A j_eff = 1/2 Kitaev material on the triangular lattice: the case of NaRuO₂

Abstract

Motivated by recent reports of a quantum-disordered ground state in the triangular lattice compound NaRuO₂, we derive a j_eff = 1/2 magnetic model for this system by means of first-principles calculations. The pseudospin Hamiltonian is dominated by bond-dependent off-diagonal Γ interactions, complemented by a ferromagnetic Heisenberg exchange and a notably antiferromagnetic Kitaev term. In addition to bilinear interactions, we find a sizable four-spin ring exchange contribution with a strongly anisotropic character, which has been so far overlooked when modeling Kitaev materials. The analysis of the magnetic model, based on the minimization of the classical energy and exact diagonalization of the quantum Hamiltonian, points toward the existence of a rather robust easy-plane ferromagnetic order, which cannot be easily destabilized by physically relevant perturbations.

Introduction

The first definition of a quantum spin liquid (QSL) state dates back to the milestone paper by P.W. Anderson in 1973¹, in which the resonating valence-bond wave function, a macroscopic liquid-like superposition of singlet states, was proposed as a variational guess for the ground state of the triangular lattice Heisenberg antiferromagnet^2,3. Another archetypal portrait of a QSL state is more recent and originated from the seminal work of Kitaev⁴, who set a bond-anisotropic spin model on the honeycomb lattice with an exact spin-liquid ground state represented in terms of Majorana fermions. Both these alternative descriptions of QSL states, which are associated with different microscopic mechanisms of frustration, left an indelible mark in the context of frustrated magnetism. On the one hand, the triangular lattice antiferromagnet is the prototypical example of a system with geometric frustration, where the presence of antiferromagnetic Heisenberg couplings over odd-sided loops of sites fights the tendency toward long-range magnetic order. On the other hand, the possibility of realizing anisotropic interactions as a consequence of the interplay between spin–orbit coupling (SOC), crystal field splitting and Hund’s coupling^5,6,7 has opened a whole new field of investigation centered around the Kitaev materials^8,9,10. Even though the original Kitaev honeycomb model has no odd-sided loops and hence no geometric frustration, it nevertheless features exchange frustration^10,11 due to the fact that bond-directional interactions with competing quantization axes cannot be satisfied simultaneously.

In this work, we investigate a recently synthesized compound, NaRuO₂, in which both paradigms of magnetic frustration described above come into play. The crystal structure of this material displays perfect triangular lattice planes of edge-sharing RuO₆ octahedra, separated by Na ions^12,13,14 (illustrated in Fig. 1). The space group of NaRuO₂ is R$\overline{3}{\rm{m}}$. The same structural arrangement is found in a family of rare-earth chalcogenides which have been recently investigated as possible spin liquid candidates¹⁵: NaYbO₂¹⁶, NaYbS₂^17,18, and NaYbSe₂¹⁹. However, at variance with the latter, in NaRuO₂ the rare-earth ion is replaced by ruthenium, which belongs to the d-block of the periodic table. In analogy to the intensively studied honeycomb compound α-RuCl₃^8,9,10, the strong SOC of ruthenium, combined with the geometry of edge-sharing ligand octahedra, is expected to realize a prime example for the Jackeli–Khaliullin mechanism to form a j_eff = 1/2 magnet with significant Kitaev interaction⁷. Resistivity measurements identified NaRuO₂ to be indeed insulating, with a small magnetization upon application of an external magnetic field, and a paramagnetic Curie temperature dependence of the magnetic susceptiblity¹³. These signatures point toward the possibility of a QSL ground state, making a microscopically motivated magnetic model for NaRuO₂ not only intriguing but also necessary.

Fig. 1: NaRuO₂ crystal structure, illustrations, and DFT results for the g-tensor values.

a Crystal structure of NaRuO₂ with an illustration of interlayer bonds J_⊥i. b Top view of the triangular Ru lattice, with an illustration of intralayer bonds and four-spin ring exchange plaquettes. The cubic coordinates employed in the magnetic model are oriented approximately along Ru–O bonds, as shown in the top right corner. c Trigonally compressed RuO₆ octahedron and corresponding ab-initio g-tensor values, with b^* defined perpendicular to the crystallographic a and c axes.

The interplay between Heisenberg exchange and Kitaev interactions on the triangular lattice^20,21 has been investigated in several works, revealing, for instance, the presence of crystals of ${{\mathbb{Z}}}_{2}$ vortices in the proximity of the magnetic phase with 120^∘ order²², and possibly a spin nematic state around the antiferromagnetic Kitaev point^23,24,25. Furthermore, an extended spin model featuring the Γ-exchange coupling²⁶, which can favor the onset of a stripy magnetic phase, has been investigated in connection with the j_eff = 1/2 iridate compound Ba₃IrTi₂O₉^27,28. In this regard, a comprehensive overview of the different phases induced by bond-anisotropic (nearest-neighbor) couplings on the triangular lattice is provided by ref. ²⁹. More recently, analogous anisotropic spin Hamiltonians have been shown to capture the effective magnetic interactions of certain transition metal dihalides^30,31,32.

