Quantum disordered ground state in the triangular-lattice magnet NaRuO2

It has long been hoped that spin liquid states might be observed in materials that realize the triangular-lattice Hubbard model. However, weak spin–orbit coupling and other small perturbations often induce conventional spin freezing or magnetic ordering. Sufficiently strong spin–orbit coupling, however, can renormalize the electronic wavefunction and induce anisotropic exchange interactions that promote magnetic frustration. Here we show that the cooperative interplay of spin–orbit coupling and correlation effects in the triangular-lattice magnet NaRuO2 produces an inherently fluctuating magnetic ground state. Despite the presence of a charge gap, we find that low-temperature spin excitations generate a metal-like term in the specific heat and a continuum of excitations in neutron scattering, reminiscent of spin liquid states previously found in triangular-lattice organic magnets. Further cooling produces a crossover into a different, highly disordered spin state whose dynamic spin autocorrelation function reflects persistent fluctuations. These findings establish NaRuO2 as a cousin to organic, Heisenberg spin liquid compounds with a low-temperature crossover in quantum disorder. Spin liquids are predicted to emerge in materials that combine strong electronic correlations with geometric frustration. Evidence has now been found for a spin liquid state in the triangular-lattice material NaRuO2.

It has long been hoped that spin liquid states might be observed in materials that realize the triangular-lattice Hubbard model. However, weak spin-orbit coupling and other small perturbations often induce conventional spin freezing or magnetic ordering. Sufficiently strong spin-orbit coupling, however, can renormalize the electronic wavefunction and induce anisotropic exchange interactions that promote magnetic frustration. Here we show that the cooperative interplay of spin-orbit coupling and correlation effects in the triangular-lattice magnet NaRuO 2 produces an inherently fluctuating magnetic ground state. Despite the presence of a charge gap, we find that low-temperature spin excitations generate a metal-like term in the specific heat and a continuum of excitations in neutron scattering, reminiscent of spin liquid states previously found in triangular-lattice organic magnets. Further cooling produces a crossover into a different, highly disordered spin state whose dynamic spin autocorrelation function reflects persistent fluctuations. These findings establish NaRuO 2 as a cousin to organic, Heisenberg spin liquid compounds with a low-temperature crossover in quantum disorder.
The interplay between electron-electron correlation effects and geometrical frustration can lead to a rich hierarchy of electronic states. The Hubbard model at half-filling shows that as the on-site Coulomb interaction (U) is reduced relative to the electron hopping energy (t), the ground state transitions from an antiferromagnetic insulating state into a non-magnetic metal. If the underlying lattice is triangular, then an intermediate non-magnetic insulating state is predicted before the onset of the metallic state [1][2][3] . The predicted properties of this intermediate phase vary depending on the theoretical approach; however, it is generally envisioned as an inherently quantum disordered magnetic state or a quantum spin liquid phase generically realized at the boundary of the Mott insulating regime.
The proposed physical manifestations of this type of spin liquid state are rare, with well-studied candidates identified in anisotropic triangular magnets built from organic molecular complexes [4][5][6][7] . In inorganic compounds, however, triangular-lattice systems identified thus far are located either deep in the insulating regime 8 , deep in the metallic state 9 or better described in the strong-coupling limit via a pure Heisenberg model 10 . One promising means of more fully exploring this materials space is to consider compounds with extended d electron orbitals, where the cooperative interplay between moderate on-site Coulomb repulsion (U) and spin-orbit coupling (λ)-driven bandwidth narrowing can generate marginally stable J eff = 1/2 Mott insulating states with U/t ≈ 1 (ref. 11). Such states are known to form in 5d transition metal Article https://doi.org/10.1038/s41567-023-02039-x samples in a slightly Ru-rich environment 20 . Furthermore, neutron powder diffraction data collected at temperatures as low as 50 mK show no signs of structural symmetry breaking or static spin correlations forming between Ru moments. This is remarkable given the large covalency and enhanced exchange expected between the extended Ru moments, which are only 3.06 Å apart.
