Abstract
Motivated by recent reports of a quantumdisordered ground state in the triangular lattice compound NaRuO_{2}, we derive a j_{eff} = 1/2 magnetic model for this system by means of firstprinciples calculations. The pseudospin Hamiltonian is dominated by bonddependent offdiagonal Γ interactions, complemented by a ferromagnetic Heisenberg exchange and a notably antiferromagnetic Kitaev term. In addition to bilinear interactions, we find a sizable fourspin 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 easyplane ferromagnetic order, which cannot be easily destabilized by physically relevant perturbations.
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Introduction
The first definition of a quantum spin liquid (QSL) state dates back to the milestone paper by P.W. Anderson in 1973^{1}, in which the resonating valencebond wave function, a macroscopic liquidlike 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 Kitaev^{4}, who set a bondanisotropic spin model on the honeycomb lattice with an exact spinliquid 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 oddsided loops of sites fights the tendency toward longrange 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 oddsided loops and hence no geometric frustration, it nevertheless features exchange frustration^{10,11} due to the fact that bonddirectional interactions with competing quantization axes cannot be satisfied simultaneously.
In this work, we investigate a recently synthesized compound, NaRuO_{2}, in which both paradigms of magnetic frustration described above come into play. The crystal structure of this material displays perfect triangular lattice planes of edgesharing RuO_{6} octahedra, separated by Na ions^{12,13,14} (illustrated in Fig. 1). The space group of NaRuO_{2} is R\(\overline{3}{\rm{m}}\). The same structural arrangement is found in a family of rareearth chalcogenides which have been recently investigated as possible spin liquid candidates^{15}: NaYbO_{2}^{16}, NaYbS_{2}^{17,18}, and NaYbSe_{2}^{19}. However, at variance with the latter, in NaRuO_{2} the rareearth ion is replaced by ruthenium, which belongs to the dblock of the periodic table. In analogy to the intensively studied honeycomb compound αRuCl_{3}^{8,9,10}, the strong SOC of ruthenium, combined with the geometry of edgesharing 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 interaction^{7}. Resistivity measurements identified NaRuO_{2} to be indeed insulating, with a small magnetization upon application of an external magnetic field, and a paramagnetic Curie temperature dependence of the magnetic susceptiblity^{13}. These signatures point toward the possibility of a QSL ground state, making a microscopically motivated magnetic model for NaRuO_{2} not only intriguing but also necessary.
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^{∘} order^{22}, and possibly a spin nematic state around the antiferromagnetic Kitaev point^{23,24,25}. Furthermore, an extended spin model featuring the Γexchange coupling^{26}, which can favor the onset of a stripy magnetic phase, has been investigated in connection with the j_{eff} = 1/2 iridate compound Ba_{3}IrTi_{2}O_{9}^{27,28}. In this regard, a comprehensive overview of the different phases induced by bondanisotropic (nearestneighbor) 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 chargetransfer salts^{33}, are characterized by nonnegligible fourspin ring exchange interactions^{34}, which incorporate higherorder contributions in the perturbationtheory 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 nontrivial chiral orders (e.g., spinvortex crystals)^{35}, or significantly affect the lowenergy magnon spectra of collinear phases^{36}, 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 surface^{39}, or a Kalmeyer–Laughlin chiral state^{40}, has been discussed. An additional level of complication arises in magnetic systems with strong spin–orbit interactions, namely the presence of spinanisotropic ringexchange interactions, which have been scarcely investigated in the past^{41}.
In this work, we perform a thorough inspection of the magnetic properties of NaRuO_{2}, from firstprinciples calculations to microscopic spin models. We provide theoretical justification for the lowenergy description of NaRuO_{2} in terms of j_{eff} = 1/2 pseudospin degrees of freedom, highlighting the importance of different sources of interactions, such as intra and interlayer exchange couplings, and bondanisotropic bilinear and ringexchange interactions stemming from the strong SOC effects. The analysis of (classical and quantum) magnetic models indicate the existence of a robust easyplane 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 NaRuO_{2} with the help of density functional theory (DFT) calculations, as detailed in the “Methods” section. The octahedral environment of the Ru 4d^{5} 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 NaRuO_{2}, we employ constrained random phase approximation (cRPA) calculations (see “Methods” section). In the nonrelativistic 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 sixband 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 experiments^{13}. We find that the combination of magnetism and Coulomb interaction is necessary to open a realistic charge gap in NaRuO_{2}, which is further enlarged by SOC.
The partial DOS resolves the dominance of ruthenium weight around the Fermi level, such that we can proceed with lowenergy modeling of this compound based on ruthenium bands.
