NaRuO₂: Kitaev-Heisenberg exchange in triangular-lattice setting

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 NaRuO₂, 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.

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 M₂L₂ 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 M₂L₂ plaquettes (see Fig. 1), this translates into directional dependence of the nearest-neighbor exchange: on differently oriented adjacent M₂L₂ 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 Khaliullin³ thus formalizes Kitaev’s effective Hamiltonian of bond-dependent anisotropic magnetic couplings⁵ proposed initially more like a heuristic device. It launched a whole new subfield in the research area of quantum magnetic materials, that of 5d⁵ and 4d⁵ honeycomb magnets⁴, with subsequent extension to 3d⁷${t}_{2g}^{5}{e}_{g}^{2}$ hexagonal networks of edge-sharing ML₆ octahedra.

Fig. 1: Orthogonal M₂L₂ plaquettes in solids.

a Rocksalt lattice, the prototype for 90° M-L-M exchange^1,2. b If two different cation varieties—alkaline (A, yellow) and magnetic TM (B) ions—form successive layers normal to the [111] axis, a rhombohedral ABL₂ structure results, with a triangular network of edge-sharing octahedra (either AL₆ or BL₆) in each layer. On each B₂L₂ plaquette (atoms not explicitly shown), the Kitaev interaction couples only (pseudo)spin components perpendicular to the respective plaquette. Red, blue, and green indicate x, y, and z projections, respectively; only two spin components are shown at each magnetic site.

Here we explore the nature of nearest-neighbor couplings in a 4d⁵ triangular-lattice magnet, NaRuO₂, and evidence the presence of sizable bond-dependent Kitaev interactions. Interestingly, those are antiferromagnetic, different from the case of honeycomb 4d⁵ and 5d⁵ 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 NaRuO₂

NaRuO₂ is rather unique, a realization of j_eff ≈ 1/2 ${t}_{2g}^{5}$ moments on a triangular network of edge-sharing RuO₆ octahedra (see Fig. 2). The basic Ru³⁺ 4d⁵ multiplet structure in the specific delafossite crystalline setting of NaRuO₂ is illustrated in Table 1, as obtained by quantum chemical⁸ 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+SOC¹¹ results are listed. The t_2g-e_g splitting is larger than in e.g., RuCl₃¹², due to larger ligand effective charges and shorter TM-ligand bonds in NaRuO₂. It is seen that the MRCI correlation treatment brings significant corrections to some of the CASSCF relative energies, in particular for the ⁶A_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 ²T_2g CF states (p-orbital-like l_eff = 1/2¹³ and s = 1/2 quantum numbers in ideal cubic environment) and speak for significant deviations from textbook^3,13j_eff = 1/2 spin-orbit moments.

Fig. 2: NaRuO₂ crystal structure.

a Sketch of successive layers of edge-sharing RuO₆ and NaO₆ octahedra. Gray, red, and yellow spheres represent Ru, O, and Na ions, respectively. b The triangular magnetic network of RuO₆ units. c To understand how the nearby surroundings affect magnetic couplings, Ru neighbors in the median plane (sites 3, 4) of a given Ru-Ru magnetic link (sites 1, 2) were removed in one set of quantum chemical computations.

Table 1 Ru³⁺4d⁵ multiplet structure, with all five 4d orbitals active in CASSCF.

Even with sizable trigonal fields, existing estimates of several meV for the Kitaev interactions in honeycomb-lattice 4d ${t}_{2g}^{5}$ compounds such as RuCl₃ (see e.g.^{12,14,15,16,17,18,19} and discussion in ref. ²⁰) and Li₂RhO₃²¹ provide strong motivation for detailed ab initio investigation of the effective couplings in NaRuO₂. To this end, further quantum chemical computations were performed for a block of two adjacent edge-sharing RuO₆ 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 [Ru₂O₁₀] magnetic unit,

$${{{{\mathcal{H}}}}}_{ij}^{(\gamma )}=J{\tilde{{{{\bf{S}}}}}}_{i}\cdot {\tilde{{{{\bf{S}}}}}}_{j}+K{\tilde{S}}_{i}^{\gamma }{\tilde{S}}_{j}^{\gamma }+\mathop{\sum}\limits_{\alpha \ne \beta }{{{\Gamma }}}_{\alpha \beta }({\tilde{S}}_{i}^{\alpha }{\tilde{S}}_{j}^{\beta }+{\tilde{S}}_{i}^{\beta }{\tilde{S}}_{j}^{\alpha }),$$

(1)

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 experimentally⁶.