Ab initio electronic structure calculations (Fig. 1c) reveal that the band structure of NaRuO 2 is highly sensitive to the structural parameters. Using the experimentally derived structure in this paper, a charge gap forms via the inclusion of both λ and U using the local density approximation (LDA). Choosing U = 1 eV, consistent with prior LDA + λ + U models of Ru 3+ systems such as RuCl 3 (ref. 21), opens a gap when incorporated with λ. However, the model and prediction of the presence or absence of a gap is highly sensitive to the local distortions of oxygen octahedra around the Ru sites-consistent with the notion of a marginally insulating state where extended hopping effects can play an important role. The resistivity data (Fig. 2a) verify that NaRuO 2 indeed possesses an insulating ground state whose electrical transport is best modelled via two-dimensional variable-range hopping; however, we note that discrimination between two-dimensional and three-dimensional forms is difficult in polycrystalline samples. If the data above 200 K are parameterized by forcing an Arrhenius fit, the apparent gap value is E g ≈ 0.22 eV.
Contrasting the electrical transport data, the temperaturedependent magnetic susceptibility data (Fig. 2b) instead show a nearly itinerant magnetic response. The weak increase in the susceptibility on cooling does not fit to a conventional Curie-Weiss form; instead, the low-temperature susceptibility is best fit to a large Pauli-like term (χ 0 ) with a weak concentration of free impurity moments (1.8 ± 0.2% of S = 1/2 moments). The χ 0 term is unusually large for an insulator, namely, χ 0 = 9.6 ± 0.2 × 10 −4 emu Oe −1 mol −1 , a value exceeding that of Pd metal with χ 0 = 7.3 × 10 −4 emu Oe −1 mol −1 at 4 K (ref. 22). As a consistency check, the large χ 0 fit via the temperature-dependent magnetization matches the linear term found in isothermal magnetization (χ 0 = 1.25 ± 0.10 × 10 −3 emu Oe −1 mol −1 ), which can be parameterized by a dominant Pauli-like linear term and a small fraction of S = 1/2 moments (1.4 ± 0.2%) (Fig. 2c). At lower temperatures, a weak cusp appears in the susceptibility data below 1.5 K (Fig. 2b, inset). There is a weak frequency dependence to the onset of this cusp ( Supplementary Fig. 2), which deviates from the expectations of a canonical spin glass, but nevertheless suggests weak iridates, where controlling the bandwidth can drive a metal-insulator transition 12 , and similarly, weak J eff = 1/2 Mott states may also be realized in 4d transition metal compounds 13 .
Specifically, Ru 3+ (4d 5 ) ions in a nearly ideal octahedral crystal field are capable of assuming a low-spin state, with both λ and U appreciable enough to stabilize a half-filled J eff = 1/2 orbital 14,15 . An ideal triangular lattice consisting of octahedrally coordinated Ru 3+ moments is also known to form in delafossite variants such as NaRuO 2 (ref. 16), which suggests an opportune setting for searching for an intermediate quantum disordered state at the boundary of the J eff = 1/2 Mott state's stability. Remarkably little is known regarding the ground state of this compound and, more generally, whether it can provide a suitable experimental window into the electronic properties of a near-critical triangular-lattice Hubbard model in the presence of strong λ.
Here we establish that NaRuO 2 hosts electrons in a unique interaction space as an ideal triangular lattice possessing a weak spinorbit-assisted Mott insulating ground state-one consistent with the expectation of a near-critical J eff = 1/2 Mott state driven by cooperative λ and U. Through studies of polycrystalline samples, our data demonstrate that despite the presence of a charge gap, the high-temperature magnetism of NaRuO 2 shows an elevated Pauli-like susceptibility, and at low temperatures where the charge gap is well established, heat capacity data show a substantial linear term akin to the Sommerfeld coefficients expected in metals. The magnetic excitations comprising this linear term generate a diffuse continuum of spin excitations, a portion of which is slowed down below a field-coupled crossover into a state with persistent fluctuations and a heat capacity quadratic in temperature. Our data demonstrate that J eff = 1/2 electrons built from extended d orbitals on a triangular lattice stabilize quantum disordered magnetic-phase behaviour, consistent with expectations of Hubbard models near the metal-insulator phase boundary 17 as well as models of Kitaev antiferromagnets 18,19 . Figure 1a shows the detailed crystal structure of NaRuO 2 . Edge-sharing RuO 6 octahedra form a triangular lattice of Ru 3+ ions separated by planes of Na ions. An ideal equilateral triangle lattice results and a slight (<5°) trigonal distortion exists in the oxygen octahedra surrounding the Ru ions. Neutron diffraction data (Fig. 1b) show fully occupied Na and O sites and no site mixing with the resolution of the measurement. The predominant defect mode within NaRuO 2 is Na Ru antisite defect formation, which can be avoided by synthesizing We note that powder samples prepared with higher-purity Na 2 O 2 as the starting reagent do not show this small fraction of Na 2 CO 3 impurity and exhibit the same properties as those reported here. The inset shows the data collected on HB-1A and reveals no magnetic scattering appearing down to 50 mK. The direct subtraction of the 1 K and 50 mK data is shown by the featureless blue line. c, LDA electronic band structure for NaRuO 2 using both λ and U = 1 eV. The inset shows the high-symmetry positions in the Brillouin zone.
