## Abstract

Kitaev exchange, a new paradigm in quantum magnetism research, occurs for 90° metal-ligand-metal links, \({t}_{2g}^{5}\) transition ions, and sizable spin-orbit coupling. It is being studied in honeycomb compounds but also on triangular lattices. While for the former it is known by now that the Kitaev intersite couplings are ferromagnetic, for the latter the situation is unclear. Here we pin down the exchange mechanisms and determine the effective coupling constants in the \({t}_{2g}^{5}\) triangular-lattice material NaRuO_{2}, recently found to host a quantum spin liquid ground state. We show that, compared to honeycomb compounds, the characteristic triangular-lattice cation surroundings dramatically affect exchange paths and effective coupling parameters, changing the Kitaev interactions to antiferromagnetic. Quantum chemical analysis combined with subsequent effective spin model simulations provide perspective onto the nature of the experimentally observed quantum spin liquid—it seemingly implies fairly large antiferromagnetic second-neighbor isotropic exchange, and the atypical proximity to ferromagnetic order is related to ferromagnetic nearest-neighbor Heisenberg coupling.

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## Introduction

Exchange through 90° metal-ligand-metal bonds represents one of the limiting cases in (super)exchange theory^{1,2}. In the simplest situation of half-filled *d*-metal orbitals, this geometry is associated with Heisenberg ferromagnetism. Away from half-filling and for degenerate orbitals, however, very intricate physics may arise, as pointed out by Jackeli and Khaliullin for \({t}_{2g}^{5}\) magnetic centers with sizable spin-orbit coupling: highly anisotropic effective interactions involving only (pseudo)spin components normal to the M_{2}L_{2} square plaquette formed by two transition-metal (TM) ions and the two bridging ligands^{3,4}. In crystals of NaCl type and derivative rhombohedral structures which imply three different possible orientations of the M_{2}L_{2} plaquettes (see Fig. 1), this translates into directional dependence of the nearest-neighbor exchange: on differently oriented adjacent M_{2}L_{2} units—i.e., normal to either *x*, *y*, or *z*—different components of the magnetic moments interact, either \({\tilde{S}}_{i}^{x}\) and \({\tilde{S}}_{j}^{x}\) (red arrows in Fig. 1b for *x*-type, normal to the *x*-axis plaquettes), \({\tilde{S}}_{i}^{y}\) and \({\tilde{S}}_{k}^{y}\) (blue arrows, *y*-type plaquettes), or \({\tilde{S}}_{i}^{z}\) and \({\tilde{S}}_{l}^{z}\) (green arrows, *z* type). The (super)exchange model of Jackeli and Khaliullin^{3} thus formalizes Kitaev’s effective Hamiltonian of bond-dependent anisotropic magnetic couplings^{5} proposed initially more like a heuristic device. It launched a whole new subfield in the research area of quantum magnetic materials, that of 5*d*^{5} and 4*d*^{5} honeycomb magnets^{4}, with subsequent extension to 3*d*^{7}\({t}_{2g}^{5}{e}_{g}^{2}\) hexagonal networks of edge-sharing ML_{6} octahedra.

Here we explore the nature of nearest-neighbor couplings in a 4*d*^{5} triangular-lattice magnet, NaRuO_{2}, and evidence the presence of sizable bond-dependent Kitaev interactions. Interestingly, those are antiferromagnetic, different from the case of honeycomb 4*d*^{5} and 5*d*^{5} magnetic lattices. Preliminary numerical tests highlight the important role of cation species around the ligands mediating (super)exchange: the sign change is related to electrostatics having to do with adjacent TM ions that in the honeycomb setting are missing. Sizable antiferromagnetic off-diagonal intersite couplings are also predicted, along with a somewhat larger ferromagnetic Heisenberg exchange. The latter seems to be consistent qualitatively with features seen in experiment: incipient ferromagnetism within a quantum-disordered ground state^{6,7}.