The edgesharing octahedral structure of NaRuO_{2} hints towards the possibility of a j_{eff} = 1/2 description of the lowenergy magnetic properties, in analogy with the intensively studied αRuCl_{3}^{8}. To check the validity of the j_{eff} = 1/2 picture for NaRuO_{2}, we calculate the nonmagnetic 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 lowenergy description in terms of j_{eff} = 1/2 pseudospins.
Magnetic model
As an appropriate magnetic model for NaRuO_{2}, 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 gtensor 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 inplane components (\({g}_{a},{g}_{{b}^{* }}\)) are larger than the outofplane one (g_{c}), as a direct consequence of the trigonal compression of the RuO_{6} 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 fourspin 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 RuO 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 caxis. For completion, in Supplementary Note 2 we translate our model to an alternative reference frame with crystallographic coordinates^{29}.
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 bonddependent terms. Considering the symmetry constraints of the R\(\overline{3}{\rm{m}}\) space group, the bilinear exchange tensor on a Z_{1}bond (as defined in Fig. 1b) follows the form
Here, J_{1} is the isotropic Heisenberg exchange, K_{1} is the Kitaev coupling, and Γ_{1} and \({\Gamma }_{1}^{{\prime} }\) the offdiagonal symmetric exchange parameters. The bilinear interactions on X_{1} and Y_{1}bonds are then related to this expression by C_{3} 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 nearestneighbor bonds with the projED method^{45} and the isotropic longerrange exchange from total energy mapping analysis (TEMA). Both methods predict a dominant intralayer FM Heisenberg J_{1} 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 nearestneighbor FM J_{1}, the shortest furtherneighbor intralayer exchange couplings are found to be nonnegligible and of similar magnitude, with J_{2} being anti and J_{3} 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.
Within the projED method, we obtain the bonddependent anisotropic couplings K_{1}, Γ_{1}, and \({\Gamma }_{1}^{{\prime} }\) at nearestneighbors, 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 K_{1} term and a dominant positive Γ_{1} 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 secondorder 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 Zbond, the hopping parameters are defined as the ligandassisted hopping t_{2} = t_{(xz; yz)}, as well as t_{1} = t_{(xz; xz)} = t_{(yz; yz)} and t_{3} = t_{(xy; xy)}, which stem predominantly from direct d orbital overlap. For the prime example of the honeycomb Kitaev material αRuCl_{3}, the indirect hopping t_{2} is dominant, somewhat close to the t_{2}only model in the JackeliKhaliullin mechanism, where the Kitaev interaction is FM (\({K}_{1}\propto 3{t}_{2}^{2}\))^{7}. In comparison, the direct hoppings t_{3} and t_{1} gain importance in NaRuO_{2}, where the nearest neighbor RuRu bond length is significantly smaller than in αRuCl_{3}. Indeed, for NaRuO_{2}, a nonrelativistic DFT calculation yields t_{1} = 80 meV, t_{2} = 125 meV and t_{3} = − 261 meV. As dictated by the geometry, t_{3} is negative and larger in magnitude than the positive t_{1}. The resulting antiferromagnetic Kitaev interaction in NaRuO_{2} can hence be directly related to the shorter nearestneighbor bond length of this triangular compound. The perturbation theory perspective also offers an explanation for the dominance of the offdiagonal symmetric exchange Γ_{1}, which scales approximately as Γ_{1} ∝ t_{2} (t_{1} − t_{3})^{50}. While the magnitude of K_{1} reduces with the competition between indirect and direct contributions, Γ_{1} increases proportional to the magnitude of the hoppings, leading to a magnetic model dominated by the offdiagonal symmetric exchange for NaRuO_{2}.
Despite the antiferromagnetic sign of the Kitaev interaction, the bilinear exchange Hamiltonian \({{{{\mathcal{H}}}}}_{2}\) features a ferromagnetically ordered ground state, due to significant J_{1} < 0 and Γ_{1} > 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 higherorder ring exchange interactions. Compared to honeycomb Kitaev materials, NaRuO_{2} could be predestined for such interactions, due to the marginally insulating Mott state reported in experiments^{13} and to its triangular lattice structure, where the shortest closed loops consist of four (instead of six) sites. Fourspin ring exchange without SOC effects has been discussed plentifully in the literature^{34,36,37,38,40} and takes the form
where the summation 〈ijkl〉 goes over plaquettes with sites i and k lying across a diagonal^{34}, 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 NaRuO_{2} 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 foursite terms in the Hamiltonian. In the most general form, an anisotropic fourspin exchange may be expressed as
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 C_{2} 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 Zbonds of bilinear exchange, it is convenient to define X, Y, and Zplaquettes, as shown in Fig. 1b. The three plaquettes are related by C_{3} rotations around the outofplane 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 ringexchange 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 ringexchange tensor \({\mathbb{K}}\) from first principles, we employ the projED method, which has been used previously to determine ringexchange couplings for organic triangular lattice compounds^{35,52}. Results on a Zplaquette 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 abinitio result. Overall, the foursite ring exchange contribution in NaRuO_{2} does not seem to be obviously negligible, with a strength of roughly 5–10% of the nearestneighbor 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 NaRuO_{2}. For instance, among the diagonal components characterizing the Zplaquette, the \({{\mathbb{K}}}_{1234}^{zzzz}\) term strongly differs from \({{\mathbb{K}}}_{1234}^{xxxx}={{\mathbb{K}}}_{1234}^{yyyy}\).