Article https://doi.org/10.1038/s41567-023-02039-x freezing in the local moment contribution to the overall susceptibility. The nature of this slowdown in the dynamics is discussed later. Exploring this further, the heat capacity data were collected down to 80 mK and are plotted in Fig. 2d. As the system is cooled far below the Debye temperature, C p (T) takes on the form C p (T) = γΤ + βΤ 3 , where β parameterizes the contribution from phonons and γ is an anomalous linear term typically observed in metals. In free-electron gases, γ represents the Sommerfeld coefficient-a term proportional to the electronic density of states occupied at the Fermi level; however, in magnetically frustrated insulators, the presence of a γ term suggests a fractionalization of electrons and the presence of a spinon Fermi surface. The γ term in NaRuO 2 is substantial (14.07 ± 0.12 mJ mole −1 K −2 ) and is distinct from the small γ values found in disordered insulators such as TiO 2 (0.1 mJ mole −1 K −2 ) (refs. 23,24). Instead, it is comparable in magnitude with those found in organic triangular-lattice spin liquid candidates 25,26 . One notable distinction is that the presence of strong spin-orbit coupling amplifies χ 0 and the resulting Wilson ratio for NaRuO 2 versus R W ≈ 1 for organic compounds.
Cooling below 2 K reveals the onset of a weak freezing transition in C p (T) matching the cusp in the low-T susceptibility. Only a small amount of entropy is associated with this T F = 2 K crossover (1.2 ± 0.2% of Rln (2)), further correlating it with the freezing of a small fraction of local moments. However, T F also marks the onset of a modified power-law behaviour C p (T) = AT α that depends on the magnetic field. Applying a magnetic field shifts this T F crossover upwards in temperature, and, as the shoulder of this freezing feature shifts to higher temperatures, the low-temperature C p (T) converges to a quadratic behaviour (α = 2). The application of a magnetic field also enhances the apparent γ term above T F , and it increases by 21% under μ 0 H = 14 T. We note that the magnetic entropy estimate (Fig. 2d, inset) becomes increasingly unreliable at higher temperatures due to an imperfect subtraction of the lattice contribution in the regime where magnetic fluctuations become substantially weaker compared with the phonon contributions. This potentially generates the weakly non-saturating response.
To investigate the nature of magnetic excitations associated with these low-energy fluctuations, inelastic neutron scattering measurements were performed. As an initial survey, temperature-subtracted data (1.8-300.0 K) are plotted in Fig. 3a, where the magnetic spectral weight at low momentum transfers appears centred near 25 meV. Energy cuts (integrated over small momentum transfers Q) of unsubtracted data are plotted for both 1.8 K and 300.0 K datasets (Fig. 3b) and show a peak in scattering at 1.8 K that is broadened, diminished and shifted downwards in energy on warming to 300.0 K. A similar subtraction of the data for a lower incident energy is also shown in Fig. 3c. The momentum distribution of the resulting gapless, low-energy continuum is centred near 1.2 Å −1 (Fig. 3d).
The broad continuum is further investigated at lower incident energies and temperatures (250 mK) (  Article https://doi.org/10.1038/s41567-023-02039-x cusp in χ′(T) and C p (T)), these low-energy spin fluctuations are nearly temperature independent. Energy cuts through both finite momentum and low-angle spectral weights are shown in Fig. 3f,g, respectively. In both cases, excitations where the system gains energy (E > 0) are nearly temperature independent, whereas excitations where the system loses energy (E < 0) populate up with increasing temperature and follow a detailed balance. This behaviour in S(Q, E) suggests quantum critical that potentially scale as a function of E/T. The low-energy data were tested for quantum critical by analysing collapse to a test function . Scaling collapse was optimized for α = 0.83 ± 0.05; although this test function is phenomenological, it reproduces the asymptotic expected in a number of quantum critical scaling relations 27,28 . The collapse of the data linked by this asymptotic form is consistent with robust quantum fluctuations present in the magnetic ground state of NaRuO 2 .