In addition to bilinear spin couplings, several magnetic materials with a triangular lattice structure, e.g., organic charge-transfer salts³³, are characterized by non-negligible four-spin ring exchange interactions³⁴, which incorporate higher-order contributions in the perturbation-theory treatment of the Hubbard model around the Mott insulating regime. While at the (semi-)classical level ring exchange can induce the formation of spirals and non-trivial chiral orders (e.g., spin-vortex crystals)³⁵, or significantly affect the low-energy magnon spectra of collinear phases³⁶, at the quantum level it is argued to potentially stabilize QSL phases^37,38; in this regard, the possible appearance of a gapless QSL with a spinon Fermi surface³⁹, or a Kalmeyer–Laughlin chiral state⁴⁰, has been discussed. An additional level of complication arises in magnetic systems with strong spin–orbit interactions, namely the presence of spin-anisotropic ring-exchange interactions, which have been scarcely investigated in the past⁴¹.

In this work, we perform a thorough inspection of the magnetic properties of NaRuO₂, from first-principles calculations to microscopic spin models. We provide theoretical justification for the low-energy description of NaRuO₂ in terms of j_eff = 1/2 pseudospin degrees of freedom, highlighting the importance of different sources of interactions, such as intra- and inter-layer exchange couplings, and bond-anisotropic bilinear and ring-exchange interactions stemming from the strong SOC effects. The analysis of (classical and quantum) magnetic models indicate the existence of a robust easy-plane ferromagnetic (FM) order, which cannot be easily destabilized by perturbations around the ab initio derived spin Hamiltonian. Based on our proposed magnetic model, we also provide theoretical inelastic neutron scattering (INS) spectra, which can be directly compared to the experiment.

Electronic properties

We begin by analyzing the electronic properties of NaRuO₂ with the help of density functional theory (DFT) calculations, as detailed in the “Methods” section. The octahedral environment of the Ru 4d⁵ sites leads to a crystal field splitting^8,42,43 with unoccupied e_g-states and occupied t_2g-states. The latter further split into j_eff = 3/2 and j_eff = 1/2 levels in the limit of strong SOC.

To estimate the Hubbard repulsion and Hund’s coupling of NaRuO₂, we employ constrained random phase approximation (cRPA) calculations (see “Methods” section). In the non-relativistic band structure, we observe a crossing between the Ru e_g band and a band with dominant Na 3s character close to the Γ point. This poses the question of which bands should be included in the Wannierization procedure prior to cRPA. One option is considering only the five 4d Ru bands, which can be expected to lead to an artificially enhanced e_g screening. The corresponding cRPA result is ${({U}_{{{{\rm{avg}}}}},{J}_{{{{\rm{avg}}}}})}_{4d}=(3.114,0.4736)$ eV. Alternatively, the crossing band may be included in a cRPA calculation based on a six-band model. This choice leads to an artificially suppressed screening, with ${({U}_{{{{\rm{avg}}}}},{J}_{{{{\rm{avg}}}}})}_{4d+3s}=(3.1865,0.4756)$ eV. Since it turns out that both approaches lead to similar results, we choose to work with the average of them: (U_avg, J_avg) = (3.15, 0.47) eV.

We employ these values as correlation corrections on the Ru 4d electrons in a relativistic band structure calculation (GGA + SOC + U, see “Methods” section) with ferromagnetically aligned magnetic moments. The resulting band structure and partial DOS are shown in Fig. 2a. We choose the spin magnetic moments to be polarized in the crystallographic a–b plane, which is the most energetically favorable orientation. Within this setting, we obtain an insulator with a charge gap of 2.1 eV which is in agreement with recent resistivity experiments¹³. We find that the combination of magnetism and Coulomb interaction is necessary to open a realistic charge gap in NaRuO₂, which is further enlarged by SOC.

Fig. 2: Electronic band structure and density of states of NaRuO₂.

a Ferromagnetic GGA+SOC+U results, with (U_avg, J_avg) = (3.15, 0.47) eV for the Ru 4d electrons, and magnetization polarized in the crystallographic a–b plane. b Non-magnetic GGA + SOC results. The partial DOS for Ru 4d orbitals projected in the j_eff basis (see Supplementary Note 1) shows a dominant j_eff = 1/2 weight close to the Fermi energy (see inset).

The partial DOS resolves the dominance of ruthenium weight around the Fermi level, such that we can proceed with low-energy modeling of this compound based on ruthenium bands.

The edge-sharing octahedral structure of NaRuO₂ hints towards the possibility of a j_eff = 1/2 description of the low-energy magnetic properties, in analogy with the intensively studied α-RuCl₃⁸. To check the validity of the j_eff = 1/2 picture for NaRuO₂, we calculate the non-magnetic band structure and project the DOS in the j_eff = {1/2, 3/2} basis obtained by considering only t_2g orbitals (see Supplementary Note 1). The results, reported in Fig. 2b, show that the character of the DOS around the Fermi energy is dominated by j_eff = 1/2 states, thus justifying a low-energy description in terms of j_eff = 1/2 pseudospins.