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 tensornorm 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 1norm 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 HeisenbergJ as in Eq. (1). Dissecting the ringexchange 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 NaRuO_{2} 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 interlayer couplings is justified by their small estimated magnitude compared to intralayer 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 outofplane 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 twodimensional 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 ringexchange 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 spinwave 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 Ru^{3+} is taken into account for the calculation of spectral intensities^{55}, such that the magnetic spectra can be directly compared to neutron scattering experiments.
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 nearestneighbor bonds^{29}. At the linear spin wave theory level we then expect the appearance of a gapless pseudoGoldstone 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 αRuCl_{3}^{58}, where they have been traced back in a spinwave description to originate from significant anharmonic effects due to Γ_{1} exchange^{46}. However, despite the dominant offdiagonal Γ_{1} exchange in the case of NaRuO_{2} (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 pseudoGoldstone modes remain at the orderingwave vector Q = 0 (Γpoint), such that a potential decay of singlemagnons into a twomagnon continuum via Γ_{1}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 powderaveraged INS spectrum that might be relevant for direct comparison of the predicted FM state to experiments^{13}. Here, the effect of interplane couplings is included, in order to obtain meaningful integration over outofplane momenta. The results from LSWT are shown in Fig. 3c and feature the gapless pseudoGoldstone mode at the smallest momenta, and a less intense gapless mode around 1.2 Å^{−1} arising from the interlayer FM stacking. Indeed, such a lowenergy mode and dominant FM fluctuations (i.e., small ∣q∣excitations) are also suggested by recent INS powder data at T = 0.25 K on NaRuO_{2}^{13}.
Discussion
In this work, we investigated the magnetic properties of NaRuO_{2}, a layered system of cornersharing RuO_{6} 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 firstprinciple methods, TEMA and projED, we derived a j_{eff} = 1/2 pseudospin Hamiltonian for NaRuO_{2}, which displays a sizable antiferromagnetic Kitaev coupling. This is a direct consequence of the comparatively smaller nearestneighbor Ru–Ru bond length in NaRuO_{2}, leading to a dominance of direct hopping mechanisms in contrast to other spin1/2 Kitaev materials to date. The strongest interactions of the model are however a symmetric Γ_{1} exchange and an FM J_{1} 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 longerrange intra and interplane exchange interactions are taken into account.
In a recent work^{59}, the nearest nearestneighbor spin Hamiltonian for NaRuO_{2} has been calculated from quantum chemistry methods. While the signs of the nearestneighbor couplings match with the ones of our model, the Γ_{1} term of ref. ^{59} is smaller than the J_{1} exchange strength, contrary to our results. The authors explore the possibility of destabilizing the FM order by an antiferromagnetic thirdneighbors exchange, which is in contrast with our firstprinciple prediction of an FM J_{3} coupling.
The parameters of the magnetic Hamiltonian, as obtained by firstprinciples calculations with the pristine structure of NaRuO_{2}, 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 longrange FM order and suggest the possibility of a marginally insulating Mott state in NaRuO_{2}^{13}, we explored the effects of higherorder spin interactions, computing fourspin ring exchange couplings from the projED method. However, although the latter turn out to be of a nonnegligible 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 NaRuO_{2}. Comparison of the powderaverage neutron scattering intensity with experimental observations shows a similar weight distribution which may signal the presence of underlying ferromagnetism in the system, although no longrange 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
Firstprinciples 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 18.0057 and the Generalized Gradient Approximation (GGA)^{61} as the exchangecorrelation functional. The correlation for the strongly localized Ru 4d electrons are corrected via the GGA + U approximation using the “atomic limit” implementation^{62}. All calculations are carried out on a 12 × 12 × 12 kgrid in the primitive unit cell. Relativistic calculations are performed within the GGA + SOC + U functional. The estimates of the t_{1}–t_{3} hoppings discussed in the “Magnetic model” section have been obtained within a nonrelativistic GGA calculation.