Table 2 Nearest-neighbor effective magnetic couplings (MRCI+SOC) for experimentally determined crystal structures of NaRuO₂/RuCl₃ and for adjusted configurations with modified median plane (MMP) cation distribution (see main text and Fig. 2c for details).

Signatures of a nearby ferromagnetic state in NaRuO₂ appear via the enhanced Van Vleck term in the magnetic susceptibility⁶, 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 capacity⁶. 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 = 0⁶. 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 Ru³⁺ ions induce glass-like freezing of the magnetic moments⁷.

What makes K antiferromagnetic

Finding an antiferromagnetic Kitaev coupling in NaRuO₂, 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 NaRuO₂ versus +1 in honeycomb ‘213’ compounds such as Na₂IrO₃ and Li₂RhO₃ or 0 in RuCl₃) have dramatic consequences on hopping matrix elements and superexchange processes. Similar additional tests for RuCl₃, where +3 ions were placed in the centers of two edge-sharing Ru₆ hexagonal rings, confirm the trend: the extra positive charge in the neighborhood of the Ru₂O₂ ‘magnetic’ plaquette (i.e., [Ru₂O₁₀] cluster of edge-sharing octahedra) invert the sign of the Kitaev coupling constant, from ferromagnetic in actual RuCl₃¹² 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 H₃LiIr₂O₆³¹, produced in that case by cations residing between the magnetic layers.

Ru₂O₂-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 exchange²). 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.

Table 3 Nearest-neighbor magnetic couplings (meV) at different levels of theory: SC (only the ${t}_{2g}^{5}$–${t}_{2g}^{5}$ configuration considered), CASSCF (also ${t}_{2g}^{4}$–${t}_{2g}^{6}$ states included), and MRCI (single and double excitations out of the Ru 4d t_2g and bridging-O 2p levels on top of CASSCF).

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(2p)→Ru(4d) 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 (J₂) and third-neighbor (J₃), in particular, we performed density-matrix renormalization group (DMRG) calculations^33,34 for a J-K-Γ-${{{\Gamma }}}^{{\prime} }$-J₂-J₃ 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 J₂ values and ferromagnetic J₃. 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 J₂ = 5, J₃ = −2—as shown in Fig. 3, at J₂ = 5, J₃ = −2 the structure factor implies a broad spectral feature for momenta near ∣Q∣ = 0, seemingly consistent with experimental observations⁶.

Fig. 3: Single-layer magnetism.

Ground-state phase diagram obtained by DMRG computations for the J-K-Γ-${{{\Gamma }}}^{{\prime} }$-J₂-J₃ model (right) and typical static spin structure factors for each phase (left). Using the quantum chemical analysis as a guide (see data in Table 3 and related discussion), J, K, Γ, and ${{{\Gamma }}}^{{\prime} }$ were set to to −5.5, 4, 4, and 1.5 meV, respectively.

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 anisotropies³⁵, 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 NaRuO₂. 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 disorder⁶ in NaRuO₂. 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 programs³⁷. 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’ RuO₆ 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 4d 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 4d and O 2p orbitals of the central RuO₆ octahedron. SOCs were accounted for following the procedure described in ref. ¹¹.

Intersite exchange in NaRuO₂

Clusters with two edge-sharing RuO₆ 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 2p levels were considered. We used the Pipek-Mezey methodology⁴⁰ 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 for¹¹, 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) scheme⁴¹.

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 group²¹, 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 RuO₆${t}_{2g}^{5}$ 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 RuCl₃ in Table 2 in the main text is different from the sign in ref. ¹² (see also footnote [48] in ref. ²⁰).