To probe the spin dynamics at even lower frequencies, the muon spin relaxation technique was used for the determination of different static local fields and the presence of magnetic fluctuations. The temperature-dependent muon polarization and its fit are presented in Fig. 4a. At 12 K, the muon spin depolarization is best described via a Gaussian Kubo-Toyabe form dominated by the contribution of nuclear moments in the sample. On cooling below 12 K, the magnetic fluctuations from the antiferromagnetically coupled electronic spins slow down into the muon time window and the time-dependent polarization is described by a generalized depolarization function P(t) = (f) GbG (∆; R; t)e (−λ GbG t) β + (1 − f)e −λ p t . This is a response comprising an uncorrelated fraction (1 − f) of paramagnetic moments whose fluctuations drive simple exponential relaxation at λ p that, on cooling, converts into a highly disordered response captured by a Gaussian-broadened Gaussian (GbG) function supplemented by spin fluctuations in the form of a stretched exponential with relaxation rate λ GbG . The GbG function 29 represents a normal distribution of Gaussian-field distributions about a central value Δ 0 with width w captured by the parameter R = w/Δ 0 . As shown in Fig. 4b, Δ 0 increases progressively with decreasing temperature below 3.0 K, and R becomes finite at approximately 2.5 K, with the value trending towards 1 at the base temperature. The absence of clear oscillatory signals suggests that the low-temperature state contains intermediate-diluted and disordered static magnetic moments. Meanwhile, this nominally static distribution of fields is modified by a slow fluctuation rate (λ GbG ) (Fig. 4c), similar to other highly frustrated materials with persistent spin fluctuations in their ground states 30 . To discriminate the presence of persistent fluctuations in this disordered ground state versus depolarization via static disorder, a longitudinal-field (LF) experiment was performed. Such a field rapidly decouples muons from slow depolarization due to purely static-field distributions, whereas depolarization persists in systems that are inherently dynamic 31 . The data in Fig. 4d indicate the persistence of fluctuations under modest applied fields, for example, contrasting the rapid decoupling seen in static-field To further distinguish the partial freezing and persistent dynamics in NaRuO 2 from conventional spin-glass behaviour, we demonstrate the magnetic-field time-scaling relations of the LF time spectra using the form P(H, t) = P(t/H γ ) (ref. 34). The polarization maps to the spin autocorrelation function; above the freezing temperature, the depolarization should scale as a function of t/H γ with the scaling exponent γ reflective of the manner through which spin correlations decay in time 35 . With the exception of the short-time-limit μ 0 H = 0.02 T data, the data scale well to this form with the analysis shown in Fig. 4e,f, and the R 2 value of the scaled data versus a test polynomial is shown in Fig. 4f (inset). This scaling analysis yields an exponent of γ ≈ 1.75, which precludes conventional spin-glass power-law correlations (γ ≤ 1) and, instead, is consistent with the expectation of decay via a stretched exponential function-identical to the type used to model the zero-field data.
The above experimental picture suggests that NaRuO 2 occupies a unique phase space where a spin-orbit-assisted Mott state gives rise to a native quantum disordered ground state. Charge fluctuations as well as anisotropic Kitaev interactions potentially play an important role in this disordered state. Ru 3+ ions sit in an octahedral crystal field quantified by a quadratic elongation parameter 36 λ quad = 1.0074, similar to that observed in Na 2 IrO 3 (ref. 37). This should promote the formation of a J eff = 1/2 wavefunction from the t 2g orbital manifold. An array of such wavefunctions on a honeycomb lattice composed of edge-sharing octahedra generates strong Kitaev exchange and is predicted to stabilize a quantum spin liquid ground state, but experimental realizations are lacking. Although materials such as Na 2 IrO 3 (ref. 38) and RuCl 3 (ref. 39) are known to stabilize strong ferromagnetic Kitaev coupling strengths, they nevertheless possess magnetically ordered ground states due to dominant Heisenberg interactions 40,41 . Similar predictions of dominant Kitaev exchange and quantum spin liquid phases have also been put forward for J eff = 1/2 electrons decorating a triangular lattice 19 . However, unlike other Kitaev candidate materials, the thermodynamic phenomenology of NaRuO 2 resembles that of highly frustrated organic spin liquids and additional interactions such as charge fluctuations may play a role. We hypothesize that this is due to the weak nature of the insulating state and added the importance of longer-range exchange interactions. These ingredients are believed to stabilize intrinsically quantum disordered ground states 42,43 as we propose is realized here.