## Results

### On-site multiplet structure and intersite couplings in NaRuO_{2}

NaRuO_{2} is rather unique, a realization of *j*_{eff} ≈ 1/2 \({t}_{2g}^{5}\) moments on a triangular network of edge-sharing RuO_{6} octahedra (see Fig. 2). The basic Ru^{3+} 4*d*^{5} multiplet structure in the specific delafossite crystalline setting of NaRuO_{2} is illustrated in Table 1, as obtained by quantum chemical^{8} complete-active-space self-consistent-field (CASSCF)^{8,9} and post-CASSCF multireference configuration-interaction (MRCI)^{8,10} embedded-cluster calculations (see Methods for computational details and Supplementary Note 1 for basis sets). To separately visualize the effects of crystal-field (CF) splittings and spin-orbit coupling (SOC), both scalar relativistic (CASSCF, MRCI) and MRCI+SOC^{11} results are listed. The *t*_{2g}-*e*_{g} splitting is larger than in e.g., RuCl_{3}^{12}, due to larger ligand effective charges and shorter TM-ligand bonds in NaRuO_{2}. It is seen that the MRCI correlation treatment brings significant corrections to some of the CASSCF relative energies, in particular for the ^{6}*A*_{1g} term. Also important is the SOC (see the last two columns in Table 1). However, the effects of trigonal fields are visible too: those remove the degeneracy of the ^{2}*T*_{2g} CF states (*p*-orbital-like *l*_{eff} = 1/2^{13} and *s* = 1/2 quantum numbers in ideal cubic environment) and speak for significant deviations from textbook^{3,13}*j*_{eff} = 1/2 spin-orbit moments.

Even with sizable trigonal fields, existing estimates of several meV for the Kitaev interactions in honeycomb-lattice 4*d* \({t}_{2g}^{5}\) compounds such as RuCl_{3} (see e.g.^{12,14,15,16,17,18,19} and discussion in ref. ^{20}) and Li_{2}RhO_{3}^{21} provide strong motivation for detailed ab initio investigation of the effective couplings in NaRuO_{2}. To this end, further quantum chemical computations were performed for a block of two adjacent edge-sharing RuO_{6} octahedra, following the procedure described in refs. ^{12,22} (see also Methods). Mapping the quantum chemical data onto the relevant effective spin model for *C*_{2h} symmetry of the [Ru_{2}O_{10}] magnetic unit,

by spin-orbit MRCI, we arrive to Kitaev, Heisenberg, and off-diagonal Γ coupling constants *K* = 2.0, *J* = −5.2, Γ ≡ Γ_{xy} = 3.6, \({{{\Gamma }}}^{{\prime} }\,\equiv \,{{{\Gamma }}}_{yz}\,=\,{{{\Gamma }}}_{zx}\,=\,1.1\) (meV). Interestingly, both *K* and Γ are antiferromagnetic, with Γ > *K*, but the largest interaction parameter is the Heisenberg *J*. The latter being ferromagnetic, it brings us away from the antiferromagnetic ground states (zigzag antiferromagnetic order, in particular) experimentally found in *j* ≈ 1/2 honeycomb systems, as depicted for instance in the phase diagrams computed for *J*-*K*-Γ triangular-lattice models in ref. ^{23}.

According to ref. ^{23}, quantum spin liquid (QSL) ground states can only be realized for relatively large, antiferromagnetic Γ. Our result for the strength of Γ represents the largest ab initio quantum chemical estimate so far in a real material setting. Yet, a nearest-neighbor *J* that is even larger in magnitude and ferromagnetic (see the values listed above and at the top of Table 2) implies that only longer-range Heisenberg couplings (analyzed in isotropic context in e.g. refs. ^{24,25,26,27}) and/or higher-order interactions such as ring exchange^{28,29,30} can pull the system to the QSL regime evidenced experimentally^{6}.

Signatures of a nearby ferromagnetic state in NaRuO_{2} appear via the enhanced Van Vleck term in the magnetic susceptibility^{6}, which approaches values observed in nearly magnetic metals like Pd. Furthermore, despite possessing a charge gap and insulating ground state, clear quasiparticle excitations generate a Sommerfeld-like term in the low-temperature heat capacity^{6}. This suggests the presence of strong spin fluctuations associated with a nearby magnetic state. These seemingly gapless spin fluctuations are robust, and are directly resolved in neutron scattering as diffuse continuum-like modes about the nuclear zone center and near **Q** = 0^{6}. Initial attempts at scaling S(Q, E) suggest the proximity of a nearby ferromagnetic quantum critical point, and small, nonmagnetic chemical perturbations to the network of Ru^{3+} ions induce glass-like freezing of the magnetic moments^{7}.