The electronic properties (band structure, density of states (DOS)) have been crosschecked with the linearized augmented planewave basis set as implemented in Wien2k^{63} 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 gtensor from the first principles. For this calculation, we consider a [RuO_{6}]^{9−} molecule within the quantum chemistry ORCA 5.03 package^{66} with the functional TPSSh, basis set def2TZVP, and complete active space selfconsistent field method for the d orbitals CASSCF(5,5). A conductorlike polarizable continuum model (CPCM)^{67} is employed with a Gaussian charge scheme, a van der Waalstype cavity, and an infinite dielectric constant.
Constrained random phase approximation
Based on the electronic structure obtained with the Wien2k package v21.1^{63} we estimate the electronic twoparticle interaction terms in NaRuO_{2} with the constrained randomphase approximation (cRPA)^{68,69}, as implemented in the FHIgap code^{70}. The integration of the Brillouin zone is done on an 8 × 8 × 8 grid. The static lowenergy 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_{k}^{65} as follows:
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 NaRuO_{2}, 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.
DFTbased derivation of the magnetic model
We extract the dominant magnetic Heisenberg couplings via the TEMA^{71,72,73}, which is a twostep process. First, we calculate the total energies within DFT (GGA + U) of different magnetic configurations of chosen supercells of NaRuO_{2}. In the second step, we fit the DFT energy of the different magnetic configurations to an effective Heisenberg spin1/2 Hamiltonian using the method of least squares. The first step is performed in the VASP 5.3 framework^{74} using spinpolarized DFT + U, where we apply the Dudarev scheme^{75}, 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 × 3Γcentered kgrid with an energy cutoff of 540 eV for the planewave 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 socalled projED technique^{45}. 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 FPLO^{60} package 18.0057. \({{{{\mathcal{H}}}}}_{{{{\rm{U}}}}}\) contains the electronic twoparticle Coulomb interaction^{45}. In a second step, the electronic Hamiltonian is solved by ED on a twosite fiveorbital 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 nearestneighbor couplings, while for longerrange 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 longerrange couplings, cannot be accounted for due to computational limitations.
We also employ the projED method to extract the fourspin ring exchange Hamiltonian \({{{{\mathcal{H}}}}}_{4}^{{{{\rm{tot}}}}}\). Due to computational limitations, the parameters are extracted by diagonalizing a foursite threeorbital 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 spinorbit coupled system like NaRuO_{2}. 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
In other words, we antialign 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 interlayer couplings, we used a threedimensional cluster of N = 6 × 6 × 6 = 216 sites.
Exact diagonalization
We perform ED of the j_{eff} = 1/2 model on twodimensional clusters of up to N = 27 sites. The INS intensity at momentum q and energy ω is given by
where f(q) is the atomic form factor of Ru^{3+}. To compute it, we employ the continued fraction method^{76}. To access a higher number of qpoints, 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. qpoints 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 spinwave theory
LSWT calculations are performed with the SpinW 3.0 library^{77}. The INS intensity is computed by taking the powder average of Eq. (7).
Data availability
The datasets generated during the current study are available under the following publicly accessible repository https://gude.unifrankfurt.de/handle/gude/232 or alternatively from the corresponding authors upon reasonable request.
Code availability
The custom codes implementing the calculations of this study are available from the authors upon reasonable request.
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Acknowledgements
We thank V. Krewald and I.I. Mazin for valuable advice regarding the ab initio calculations. We thank also D. Ceresoli, M. Imada, J. Kuneš, and P.A. Maksimov for fruitful comments and discussions. R.V., A.R., K.R., D.A.S.K., and F.F. gratefully acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for funding through Project No. 411289067 (VA117/151), TRR 288—422213477 (project A05) and CRC 1487—443703006 (project A01). L.B. was supported by the DOE, Office of Science, Basic Energy Sciences under Award No. DEFG0208ER46524. S.D.W. acknowledges support by DOE, Office of Science, Basic Energy Sciences under Award No. DESC0017752.
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R.V., S.D.W., and L.B. conceived the project. Density functional theory calculations were performed by A.R., K.R., cRPA calculations by S.B., projED calculations by K.R., and calculations on magnetic models by D.A.S.K. and F.F. All authors contributed to the writing of the paper.
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Razpopov, A., Kaib, D.A.S., Backes, S. et al. A j_{eff} = 1/2 Kitaev material on the triangular lattice: the case of NaRuO_{2}. npj Quantum Mater. 8, 36 (2023). https://doi.org/10.1038/s41535023005676
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DOI: https://doi.org/10.1038/s41535023005676
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