Inorganic compounds with simple structures and large-scale magnetic exchange energy rarely realize quantum disordered or spin-liquid-type ground states due to low-energy interactions, such as Dzyaloshinskii-Moriya interactions or weak lattice distortions that lift magnetic frustration. Even rarer is a quantum spin liquid candidate that manifests a transition/crossover between quantum disordered phases evident in the breakdown of the γ term in NaRuO 2 . The unique interplay between the marginally insulating Mott state and anisotropic exchange fostered by cooperative spin-orbit coupling and on-site Coulomb interactions suggests that NaRuO 2 occupies a unique energy landscape-one where Kitaev coupling in the Hubbard model fosters nearby quantum disordered states. The result is a material that provides an appealing opportunity for pushing the manifestation of exotic phenomena such as non-Abelian quasiparticle excitations associated with a native spin liquid state up to higher temperatures for future quantum information applications.

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Sample preparation
Polycrystalline NaRuO 2 was synthesized using mechanochemical methods. Sodium peroxide (Na 2 O 2 ) beads (Sigma, 97.00%), ruthenium dioxide (RuO 2 ) powder (Alfa, 99.95%) and sodium metal (Alfa 99.8%) are combined in a pre-seasoned tungsten carbide ball-mill vial under argon. The synthesis generally needs excess Na and O during the reaction; however, we note that NaRuO 2 exhibits an unusually large degree of off-stoichiometry in the Na-rich regime. To optimize the purity and Na stoichiometry, we have designed the synthesis to minimize Na Ru antisite defects by forcing NaRuO 2 to crystallize in equilibrium with Ru and NaRu 2 O 4 , which corresponds to the Ru-rich edge of the phase diagram. Thus, the stoichiometry has been empirically tuned to Na 1.07 (Na 2 O 2 ) 0.35 (RuO 2 ) 1. 35 . A more detailed analysis of the phase diagram and the resulting Na Ru off-stoichiometry can be found elsewhere 20 . The resulting mixture is milled for 60 min in a Spex 8000D Mixer/ Mill using four 7.9 mm tungsten carbide balls. The reaction generates a substantial amount of heat, and precursor powders produced through milling are effectively amorphous. The resulting precursor powder is lightly ground in an agate mortar under argon, sieved through a 100 µm sieve and loaded into 2 ml CoorsTek alumina crucibles. Small quantities of precursor are also pressed into discs with a Carver press and subsequently buried in the precursor powder. Note that the unreacted powders are extremely hydroscopic and mildly pyrophoric, so the crucibles are sealed under approximately 1 bar of argon in fused silica ampoules and immediately placed within a preheated furnace at 900 °C. The samples are annealed for 30 min and are then immediately air-quenched before extracting powders within an argon glovebox with water and O 2 levels of <0.5 ppm. During annealing, both pellet and powder transform into the desired phase, whereas the pellet simultaneously densifies (sinters). This ensures that sample preparation is consistent throughout the measurement probes. The resulting powders are largely phase pure, with trace amounts of Ru metal (<1%) and Na 2 CO 3 (<3%) due to an impurity in the starting Na 2 O 2 reagent. Product powders are black and remain highly moisture sensitive. NaRhO 2 powder was prepared and sintered as a phonon reference in heat capacity measurements for estimating the magnetic entropy. It was prepared following the methods reported elsewhere 44 and the phase purity was confirmed via X-ray diffraction ( Supplementary Fig. 5).