### What makes *K* antiferromagnetic

Finding an antiferromagnetic Kitaev coupling in NaRuO_{2}, opposite to the ferromagnetic *K* generally found in honeycomb Kitaev-Heisenberg magnets (see^{12,21,31} for quantum chemical results, refs. ^{14,15,16,17} for estimates based on effective-model (super)exchange theory, and refs. ^{18,19} for values obtained by fitting experimental spectra) is intriguing. To shed light on this aspect, we performed the following numerical experiment: in a new set of quantum chemical computations, the two magnetic-plane cations in the immediate neighborhood of the bridging ligands, forming a line perpendicular to the link of nearest-neighbor Ru sites (see Fig. 2c), were removed and their charge redistributed within the point-charge array modeling the extended solid-state surroundings. Remarkably, without those nearby positive ions, three of the effective magnetic couplings change sign (see Table 2). This suggests that strong orbital polarization effects at the ligand sites, induced by the extra nearby positive charge on the triangular lattice (+3 effective nearby charges in NaRuO_{2} versus +1 in honeycomb ‘213’ compounds such as Na_{2}IrO_{3} and Li_{2}RhO_{3} or 0 in RuCl_{3}) have dramatic consequences on hopping matrix elements and superexchange processes. Similar additional tests for RuCl_{3}, where +3 ions were placed in the centers of two edge-sharing Ru_{6} hexagonal rings, confirm the trend: the extra positive charge in the neighborhood of the Ru_{2}O_{2} ‘magnetic’ plaquette (i.e., [Ru_{2}O_{10}] cluster of edge-sharing octahedra) invert the sign of the Kitaev coupling constant, from ferromagnetic in actual RuCl_{3}^{12} to antiferromagnetic in the presence of positive charge in the centers of the two hexagons sharing the magnetic Ru-Ru link (see lowest lines in Table 2). Strong polarization effects of this type were earlier pointed out in the case of the Kitaev-Heisenberg honeycomb iridate H_{3}LiIr_{2}O_{6}^{31}, produced in that case by cations residing between the magnetic layers.

### Ru_{2}O_{2}-unit correlations. Extended magnetic lattice

Insights into the interplay between spin-orbit couplings and electron correlation effects on a plaquette of two nearest-neighbor Ru ions and two bridging ligands can be obtained by analyzing computational data obtained at lower levels of approximation, i.e., spin-orbit calculations including only the \({t}_{2g}^{5}\)–\({t}_{2g}^{5}\) electron configuration and CASSCF+SOC calculations that also account for excited-state \({t}_{2g}^{4}\)–\({t}_{2g}^{6}\) configurations since all possible occupations are considered within the six-orbital (three *t*_{2g} orbitals per Ru site) active space in the latter case.

As illustrated in Table 3, anisotropic effective intersite interactions comparable in size with the isotropic Heisenberg *J* are already found at the single-configuration (SC) level, i.e., when accounting for just \({t}_{2g}^{5}\)–\({t}_{2g}^{5}\) Coulomb exchange (sometimes referred to as direct exchange^{2}). Large anisotropic Coulomb exchange as found in the SC calculation represents physics not addressed so far in the literature—Kitaev magnetism is presently exclusively explained through TM-TM kinetic exchange and TM-L-TM superexchange.

CASSCF, through the inclusion of intersite *t*_{2g}→ *t*_{2g} excitations (i.e., TM-TM kinetic exchange), brings sizable corrections to *K*, *J*, and Γ in particular. By considering next, in the spin-orbit MRCI treatment, all possible single and double excitations out of the Ru *t*_{2g} and bridging-ligand orbitals (Ru(*t*_{2g})→Ru(*e*_{g}) and O(2*p*)→Ru(4*d*) transitions, among others), the most significant post-CASSCF corrections are found to occur for *K* and Γ. This suggests that more sophisticated calculations, e.g., MRCI+SOC based on larger active spaces in the prior CASSCF, would bring significant additional corrections to *K* and Γ mostly, i.e., would only enhance antiferromagnetic fluctuations on the extended lattice.