Magnetic model

As an appropriate magnetic model for NaRuO₂, we consider a j_eff = 1/2 pseudospin Hamiltonian. To relate the pseudospin S of the magnetic Hamiltonian to the magnetic moment, ${{{\bf{M}}}}={\mu }_{{{{\rm{B}}}}}{\mathbb{G}}\cdot {{{\bf{S}}}}$, we calculate the gyromagnetic g-tensor from first principles (see “Methods” section). We find it to be approximately diagonal with $({g}_{a},{g}_{{b}^{* }},{g}_{c})=(2.46,2.43,1.57)$ in crystallographic coordinates (with b^* perpendicular to a and c). With respect to the triangular plane, the in-plane components (${g}_{a},{g}_{{b}^{* }}$) are larger than the out-of-plane one (g_c), as a direct consequence of the trigonal compression of the RuO₆ octahedra along the crystallographic c axis (see Fig. 1c)⁴⁴.

For the magnetic interactions between the pseudospins, we consider a Hamiltonian consisting of a bilinear exchange term ${{{{\mathcal{H}}}}}_{2}$ and a four-spin ring exchange term ${{{{\mathcal{H}}}}}_{4}$. We express this model in the conventionally used cubic coordinates for Kitaev materials, which consist of orthogonalized axes oriented approximately along the Ru-O bonds, as illustrated in the top right corner of Fig. 1b. We denote the three components of the pseudospin at site i as ${S}_{i}^{\mu }$, with μ = {x, y, z}. In this framework, the [111] pseudospin direction is parallel to the crystallographic c-axis. For completion, in Supplementary Note 2 we translate our model to an alternative reference frame with crystallographic coordinates²⁹.

The bilinear contribution to the magnetic Hamiltonian ${{{{\mathcal{H}}}}}_{2}={\sum }_{i < j}{\sum }_{\mu \nu }{{\mathbb{J}}}_{i,j}^{\mu \nu }{S}_{i}^{\mu }{S}_{j}^{\nu }$ contains, especially for nearest neighbors, anisotropic bond-dependent terms. Considering the symmetry constraints of the R$\overline{3}{\rm{m}}$ space group, the bilinear exchange tensor on a Z₁-bond (as defined in Fig. 1b) follows the form

$${{\mathbb{J}}}_{{{{{\rm{Z}}}}}_{1-{{{\rm{bond}}}}}}=\left(\begin{array}{ccc}{J}_{1}&{\Gamma }_{1}&{\Gamma }_{1}^{{\prime} }\\ {\Gamma }_{1}&{J}_{1}&{\Gamma }_{1}^{{\prime} }\\ {\Gamma }_{1}^{{\prime} }&{\Gamma }_{1}^{{\prime} }&{J}_{1}+{K}_{1}\end{array}\right).$$

(1)

Here, J₁ is the isotropic Heisenberg exchange, K₁ is the Kitaev coupling, and Γ₁ and ${\Gamma }_{1}^{{\prime} }$ the off-diagonal symmetric exchange parameters. The bilinear interactions on X₁- and Y₁-bonds are then related to this expression by C₃ spin rotations around the [111] axis, amounting to cyclic permutation of (x, y, z) spin components.

We performed DFT calculations to obtain the magnetic exchange parameters as described in the “Methods” section. In Table 1 we show the dominant magnetic couplings extracted for nearest-neighbor bonds with the projED method⁴⁵ and the isotropic longer-range exchange from total energy mapping analysis (TEMA). Both methods predict a dominant intralayer FM Heisenberg J₁ coupling. However, for a quantitative agreement of the results, larger U values in the DFT + U calculations with respect to projED should be considered in this case. This is to be expected since the two techniques rely on different implementations of the Coulomb terms (see “Methods”). The readjustment can be directly taken into account by considering a scaling factor ${J}_{1}^{{{{\rm{projED}}}}}/{J}_{1}^{{{{\rm{TEMA}}}}}\approx 0.65$ for the Heisenberg exchange couplings obtained by TEMA. This is justified by the fact that the ratio between different Heisenberg exchange couplings does not vary significantly as a function of U for this system. In addition to the strong nearest-neighbor FM J₁, the shortest further-neighbor intralayer exchange couplings are found to be non-negligible and of similar magnitude, with J₂ being anti- and J₃ ferromagnetic. The interlayer Heisenberg couplings J_⊥1, J_⊥2, and J_⊥3 (shown in Fig. 1a) are one magnitude smaller than the intralayer ones, with J_⊥1 and J_⊥2 being ferromagnetic and J_⊥3 antiferromagnetic.

Table 1 Magnetic exchange parameters.

Within the projED method, we obtain the bond-dependent anisotropic couplings K₁, Γ₁, and ${\Gamma }_{1}^{{\prime} }$ at nearest-neighbors, where the cRPA values (U_avg, J_avg) = (3.15, 0.47) eV are employed. Compared to previously estimated magnetic parameters for other Ru 4d systems^46,47,48,49, it is interesting to note that we have strongly antiferromagnetic Kitaev K₁ term and a dominant positive Γ₁ as the largest coupling. These are contributions that, to the best of our knowledge, have not been observed in a real material with effective spin 1/2 so far.

