splitting of the Na spectra into triplets, in the magnetically ordered phase, indicates breaking of the cubic point symmetry, caused by local distortions of electronic charge distribution. These distortions, marking the broken local point symmetry (BLPS) phase, occur above the transition into the magnetic state, as depicted in Fig. 1b. To confirm this finding, we measured low T spectra as a function of strength and orientation of the applied magnetic field, as we describe next.

Figure 2: Local cubic symmetry breaking in the ordered phase. (a) 23Na spectra in low temperature ordered state as a function of the angle between the applied magnetic field and [001] crystalline axis. In this case, H was rotated in the (1 0) plane of the crystal which contains three high symmetry directions: [001], [111] and [110]. (b) The mean peak-to-peak splitting (δ q ) between any two adjacent peaks of the triplets I and II. Error bars reflect the scattering of deduced (δ q ) values. Solid line is the fit to , where θ denotes the angle between the principal axis of the EFG (V zz ) and the applied magnetic field (H) as depicted for nuclei with spin I=3/2 and magnetic moment μ (see Supplementary Note 5). (c) Schematic of a proposed lattice distortion involving the O2− ions. The distortion breaks cubic symmetry at the Na site giving rise to finite electric field gradient (EFG). Different types and colours of double arrows illustrate unequal magnitude of oxygen displacements. Displacements along different axes can comprise either from compression or elongation of the oxygen bonds (see Supplementary Note 5). Depicted distortion magnitude is not to scale with the distance between nearest neighbour Os atoms, and is amplified for clarity. (d) Schematic of the energy levels of a spin-3/2 nucleus in a finite magnetic field and in the presence of quadrupolar interaction with EFG, generated by surrounding electronic charges, and resulting NMR spectra. In the absence of quadrupolar interaction spectrum consists of a single narrow line at frequency ω and of width Δ ω . In the presence of quadrupolar interaction, the centre transition remains at frequency ω, while the satellite transitions appear at frequencies shift by ±δ q , proportional to the magnitude of the EFG. For small values of the EFG, satellite transition cannot be resolved and only line broadening is observed. Strictly speaking, there is also a broadening due to the distribution of magnitude of the EFG itself, but this is manifested only on the satellites and not on the central transition. In our case, this distribution can be neglected as all the lines show the same width. Full size image

Specifically, we examine δ q , the average separation between two adjacent quadrupolar satellite lines, as a function of strength (H) and orientation (H) of the applied magnetic field. The size of δ q is proportional to the magnitude of the EFG and the square of the spin operator projected along the principal axis of the EFG (V zz ) (see Supplementary Note 5). In strong applied field, as is the case in our experiment, the size of δ q is controlled by the projection of along H, proportional to the cos2 (θ) of the angle θ between H and V zz , as shown in Fig. 2b. Evidently, δ q is at its maximum for H applied in the direction of V zz , which in our experiment corresponds to [001] direction. Therefore, if the splitting δ q originates from quadrupole interactions, δ q should remain constant as strength of the the applied field is changed and should follow function of θ as its orientation is varied21. We compare δ q in fields ranging from 7 to 15 T at 4 K, deep in the LRO phase (see Fig. 1c). Comparison reveals that δ q varies by <2%, which is of the order of the error bars. Furthermore, we measured the spectra at 15 T and 8 K as a function of the angle (θ) between H and [001] crystalline axis as plotted in Fig. 2a. The angle dependence of the splitting δ q is displayed in Fig. 2b. It indeed follows the exact functional dependence expected for the δ q originating from quadrupole interactions. Both the observed insensitivity of δ q to the strength of the magnetic field and its dependence on θ indicate that δ q splitting originates from structural/orbital distortions for which the principal axes of the EFG coincide with those of the crystal (see Methods). Moreover, the insensitivity of δ q to the strength of the magnetic field rules out the possibility that the detected distortions originate from trivial magnetostriction effects on the crystal (see Supplementary Discussion 2). Our finding, that structural distortion is present in the LRO phase, is in contrast to the predictions made by the first-principles density functional theory calculation19,22. This is important in so far that it clearly shows that quantum models based on complex multipolar interaction generating high-order spin exchange is consistent with the observed nature of emergent phases in Mott insulators with the strong SOC3,20.

Lattice distortions and orbital order

To resolve the microscopic nature of the observed lattice distortions and determine their magnitude, we performed detailed numerical calculations of the EFG, and thus δ q , based on the point charge approximation21 (see Supplementary Note 5). We found that our observation, revealing equal δ q on two magnetically inequivalent Na sites, can be best explained by a scenario involving distortions of the O2− octahedra, surrounding Na+ ions as depicted in Fig. 2c. In this scenario, one structurally distinct Na site in non-cubic environment is generated. As it results from our calculations, distortions that generate orthorhombic local symmetry at the Na site are required to account for both the amplitude of the detected splitting (δ q ≈190 kHz) and its dependence on the field orientation. Thus, in the ordered phase, our observations are explained by the orthorhombic distortions that comprise of dominant deformations along [001] and one of the crystalline axis in the (110) plane. Even though we find several possible distortions which can induce the observed splitting, we emphasize that they all involve a symmetry-lowering transition to an orthorhombic point symmetry in the LRO phase. Considering the observed amplitude of δ q , we deduce that a typical magnitude of the distortion along any particular direction in the LRO phase does not exceed 0.8% of the respective lattice constant. Above T c , in the BLPS phase, the width of the NMR spectra allows us to place an upper limit on distortions. We infer that the limit equals to 0.02% of the respective lattice constant, as any deformations that exceed this value would cause visible splitting of the NMR spectra in the PM state.