An interesting observation is that, adding the Coulomb-exchange contributions computed at SC level for *K* and *J* (−1.0 and −0.7 meV, respectively, first line in Table 3) to the *K* and *J* estimates derived from effective-model calculations where Coulomb exchange is neglected (*K* = 2.9 and *J* = −4.2 meV)^{32} brings the latter in rather good agreement with the MRCI values (2.0 and −5.2 meV, respectively), although this is not the case for Γ and \({{{\Gamma }}}^{{\prime} }\).

To obtain first impressions on the role of longer-range Heisenberg interactions, second-neighbor (*J*_{2}) and third-neighbor (*J*_{3}), in particular, we performed density-matrix renormalization group (DMRG) calculations^{33,34} for a *J*-*K*-Γ-\({{{\Gamma }}}^{{\prime} }\)-*J*_{2}-*J*_{3} model on fragments of 19, 27, and 37 sites of the triangular lattice. In order to prevent artifacts, e.g., artificial stabilization (or destabilization) of particular magnetic states, we applied open boundary conditions. The validity of this material model is discussed in Supplementary Note 2 and Supplementary Note 3. Setting *J*, *K*, Γ, and \({{{\Gamma }}}^{{\prime} }\) to –5.5, 4, 4, and 1.5 meV, respectively, we obtain a very rich phase diagram, see Fig. 3. Notably, a QSL phase is found for antiferromagnetic *J*_{2} values and ferromagnetic *J*_{3}. The corresponding structure factor has no Bragg peaks but exhibits a characteristic pattern. It indicates that the QSL emerges from the competition of adjacent ordered states. The most remarkable spot is the region where the QSL neighbors ferromagnetic order and a spiral phase, namely, around *J*_{2} = 5, *J*_{3} = −2—as shown in Fig. 3, at *J*_{2} = 5, *J*_{3} = −2 the structure factor implies a broad spectral feature for momenta near ∣**Q**∣ = 0, seemingly consistent with experimental observations^{6}.

## Discussion

While the way nearby cations are structurally arranged can strongly affect on-site electronic-structure features such as subshell level splittings^{22,35,36}, single-ion anisotropies^{35}, and *g* factors^{22,36}, detailed ab initio investigations of the effect on intersite magnetic couplings are less numerous. Here we show that, compared to honeycomb compounds, the characteristic triangular-lattice cation surroundings dramatically affect superexchange paths and effective coupling parameters. In particular, with respect to honeycomb *j* ≈ 1/2 systems, the Kitaev interaction constant changes its sign in triangular-lattice NaRuO_{2}. By providing insight into the signs and strengths of the nearest-neighbor magnetic interactions in this material, our work sets the frame within which the role of longer-range effective spin couplings should be addressed. Interestingly, while giving rise to antiferromagnetic order on honeycomb Kitaev-Heisenberg lattices, the latter appear to be decisive in realizing quantum disorder^{6} in NaRuO_{2}. Last but not least, we establish the role of anisotropic Coulomb exchange (in trianguar-lattice setting), a mechanism not addressed so far in Kitaev-Heisenberg magnetism.

## Methods

### Ru-site multiplet structure

All quantum chemical computations were carried out using the molpro suite of programs^{37}. Crystallographic data as reported by Ortiz et al.^{6} were utilized. For each type of embedded cluster, the crystalline environment was modeled as a large array of point charges which reproduces the crystalline Madelung field within the cluster volume; we employed the ewald program^{38,39} to generate the point-charge embeddings.