The microscopic origin of the unusual antiferromagnetic sign of the Kitaev interaction encountered here can be understood as follows: from the perspective of second-order perturbation theory in a perfect octahedral environment (considering only the occupied t_2g orbitals)⁵⁰, the Kitaev interaction scales as ${K}_{1}\propto {({t}_{1}-{t}_{3})}^{2}-3\,{t}_{2}^{2}$. On a Z-bond, the hopping parameters are defined as the ligand-assisted hopping t₂ = t_(xz; yz), as well as t₁ = t_(xz; xz) = t_(yz; yz) and t₃ = t_(xy; xy), which stem predominantly from direct d orbital overlap. For the prime example of the honeycomb Kitaev material α-RuCl₃, the indirect hopping t₂ is dominant, somewhat close to the t₂-only model in the Jackeli-Khaliullin mechanism, where the Kitaev interaction is FM (${K}_{1}\propto -3{t}_{2}^{2}$)⁷. In comparison, the direct hoppings t₃ and t₁ gain importance in NaRuO₂, where the nearest neighbor Ru-Ru bond length is significantly smaller than in α-RuCl₃. Indeed, for NaRuO₂, a non-relativistic DFT calculation yields t₁ = 80 meV, t₂ = 125 meV and t₃ = − 261 meV. As dictated by the geometry, t₃ is negative and larger in magnitude than the positive t₁. The resulting antiferromagnetic Kitaev interaction in NaRuO₂ can hence be directly related to the shorter nearest-neighbor bond length of this triangular compound. The perturbation theory perspective also offers an explanation for the dominance of the off-diagonal symmetric exchange Γ₁, which scales approximately as Γ₁ ∝ t₂ (t₁ − t₃)⁵⁰. While the magnitude of K₁ reduces with the competition between indirect and direct contributions, Γ₁ increases proportional to the magnitude of the hoppings, leading to a magnetic model dominated by the off-diagonal symmetric exchange for NaRuO₂.

Despite the antiferromagnetic sign of the Kitaev interaction, the bilinear exchange Hamiltonian ${{{{\mathcal{H}}}}}_{2}$ features a ferromagnetically ordered ground state, due to significant J₁ < 0 and Γ₁ > 0 interactions, as discussed in more detail below. To seek out possible additional interactions that might destabilize the FM ground state, we consider the effects of higher-order ring exchange interactions. Compared to honeycomb Kitaev materials, NaRuO₂ could be predestined for such interactions, due to the marginally insulating Mott state reported in experiments¹³ and to its triangular lattice structure, where the shortest closed loops consist of four (instead of six) sites. Four-spin ring exchange without SOC effects has been discussed plentifully in the literature^{34,36,37,38,40} and takes the form

$$\begin{array}{l}{{{{\mathcal{H}}}}}_{4}^{{{{\rm{iso}}}}}=\frac{{K}^{{{{\rm{iso}}}}}}{{S}^{2}}\mathop{\sum}\limits_{\langle ijkl\rangle }({{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j})({{{{\bf{S}}}}}_{k}\cdot {{{{\bf{S}}}}}_{l})+({{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{l})({{{{\bf{S}}}}}_{j}\cdot {{{{\bf{S}}}}}_{k})\\ \qquad \quad \qquad \qquad-\,({{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{k})({{{{\bf{S}}}}}_{j}\cdot {{{{\bf{S}}}}}_{l}),\end{array}$$

(2)

where the summation 〈ijkl〉 goes over plaquettes with sites i and k lying across a diagonal³⁴, K^iso is the coupling constant, and the superscript “iso” denotes that this is the conventional isotropic (i.e., SU(2)-symmetric) ring exchange. Note that the prefactor conventions for the ring exchange interaction vary in the literature^35,39,40,51. Here, we choose to include a factor $\frac{1}{{S}^{2}}$ in Eqs. (2), (3). Thus, in the convention without this prefactor (e.g., ref. ⁴⁰), the ring exchange coupling constants we compute below would appear four times larger.

However, for NaRuO₂ there is no reason why the ring exchange between the pseudospin ${j}_{{{{\rm{eff}}}}}=\frac{1}{2}$ moments should follow the form of ${{{{\mathcal{H}}}}}_{4}^{{{{\rm{iso}}}}}$, since SOC is expected to induce anisotropic four-site terms in the Hamiltonian. In the most general form, an anisotropic four-spin exchange may be expressed as

$${{{{\mathcal{H}}}}}_{4}^{{{{\rm{tot}}}}}=\frac{1}{{S}^{2}}\mathop{\sum}\limits_{\langle ijkl\rangle }\mathop{\sum}\limits_{\mu \nu \rho \eta }{{\mathbb{K}}}_{ijkl}^{\mu \nu \rho \eta }\,({S}_{i}^{\mu }{S}_{j}^{\nu }{S}_{k}^{\rho }{S}_{l}^{\eta }),$$

(3)

where the tensor ${\mathbb{K}}$ contains the coupling constants. The presence of inversion symmetry with respect to the center of each plaquette, together with a C₂ rotation axis parallel to the shortest diagonal, reduces the 81 entries of ${\mathbb{K}}$ for one plaquette to 24 independent parameters. Furthermore, analogous to the X-, Y-, and Z-bonds of bilinear exchange, it is convenient to define X-, Y-, and Z-plaquettes, as shown in Fig. 1b. The three plaquettes are related by C₃ rotations around the out-of-plane axis and hence the tensor of one plaquette type fully encodes ${{{{\mathcal{H}}}}}_{4}^{{{{\rm{tot}}}}}$.

Note that in contrast to conventional ring exchange, care has to be taken for the order of the site numbering within a plaquette. For example, for a single plaquette (with the site labeling illustrated in Fig. 1b), swapping two sites across a diagonal is not a symmetry of the ring-exchange tensor, i.e., ${{\mathbb{K}}}_{1234}^{\mu \nu \rho \eta }\,\ne \,{{\mathbb{K}}}_{1432}^{\mu \nu \rho \eta }$, even in presence of the aforementioned symmetries.