X-ray synchrotron diffraction
High-resolution synchrotron powder diffraction data were collected using beamline 11-BM at the Advanced Photon Source, Argonne National Laboratory, using an average wavelength of 0.457925 Å. Both 300 and 5 K measurements were performed to check for any crystallographic phase transformations, as well as for a better analysis of the thermal parameters and occupancies at base temperatures. Discrete detectors covering an angular range from −6.000° to 16.000° in 2θ are scanned over a 34.000° range, with data points collected every 0.001° and a scan speed of 0.010° s -1 . Due to the air sensitivity of the materials, small quantities of NaRuO 2 were diluted with amorphous SiO 2 in a glovebox and sealed under argon in flame-tipped amorphous SiO 2 capillaries. These capillaries were nested within Kapton sleeves and held in place with a small amount of modelling clay. The resulting data and pattern refinement are shown in Supplementary Fig. 1

Neutron diffraction
Neutron powder diffraction measurements were performed on the fixed-incident-energy triple-axis spectrometer HB-1A (λ = 2.37 Å) of the High Flux Isotope Reactor at Oak Ridge National Laboratory. Six grams of phase-pure polycrystalline NaRuO 2 was loaded in a cylindrical Cu can, which was then placed in a dilution insert of an orange cryostat, providing a base temperature of 50 mK. The collimation configuration of 40′-40′-40′-80′ yielded an energy resolution (full-width at half-maximum) at the elastic line of ~1 meV. The combination of the double-bounce monochromator system and the placement of the pyrolytic graphite crystal analyser for energy discrimination before the single He-3 detector provided an excellent signal-to-noise ratio. Additional contamination from higher-order wavelength contamination was minimized with the use of two pyrolytic graphite filters.
High-resolution neutron powder diffraction experiments were performed on the high-resolution neutron powder diffractometer BT-1 of the NIST Center for Neutron Research at the National Institute of Standards and Technology. A Ge(311) monochromator provided a λ = 2.0772 Å with the maximum neutron flux, allowing for full diffraction patterns to be collected in 12 h. Then, 4.5 g of phase-pure polycrystalline NaRuO 2 was loaded into a vanadium can under a helium atmosphere and placed into a top-loading flow-type orange cryostat, providing a base temperature of 1.6 K. Rietveld refinements of the neutron diffractograms were performed with TOPAS-Academic V6 with the results shown in Supplementary Table 1.

Inelastic neutron scattering
High-energy inelastic neutron scattering experiments were performed on the direct-geometry time-of-flight chopper spectrometer SEQUOIA at the Spallation Neutron Source at the Oak Ridge National Laboratory 45 . Then, 8 g of phase-pure polycrystalline NaRuO 2 was loaded in a cylindrical Al can that was placed in a top-loading helium cryostat, providing a base temperature of 1.8 K. Powder-averaged (Q, E) spectra were collected with incident energies of E i = 80 and 42 meV, operating in the high-flux mode, providing an elastic resolution of ~7.8% and ~2.3% of E i , respectively. Background contributions to the inelastic spectra were approximated by the measurement of an empty aluminium can, which was measured under identical experimental conditions for approximately one-third of the counting time allocated for the sample. Scattering maps (Fig. 3a,c)  Low-energy inelastic neutron scattering experiments were performed on the direct-geometry time-of-flight chopper spectrometer CNCS of the Spallation Neutron Source. Approximately 9 g NaRuO 2 was loaded under an inert atmosphere in a cylindrical Cu can that was placed on a HelioxVT A-100 mm 3 He insert in a top-loading 100 mm orange cryostat, providing a base temperature of 250 mK. Powder-averaged (Q, E) spectra were collected with incident energies of 3.32 and 1.55 meV, operating in the high-flux mode, providing an elastic resolution of 0.10 and 0.03 meV, respectively 46 . Background contributions to the inelastic spectra were approximated by an empty Cu can, which was measured under identical experimental conditions with equal counting times that were allocated for the sample and removed from the data. Normalization with a vanadium standard was performed to account for variations in the detector response and solid-angle coverage. Additional background subtracted maps of S(Q, E) used for the scaling analysis data are presented in Supplementary Figs. 6 and 7.

Magnetic susceptibility
The temperature dependence of the zero-field-cooled (ZFC) and field-cooled d.c. magnetization of 17.6 mg NaRuO 2 powder placed in a brass holder was measured on a 7 T Quantum Design Magnetic Property Measurement System superconducting quantum interference device magnetometer. Data were continuously collected in the sweep mode with a ramp rate of 2 K min -1 in the presence of an external d.c. field of 1,000 Oe (1 Oe = (1,000/4π) A m -1 ). The fit Pauli-like χ 0 was corrected for core diamagnetism (-53.8 × 10 -6 emu mol -1 Oe -1 ) (ref. 47).