In the PM state, BLPS phase is characterized by significant broadening of the NMR spectra. This broadening grows rapidly on decreasing temperature towards T c . The angle dependence of the broadening does not coincide with either that of the internal uniform or staggered fields, indicating that the broadening predominantly originates from lattice distortions. Because in the BLPS phase we do not observe well defined splitting, but rather convoluted broadening, the exact dependence of δ q on the field orientation is unknown. Consequently, dominant tetragonal distortions along [001] direction can in principle account for the line broadening in the PM phase (see Methods). Thus, the BLPS phase can be viewed as the PM phase in which the cubic point symmetry is broken by either dominant tetragonal deformations of the oxygen tetrahedra along [001] direction or orthorhombic distortions, as is the case in the LRO phase (see Supplementary Note 5 and Supplementary Discussion 1). Hence, it is possible that the solid line, indicating T c into LRO magnetic phase, in Fig. 1b denotes symmetry lowering transition from tetragonal-to-orthorhombic phase as well.

LRO magnetic state

We emphasize that detected lattice distortions lead to only one magnetically distinct Na site. Thus, we conclude that the observed magnetically distinct Na sites (labelled as I and II in Fig. 1) must originate from the novel type of magnetism and not from trivial artifacts of lattice distortions. In order to deduce the microscopic nature of the LRO magnetism, we next discuss the temperature and field evolution of the local fields. We used the NMR shift data to infer the local uniform and staggered fields, where the average is taken over the triplet I and II, as denoted (see Supplementary Notes 1 and 2). We note that H u corresponds to the local field as determined by the first moment of the entire spectra. Temperature evolution of these local fields is displayed in Fig. 3. Well below T c , H u increases with increasing H, while H stag (observable only by local probes) remains constant. Interestingly, both H u and H stag are of the same order of magnitude. Presence of H stag implies that the LRO state contains two-sublattice magnetization with significant antiparallel components. Such magnetization naturally accounts for the appearance of two magnetically inequivalent Na sites, that is the appearance of distinct local fields 〈H I 〉 and 〈H II 〉. As visible in Fig. 4a, the internal field at the Na site consists of a sum of projection along H of total of six nearest-neighbour Os moments, four of which have equal magnetic moment projections as they belong to the same sub-layer. That is, the internal field at the Na site in one plane consists of a sum of the projection of the four Os moments on the same layer, and thus equal projected moments from one sub-lattice labelled A, and two Osmia above/below in neighbouring layers with projected magnetic moments pointing in a different direction (sub-lattice labelled B), and thus producing different local field than A moments at the Na site. Na nuclei in the next plane will then sense four type B and two type A projected Os moments. This generates two sets of inequivalent Na sites and causes the magnetic splitting in spectrum between triplet I and II, as two types of moments induce different local fields at the Na site.

Figure 3: Temperature dependence of the uniform and staggered fields. Absolute value of the uniform internal field, H u , (a) and the staggered internal field, H stag , (b) as a function of reduced temperature for various . Typical error bars are on the order of a few per cent and not shown for clarity. Lines are guide to the eyes. Full size image

Figure 4: Resolution of the spin orientation in the ordered phase. (a) The magnitude of the internal field associated with triplet I (H I ) and II (H II ) (solid symbols), and the first moment of the entire spectrum (uniform field, H u ) as a function of the angle between the applied magnetic field and [001] crystalline axis at 8 K and 15 T. Open symbols depict the angular dependence of the staggered field, H stag . Typical error bars are on the order of a few per cent and not shown for clarity. Dotted lines are guide to the eyes. Solid lines are calculated fields from the spin model sketched in part (d), as described in the text. (b) Schematic of the net magnetization in the XY plane in the spin model, consistent with our data and proposed in ref. 20. Arrows of different shades depict spins from two sub-lattices, labelled as A and B. These spins of equal magnitude are canted by angle ±φ with respect to [110] direction. (c) Comparison of the angular dependence of bulk magnetization per FU (formula unit) from ref. 12 (open symbols) and H u determined from our NMR measurements. Lines are guide to the eyes. (d) Schematic of the spin model consistent with our data and proposed in ref. 20. Different shades and orientation of arrows indicate distinct ionic (spin) environments on Os site. Planes containing moments from sub-lattice A and B are shown. These equal magnitude moments in two sub-lattices are canted with respect to [110] direction. Full size image

Next, we inspect the local fields as the applied magnetic field was rotated in the (1 0) plane of the crystal, as illustrated in Fig. 4a. This is essential for ensuring that we understand what components of anisotropic magnetic susceptibility are being measured. For , H stag , as well as 〈H II 〉, reaches its maximum value, while both H u and 〈H I 〉 are at their minimum. In principle, H u should scale as bulk magnetization (M). As evident in Fig. 4c, this is not the case here. This finding reveals that despite the fact that the net magnetization is aligned with the [110] axis, the local fields, that is, spin expectation values, are not. This very fact was predicted to arise as a direct consequence of lattice distortions driven by complex interactions in this class of materials3.