To compute the ring-exchange tensor ${\mathbb{K}}$ from first principles, we employ the projED method, which has been used previously to determine ring-exchange couplings for organic triangular lattice compounds^35,52. Results on a Z-plaquette are given in Table 2, and details of the calculation are outlined in the “Methods” section. We do not attempt to create a minimal model here and show agnostically the full ab-initio result. Overall, the four-site ring exchange contribution in NaRuO₂ does not seem to be obviously negligible, with a strength of roughly 5–10% of the nearest-neighbor bilinear exchange parameters. As anticipated, the shown results deviate substantially from the conventional isotropic ring exchange ${{{{\mathcal{H}}}}}_{4}^{{{{\rm{iso}}}}}$. This is not surprising because of the strong SOC in NaRuO₂. For instance, among the diagonal components characterizing the Z-plaquette, the ${{\mathbb{K}}}_{1234}^{zzzz}$ term strongly differs from ${{\mathbb{K}}}_{1234}^{xxxx}={{\mathbb{K}}}_{1234}^{yyyy}$.

Table 2 Four-spin ring exchange couplings for NaRuO₂.

To quantify the degree of anisotropy of the total ring exchange Hamiltonian encoded in Table 2, we express it as a sum consisting of the conventional isotropic ring exchange from Eq. (2) and a purely anisotropic contribution: ${{\mathbb{K}}}^{{{{\rm{tot}}}}}={{\mathbb{K}}}^{{{{\rm{iso}}}}}+{{\mathbb{K}}}^{{{{\rm{ani}}}}}$. The choice of this splitting is not unique, but we choose the coupling constant K^iso in ${\left({{\mathbb{K}}}^{{{{\rm{iso}}}}}\right)}_{1234}^{\mu \nu \rho \eta }={K}^{{{{\rm{iso}}}}}\left({\delta }_{\mu \nu }{\delta }_{\rho \eta }+{\delta }_{\mu \eta }{\delta }_{\nu \rho }-{\delta }_{\mu \rho }{\delta }_{\nu \eta }\right)$ (cf. Eq. (2)) such that the tensor-norm of the anisotropic part, $\parallel {{\mathbb{K}}}^{{{{\rm{ani}}}}}\parallel =\parallel {{\mathbb{K}}}^{{{{\rm{tot}}}}}-{{\mathbb{K}}}^{{{{\rm{iso}}}}}\parallel$, is minimized. Here, the tensor 1-norm is used ($\parallel {\mathbb{K}}\parallel ={\sum }_{\mu \nu \rho \eta }| {{\mathbb{K}}}_{1234}^{\mu \nu \rho \eta }|$). This choice is motivated by an analogy to the case of the bilinear Hamiltonian, where the same procedure splits the bilinear exchange tensor ${{\mathbb{J}}}_{ij}^{\mu \nu }$ into an (isotropic) Heisenberg exchange part and an anisotropic part, arriving at the same definition of Heisenberg-J as in Eq. (1). Dissecting the ring-exchange interaction in this way leads to K^iso = − 0.06 meV and $\parallel {{\mathbb{K}}}^{{{{\rm{ani}}}}}\parallel \,/\,\parallel {{\mathbb{K}}}^{{{{\rm{iso}}}}}\parallel =5.6$, which shows that ring exchange in NaRuO₂ is dominated by the anisotropic contributions.

Properties of the magnetic model

We investigate the ground state of the magnetic model given in Table 1 by applying two different methods. We consider the classical ground state via an iterative minimization method of the energy^53,54 and then we include quantum fluctuations by tackling the Hamiltonian with exact diagonalization (ED) on finite clusters with up to 27 sites (see “Methods” section and Supplementary Note 3).

First, we consider the classical ground state of the ${{{{\mathcal{H}}}}}_{2}$ Hamiltonian restricted to the triangular lattice plane. The omission of inter-layer couplings is justified by their small estimated magnitude compared to intra-layer couplings (cf. Table 1). The minimum of the classical energy is provided by an FM spin arrangement, with spins lying in the triangular lattice plane. The FM nature of the ground state turns out to be stable upon different perturbations of the Hamiltonian around the ab initio model (as discussed in Supplementary Note 4) and upon the inclusion of anisotropic ring exchange or out-of-plane interactions. For what concerns the latter, the classical energy minimum yields an FM ground state, with spins being parallel to each other both within and between layers, consistent with total energy calculations within DFT. However, the configuration with ferromagnetically stacked FM layers is lower in energy than the one with antiferromagnetically stacked FM layers only by ~0.1 meV/Ru. ED calculations, performed on various two-dimensional clusters with different shapes and numbers of sites, confirm the stability of the FM ground state when quantum effects come into play. The addition of the ring-exchange interaction ${{{{\mathcal{H}}}}}_{4}^{{{{\rm{tot}}}}}$ does not destabilize the FM order, neither for classical nor for quantum spins, but leads to a small tilt of the ordered moment out of the triangular lattice plane (by less than 1° in our model).