Article https://doi.org/10.1038/s41567-023-02039-x Isothermal magnetization measurements of the same 17.6 mg NaRuO 2 sample at 1.8 K and 300.0 K were performed on a Quantum Design 14 T Dynacool Physical Property Measurement System (PPMS) employing the vibrating sample magnetometer option. ZFC data were continuously collected in the sweep mode with a ramp rate of 100 Oe s -1 .
The temperature dependence of the a.c. susceptibility of 10.1 mg NaRuO 2 powder was measured on a Quantum Design 14 T Dynacool PPMS employing the a.c. susceptibility option for the dilution refrigerator. A portion of a sintered pellet with approximate dimensions of 1.0 mm × 1.0 mm × 0.5 mm was adhered to a quartz sample mounting post with a thin layer of GE varnish. All the a.c. measurements were collected in the stable mode under ZFC conditions in an external d.c. magnetic field of 1 T and ramp rate of 0.08 K min -1 .

Heat capacity
The temperature and field dependence of the specific heat capacity of a 5.62 mg fragment of a phase-pure NaRuO 2 sintered pellet and 6.21 mg of its non-magnetic analogue NaRhO 2 were measured on a Quantum Design 14 T Dynacool PPMS employing the heat capacity option. Apiezon N grease was used to optimize thermal coupling between the sample and calorimeter stage. All the measurements were performed on heating, whereas all the measurements in the field were done under ZFC conditions. The phonon contribution was removed for estimating the magnetic entropy only, and this was achieved by the subtraction of a NaRhO 2 standard sample.

SPS
NaRuO 2 pellets (diameter, 10 mm; thickness, 2 mm) were prepared from phase-pure NaRuO 2 powder using field-assisted sintering (FCT Systeme). The pellets were pressed at 850 °C and 90 MPa for 60 min in an Ar-filled chamber with a pressure of 30 hPa, using a heating rate of 150 °C min -1 and cooling rate of 40 °C min -1 . All the pellets were subsequently ground to a 2,000 grit finish before resistivity measurements. Note that the SPS samples were primarily used to test the influence of grain-boundary effects by producing samples with a variety of experimental densities (Supplementary Fig. 4).

Resistivity
NaRuO 2 sintered pellets were sectioned into rectangular bars with approximate dimensions of 1.0 mm × 2.0 mm × 0.5 mm. Electrical contacts were made in the standard four-point geometry with contacts being made with a combination of gold wire and silver paint. The paint used was DuPont cp4929N-100, and the gold wire used for leads is Alfa Aesar 0.05 mm Premion 99.995%. Thermal contact and electrical isolation were ensured using layers of GE varnish and cigarette paper. The temperature dependence of electrical resistivity was measured with the electrical transport option on a 9 T Quantum Design Dynacool PPMS using a drive current of 10 µA and drive frequency of 100 Hz. Data were continuously collected in the sweep mode with a ramp rate of 2 K min -1 .

Muon spin relaxation measurements
The muon depolarization data were taken on the general-purpose surface muon spectrometer on the πM3.2 beamline at the Paul Scherrer Institute. For the zero-field data, the muon beam spin polarization was oriented at 45° to the muon momentum, whereas for the LF studies, the polarization was anti-parallel to the momentum. A pressed powder disc (diameter, 10 mm; thickness, 2 mm) was wrapped in thin Mylar foil and suspended in a gas-flow cryostat. Data were fit using the musrfit program 48 and the supporting fit parameters are presented in Supplementary Fig. 3.

Electronic structure calculations
First-principles electronic calculations based on density functional theory were performed using the Vienna ab initio simulation package (version 5.4.4) 49,50 with the LDA functional and projector-augmented wave pseudopotentials 51,52 . The plane-wave energy cutoff was set to 400 eV and a Γ-centred 15 × 15 × 15 k-point mesh was automatically generated within the Vienna ab initio simulation package. Initial biasing of spin polarization and tetrahedral smearing with Blöchl corrections were used for the self-consistent static calculation 53 . The electronic structure was calculated via a non-self-consistent run using the charge density from the static calculation with spin-orbit coupling and Hubbard U correction of 1 eV. A k-point path for the band structure calculation was generated using the AFLOW online tool 54 . All the calculations had energy convergence better than 10 −6 eV.