To clarify the Ru-site multiplet structure, a cluster consisting of one ‘central’ RuO_{6} octahedron, the six in-plane adjacent octahedra, and 12 nearby Na cations was designed. The quantum chemical study was initiated as CASSCF calculations^{8,9} with an active orbital space containing the five 4*d* orbitals of the central Ru ion. Post-CASSCF correlation computations were performed at the MRCI level, with single and double excitations^{8,10} out of the Ru 4*d* and O 2*p* orbitals of the central RuO_{6} octahedron. SOCs were accounted for following the procedure described in ref. ^{11}.

### Intersite exchange in NaRuO_{2}

Clusters with two edge-sharing RuO_{6} octahedra in the central region were considered in order to derive the intersite effective magnetic couplings. The eight in-plane octahedra directly linked to the two-octahedra central unit and 22 nearby Na ions were also explicitly included in the quantum chemical computations but using much more compact basis sets.

CASSCF computations were carried out with six (Ru *t*_{2g}) valence orbitals and ten electrons as active (abbreviated as (10e,6o) active space); the *t*_{2g} orbitals of the adjacent TM ions were part of the inactive orbital space. In the subsequent MRCI correlation treatment, single and double excitations out of the central-unit Ru *t*_{2g} and bridging-O 2*p* levels were considered. We used the Pipek-Mezey methodology^{40} to obtain localized central-unit orbitals.

The CASSCF optimization was performed for the lowest nine singlet and lowest nine triplet states associated with the (10e,6o) setting. Those were the states for which SOCs were further accounted for^{11}, either at SC, CASSCF, or MRCI level, which finally yields a number of 36 spin-orbit states in each case. The SC label in Table 3 in the main text stands for a CASCI in which intersite excitations are not considered. This is also referred to as occupation-restricted multiple active space (ORMAS) scheme^{41}.

The lowest four spin-orbit eigenstates from the molpro output (eigenvalues lower by ~200 meV or more than the eigenvalues of higher-lying excited states, as illustrated for example in Table 1) were mapped onto the eigenvectors of the effective spin Hamiltonian (1), following the procedure described in refs. ^{12,22}: those four expectation values and the matrix elements of the Zeeman Hamiltonian in the basis of the four lowest-energy spin-orbit eigenvectors are put in direct correspondence with the respective eigenvalues and matrix elements of (1). Having two of the states in the same irreducible representation of the *C*_{2h} point group^{21}, such one-to-one mapping translates into two possible sets of effective magnetic couplings. The relevant array is chosen as the one whose *g* factors fit the *g* factors obtained for a single RuO_{6}\({t}_{2g}^{5}\) octahedron.

We used the standard coordinate frame usually employed in the literature, different from the rotated frame employed in earlier quantum chemical studies^{12,21,31} that affects the sign of Γ. This is the reason the sign of Γ for RuCl_{3} in Table 2 in the main text is different from the sign in ref. ^{12} (see also footnote [48] in ref. ^{20}).

## Data availability

Quantum chemical data (atomic coordinates, point-charge data, molpro output) have been deposited in the RADAR database under the https://doi.org/10.22000/1736.

## Code availability

Relevant codes are available from the corresponding author upon reasonable request.

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## Acknowledgements

P. B. and L. H. acknowledge financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG), Project ID 441216021, and technical assistance from U. Nitzsche. S. N. acknowledges financial support via project A05 of the Collaborative Research Center 1143 of the DFG (Project ID 247310070). S. D. W. acknowledges support via DOE, Office of Science, Basic Energy Sciences under Award DE-SC0017752. P. B. thanks M. S. Eldeeb for discussions. We also acknowledge instructive discussions with I. Rousochatzakis, U. K. Rößler, and G. Khaliullin.

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P.B. performed the ab initio quantum chemical computations, with assistance from L.H. and N.A.B. S.N. carried out the model-Hamiltonian calculations for the extended lattice and subsequent analysis. S.D.W. and L.H. initiated this study.

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Bhattacharyya, P., Bogdanov, N.A., Nishimoto, S. *et al.* NaRuO_{2}: Kitaev-Heisenberg exchange in triangular-lattice setting.
*npj Quantum Mater.* **8**, 52 (2023). https://doi.org/10.1038/s41535-023-00582-7

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DOI: https://doi.org/10.1038/s41535-023-00582-7