After having established the FM character of the ground state, we move on to compute excitations, namely the INS intensity predicted by the magnetic model. We employ linear spin-wave theory (LSWT), complemented by ED to investigate effects beyond LSWT. The results are summarized in Fig. 3, where the ab initio magnetic form factor for Ru³⁺ is taken into account for the calculation of spectral intensities⁵⁵, such that the magnetic spectra can be directly compared to neutron scattering experiments.

Fig. 3: Computed neutron scattering intensity within the magnetic model.

a Momentum path in the Brillouin zone of the triangular lattice, b INS intensity for the Hamiltonian ${{{{\mathcal{H}}}}}_{2}$ without inter-plane interaction along the high symmetry lines in momentum space. The color plot shows ED results combined from different clusters up to 27 sites. Due to elsewise overbearing intensity at q = 0, the plotted intensity is adjusted for q = 0, by reducing the plotted intensity and broadening at q = 0 (see “Methods”). The overlayed band dispersion is the single-magnon energy from LSWT. As the ED ground state is a superposition of different degenerate magnetic domains, the compared LSWT energy is plotted as the average of three calculations, corresponding to expansions around in-plane ferromagnetic order, with polarization perpendicular to X₁-, Y₁- or Z₁-bonds. c Calculated powder-averaged INS spectrum from LSWT, for the model in Table 1, including inter-plane exchange couplings. Intensities above ~3.5% of the maximum intensity are cut off due to large intensity at q ≈ 0.

As previously mentioned, the magnetic moments of the FM ground state of the ${{{{\mathcal{H}}}}}_{2}$ Hamiltonian lie within the triangular lattice plane, without picking any preferred direction on the classical level. However, this continuous symmetry is accidental and is lifted by quantum fluctuations, which select the configurations in which the moments are perpendicular to one of the nearest-neighbor bonds²⁹. At the linear spin wave theory level we then expect the appearance of a gapless pseudo-Goldstone mode, which becomes gapped when quantum effects beyond the lowest order are considered^56,57. The ED results, compared to the LSWT prediction in Fig. 3b, confirm this picture and find a gap at q = 0 of the order of 1–2 meV.

A further effect beyond LSWT that one might expect here is the appearance of strong scattering continua even in magnetically ordered phases. Such continua were observed in the honeycomb Kitaev material α-RuCl₃⁵⁸, where they have been traced back in a spin-wave description to originate from significant anharmonic effects due to Γ₁ exchange⁴⁶. However, despite the dominant off-diagonal Γ₁ exchange in the case of NaRuO₂ (cf. Table 1), no substantial scattering continuum is found here and the ED spectrum qualitatively follows the sharp modes of LSWT, as shown in a comparison in Fig. 3b. This can be understood as a consequence of the fact that in the present FM state, the pseudo-Goldstone modes remain at the ordering-wave vector Q = 0 (Γ-point), such that a potential decay of single-magnons into a two-magnon continuum via Γ₁-exchange is kinematically not allowed. We note that the inclusion of ring exchange increases the magnon energies by ~2 meV, but does not qualitatively change the main features of the spectrum.

We also compute the powder-averaged INS spectrum that might be relevant for direct comparison of the predicted FM state to experiments¹³. Here, the effect of inter-plane couplings is included, in order to obtain meaningful integration over out-of-plane momenta. The results from LSWT are shown in Fig. 3c and feature the gapless pseudo-Goldstone mode at the smallest momenta, and a less intense gapless mode around 1.2 Å⁻¹ arising from the inter-layer FM stacking. Indeed, such a low-energy mode and dominant FM fluctuations (i.e., small ∣q∣-excitations) are also suggested by recent INS powder data at T = 0.25 K on NaRuO₂¹³.

Discussion

In this work, we investigated the magnetic properties of NaRuO₂, a layered system of corner-sharing RuO₆ octahedra, which constitutes a prime example of the realization of anisotropic spin couplings, such as the Kitaev interaction, on a triangular lattice structure. By combining two complementary first-principle methods, TEMA and projED, we derived a j_eff = 1/2 pseudospin Hamiltonian for NaRuO₂, which displays a sizable antiferromagnetic Kitaev coupling. This is a direct consequence of the comparatively smaller nearest-neighbor Ru–Ru bond length in NaRuO₂, leading to a dominance of direct hopping mechanisms in contrast to other spin-1/2 Kitaev materials to date. The strongest interactions of the model are however a symmetric Γ₁ exchange and an FM J₁ Heisenberg term. The spin Hamiltonian with bilinear interactions possesses a rather robust FM order, oriented parallel to the triangular lattice plane formed by ruthenium ions, also when longer-range intra- and inter-plane exchange interactions are taken into account.

In a recent work⁵⁹, the nearest nearest-neighbor spin Hamiltonian for NaRuO₂ has been calculated from quantum chemistry methods. While the signs of the nearest-neighbor couplings match with the ones of our model, the Γ₁ term of ref. ⁵⁹ is smaller than the J₁ exchange strength, contrary to our results. The authors explore the possibility of destabilizing the FM order by an antiferromagnetic third-neighbors exchange, which is in contrast with our first-principle prediction of an FM J₃ coupling.

The parameters of the magnetic Hamiltonian, as obtained by first-principles calculations with the pristine structure of NaRuO₂, place this material deep inside an extended FM phase, which cannot be easily destabilized by perturbing the Hamiltonian around the ab initio point. Since the experimentally available results do not show the conventional features of long-range FM order and suggest the possibility of a marginally insulating Mott state in NaRuO₂¹³, we explored the effects of higher-order spin interactions, computing four-spin ring exchange couplings from the projED method. However, although the latter turn out to be of a non-negligible size, they seem insufficiently strong to melt the FM order. Nevertheless, the nature of the ring exchange interaction is strongly anisotropic and its consequences warrant further investigation, also in the context of other Kitaev materials.

Furthermore, employing LSWT calculations and ED on finite clusters, we computed the INS spectra for NaRuO₂. Comparison of the powder-average neutron scattering intensity with experimental observations shows a similar weight distribution which may signal the presence of underlying ferromagnetism in the system, although no long-range magnetic order was observed in experiments^13,14. This raises the question of the role of disorder in the material, which may be addressed in future investigations.

Methods

First-principles calculations

All first principles calculations employ the crystal structure published in ref. ¹³. For the calculation of electronic properties we use the full potential local orbital (FPLO)⁶⁰ package 18.00-57 and the Generalized Gradient Approximation (GGA)⁶¹ as the exchange-correlation functional. The correlation for the strongly localized Ru 4d electrons are corrected via the GGA + U approximation using the “atomic limit” implementation⁶². All calculations are carried out on a 12 × 12 × 12 k-grid in the primitive unit cell. Relativistic calculations are performed within the GGA + SOC + U functional. The estimates of the t₁–t₃ hoppings discussed in the “Magnetic model” section have been obtained within a non-relativistic GGA calculation.

The electronic properties (band structure, density of states (DOS)) have been cross-checked with the linearized augmented plane-wave basis set as implemented in Wien2k⁶³ version 19.1, with Ru 4d correlation correction included via the SIC method^64,65 with effective Coulomb repulsion U_eff = 2 eV.

We also compute the gyromagnetic g-tensor from the first principles. For this calculation, we consider a [RuO₆]⁹⁻ molecule within the quantum chemistry ORCA 5.03 package⁶⁶ with the functional TPSSh, basis set def2-TZVP, and complete active space self-consistent field method for the d orbitals CASSCF(5,5). A conductor-like polarizable continuum model (C-PCM)⁶⁷ is employed with a Gaussian charge scheme, a van der Waals-type cavity, and an infinite dielectric constant.

Constrained random phase approximation

Based on the electronic structure obtained with the Wien2k package v21.1⁶³ we estimate the electronic two-particle interaction terms in NaRuO₂ with the constrained random-phase approximation (cRPA)^68,69, as implemented in the FHI-gap code⁷⁰. The integration of the Brillouin zone is done on an 8 × 8 × 8 grid. The static low-energy limit of the partially screened interaction is projected onto the relevant orbitals, where screening processes in the same window are excluded. The spherical symmetric expressions for d electrons in the atomic limit are based on Slater integrals F_k⁶⁵ as follows:

$${U}_{{{{\rm{avg}}}}}=\frac{1}{{(2l+1)}^{2}}\mathop{\sum}\limits_{\alpha \beta }{U}_{\alpha \beta }={F}_{0}$$

(4)

$${J}_{{{{\rm{avg}}}}}=\frac{7}{5}\frac{1}{2l(l+1)}\mathop{\sum}\limits_{\alpha \ne \beta }{J}_{\alpha \beta }=\frac{{F}_{2}+{F}_{4}}{14},$$

(5)

where α, β are orbital indices and l is the angular momentum quantum number.

As mentioned in the “Results” section, due to a band crossing of the Ru e_g bands with a Na 3s band in NaRuO₂, there are two sensible ways to select the relevant orbitals considered in the Wannier projection. We denote results based on the five Ru 4d orbitals as ${({U}_{{{{\rm{avg}}}}},{J}_{{{{\rm{avg}}}}})}_{4d}$ and results including also the Na 3s band as ${({U}_{{{{\rm{avg}}}}},{J}_{{{{\rm{avg}}}}})}_{4d+3s}$. Since both these options lead to very similar results, further calculations in the main text adopt the average of both cRPA results.

DFT-based derivation of the magnetic model

We extract the dominant magnetic Heisenberg couplings via the TEMA^71,72,73, which is a two-step process. First, we calculate the total energies within DFT (GGA + U) of different magnetic configurations of chosen supercells of NaRuO₂. In the second step, we fit the DFT energy of the different magnetic configurations to an effective Heisenberg spin-1/2 Hamiltonian using the method of least squares. The first step is performed in the VASP 5.3 framework⁷⁴ using spin-polarized DFT + U, where we apply the Dudarev scheme⁷⁵, with effective Coulomb repulsion U_eff = 3.5 eV. Here, we consider 14 different magnetic configurations. The calculations are performed within a 3 × 2 × 1 supercell on a 5 × 8 × 3Γ-centered k-grid with an energy cut-off of 540 eV for the plane-wave basis set. The quality of the TEMA for the considered model is discussed in Supplementary Note 5. We have checked that different values of the effective Coulomb repulsion don’t significantly affect the ratio between the various exchange couplings.

As a second method, we employ the so-called projED technique⁴⁵. The approach consists of two main steps. First, an effective 4d electronic Hamiltonian ${{{{\mathcal{H}}}}}_{{{{\rm{tot}}}}}={{{{\mathcal{H}}}}}_{{{{\rm{hop}}}}}+{{{{\mathcal{H}}}}}_{{{{\rm{U}}}}}$ is constructed, where ${{{{\mathcal{H}}}}}_{{{{\rm{hop}}}}}$ consists of complex electronic hopping parameters, determined from first principles via Wannier projection of a relativistic band structure calculation (GGA + SOC). Here, we extract the Wannier functions by using the FPLO⁶⁰ package 18.00-57. ${{{{\mathcal{H}}}}}_{{{{\rm{U}}}}}$ contains the electronic two-particle Coulomb interaction⁴⁵. In a second step, the electronic Hamiltonian is solved by ED on a two-site five-orbital cluster and its low energy states are projected onto spin operators, arriving at the desired effective spin Hamiltonian, e.g., ${{{{\mathcal{H}}}}}_{2}={\mathbb{P}}{{{{\mathcal{H}}}}}_{{{{\rm{tot}}}}}{\mathbb{P}}={\sum }_{i < j}{\sum }_{\mu \nu }{{\mathbb{J}}}_{i,j}^{\mu \nu }{S}_{i}^{\mu }{S}_{j}^{\nu }$. Note that here S is a pseudospin with j_eff = 1/2. We employ the projED method for the calculation of nearest-neighbor couplings, while for longer-range interactions we resort to TEMA results. This choice is motivated by the fact that, within projED, the indirect hoppings over multiple sites, which are expected to become more and more important for longer-range couplings, cannot be accounted for due to computational limitations.

We also employ the projED method to extract the four-spin ring exchange Hamiltonian ${{{{\mathcal{H}}}}}_{4}^{{{{\rm{tot}}}}}$. Due to computational limitations, the parameters are extracted by diagonalizing a four-site three-orbital electronic Hamiltonian involving only Ru t_2g orbitals. We adopt this approximation since the aim of this work is to estimate the general form and order of magnitude of the ring exchange interaction in a strongly spin-orbit coupled system like NaRuO₂. Possible refinements of this approach are beyond the scope of this work and will be pursued in future studies.

Iterative minimization (classical spins)

We obtain the classical ground state of the spin Hamiltonian ${{{\mathcal{H}}}}$ by performing a numerical minimization of the energy on a finite lattice with periodic boundary conditions. We employ an iterative method in which the orientation of the spins (unit vectors at the classical level) is initialized with random values and updated by performing local moves. A single update is performed by selecting a random site i and changing its spin orientation according to

$${{{{\bf{S}}}}}_{i}\mapsto -\frac{{{{{\bf{h}}}}}_{i}}{\parallel {{{{\bf{h}}}}}_{i}\parallel }\,\,\,{{\mbox{where}}}\,\,\,{{{{\bf{h}}}}}_{i}=\left(\frac{\partial {{{\mathcal{H}}}}}{\partial {S}_{i}^{x}},\frac{\partial {{{\mathcal{H}}}}}{\partial {S}_{i}^{y}},\frac{\partial {{{\mathcal{H}}}}}{\partial {S}_{i}^{z}}\right).$$

(6)

In other words, we anti-align the spin at site i to the effective field h_i created by the interactions with the other spins in the lattice. The procedure is repeated several times until the minimum energy is reached. To try to mitigate the possibility of ending up in local energy minima, we perform a number of different calculations starting from different random initializations. Most numerical results have been obtained on a triangular lattice of N = 12 × 12 = 144 sites. For calculations involving inter-layer couplings, we used a three-dimensional cluster of N = 6 × 6 × 6 = 216 sites.

Exact diagonalization

We perform ED of the j_eff = 1/2 model on two-dimensional clusters of up to N = 27 sites. The INS intensity at momentum q and energy ω is given by

$${{{\mathcal{I}}}}({{{\bf{q}}}},\omega )\propto {f}^{2}({{{\bf{q}}}})\int{{{\rm{d}}}}t\,{e}^{-i\omega t}\mathop{\sum}\limits_{\mu ,\nu }\left({\delta }_{\mu ,\nu }-\frac{{k}_{\mu }{k}_{\nu }}{{k}^{2}}\right)\left\langle {S}_{-{{{\bf{k}}}}}^{\mu }(t){S}_{{{{\bf{k}}}}}^{\nu }(0)\right\rangle$$

(7)

where f(q) is the atomic form factor of Ru³⁺. To compute it, we employ the continued fraction method⁷⁶. To access a higher number of q-points, we plot together results coming from clusters of different shapes and sizes up to N = 27 (clusters shown in Supplementary Note 3), similar to as done in, e.g., ref. ⁴⁶. On all clusters and in the LSWT results, the intensity at small momenta q ≈ 0 is found so large that in a simple color plot, the intensity at q away from q ≈ 0 would be almost invisible. We, therefore, opted for Fig. 3b to plot a smaller broadening and a reduced intensity only at the q = 0 point. Without this adjustment, the intensity at q = 0 would appear ~21 times larger, rendering the rest of the dispersion invisible to the eye. q-points with q ≠ 0 are broadened by 1 meV Gaussians and q = 0 by 0.5 meV, in order to make the gap in ED better visible by eye.

Linear spin-wave theory

LSWT calculations are performed with the SpinW 3.0 library⁷⁷. The INS intensity is computed by taking the powder average of Eq. (7).