Abstract
Large anisotropic exchange in 5d and 4d oxides and halides open the door to new types of magnetic ground states and excitations, inconceivable a decade ago. A prominent case is the Kitaev spin liquid, host of remarkable properties such as protection of quantum information and the emergence of Majorana fermions. Here we discuss the promise for spinliquid behavior in the 4d^{5} honeycomb halide αRuCl_{3}. From advanced electronicstructure calculations, we find that the Kitaev interaction is ferromagnetic, as in 5d^{5} iridium honeycomb oxides, and indeed defines the largest superexchange energy scale. A ferromagnetic Kitaev coupling is also supported by a detailed analysis of the fielddependent magnetization. Using exact diagonalization and densitymatrix renormalization group techniques for extended KitaevHeisenberg spin Hamiltonians, we find indications for a transition from zigzag order to a gapped spin liquid when applying magnetic field. Our results offer a unified picture on recent magnetic and spectroscopic measurements on this material and open new perspectives on the prospect of realizing quantum spin liquids in d^{5} halides and oxides in general.
Introduction
Quantum spin liquids (SL’s) are states of matter that cannot be described by the broken symmetries associated with conventional magnetic ground states^{1}. Whereas there is a rich variety of mathematical models that exhibit SL behavior, finding materials in which a quantum SL state is realized is an intensely pursued goal in present day experimental condensedmatter physics^{2,3,4}. Of particular interest is the Kitaev Hamiltonian on the honeycomb lattice^{5}, which is a mathematically wellunderstood twodimensional model exhibiting various topological SL states. Its remarkable properties include protection of quantum information and the emergence of Majorana fermions^{5,6}.
The search to realize the Kitaev model of effectively spin1/2 particles on the honeycomb lattice was centered until recently mainly on honeycomb iridate materials^{7,8} of the type A_{2}IrO_{3}, where A is either Na or Li. In these systems though longrange magnetic order develops at low temperatures, for all known different crystallographic phases^{9,10,11,12,13}. The SL regime is most likely preempted in the iridates by the presence of significant residual Heisenbergtype J couplings, by longerrange spin interactions, or by having crystallographically distinct IrIr bonds with dominant J’s on some of those, if not a combination of these factors^{14,15,16,17}.
Also of interest in this context is ruthenium trichloride, RuCl_{3}, in its honeycomb (α) crystalline phase^{18,19,20,21,22,23,24,25,26}. Very recent Raman and neutron scattering measurements suggest that the 4d^{5} halide honeycomb system is closer to the Kitaev limit^{22,23}. But also this material orders antiferromagnetically at low temperatures, as the 5d^{5} iridium oxides do, and precisely how close to the idealized Kitaev model αRuCl_{3} is, remains a question to be clarified.
Here we present results of combined quantum chemistry electronicstructure computations and exactdiagonalization (ED) calculations for extended KitaevHeisenberg spin Hamiltonians using as starting point for the ED study the magnetic couplings derived at the quantum chemistry level. Our results for the Ru^{3+} 4dshell electronic structure show sizable trigonal splitting of the 4d t_{2g} levels and therefore a spinorbit ground state that significantly deviates from the j_{eff} = 1/2 picture^{7}. The trigonally distorted environment further gives rise to strong anisotropy of the computed g factors, consistent with experimental observations^{20,27}. Calculating the magnetic interactions between two adjacent 1/2pseudospins, we find that the nearestneighbor (NN) Kitaev exchange K is ferromagnetic (FM), in any of the αRuCl_{3} crystalline structures reported so far. It is however significantly weaker than in 5d^{5} Ir oxides and even than in 4d^{5} Li_{2}RhO_{3}, which points at a rather different balance between the various superexchange processes in the halide and in the oxides.
The resulting magnetic phase diagram that we compute as function of longerrange second and thirdneighbor magnetic couplings is very rich, due to the comparable size of the various residual interactions. While a SL state does show up in this phase diagram, it arises in a setting different from Kitaev’s original SL regime, as it emerges from an interplay of Kitaev physics and geometrically frustrated magnetism. We additionally find that applying an external magnetic field whilst the system is in the longrange ordered zigzag ground state can induce a phase transition into a quantum SL. In order to make direct contact with experimental observations, we calculate by ED the fielddependent magnetization in the presence of longerrange magnetic interactions and compare that to the measurements. This comparison makes clear that the ED and experimental data can only be matched when J is small and antiferromagnetic (AF) and K significantly stronger and FM, in accordance with the results from the ab initio quantum chemistry calculations.
The magnitude of our computed K compares well with recent estimates based on neutron scattering^{23} and Raman^{22} data. However, our finding that K is FM brings into question the interpretation of the neutron scattering experiments in terms of a pure KitaevHeisenberg model with AF K but without longerrange magnetic couplings which we find to be essential for an understanding of the magnetic properties of αRuCl_{3}.
Spinorbit ground state and excitations
We start our discussion with the analysis of the Ru^{3+} 4dshell electronic structure. As in the 5d^{5} iridates, the magnetic moments in αRuCl_{3} are related to the one hole in the transitionmetal t_{2g} subshell, described by the effective L = 1 angularmomentum and S = 1/2 spin quantum numbers. Even if the spinorbit coupling (SOC) for 4d electrons is weaker than in the Ir 5d orbitals, it still splits the states into a j_{eff} = 1/2 sector, where the hole resides, and a j_{eff} = 3/2 manifold that is filled. But for noncubic environment, these j_{eff} = 1/2 and j_{eff} = 3/2 components may display some degree of admixture.
Three different crystallographic structures^{28,29,30} have been reported for αRuCl_{3}, each of those displaying finite amount of trigonal compression of the Cl_{6} octahedra. To shed light on the nature of the 1/2pseudospin in αRuCl_{3} we first discuss in this section results of ab initio manybody calculations at the completeactivespace selfconsistentfield (CASSCF) and multireference configurationinteraction (MRCI) levels of theory^{31} for embedded atomic clusters having one RuCl_{6} octahedron as reference unit.
As shown in Table 1, the degeneracy of the Ru t_{2g} levels is completely removed, with CASSCF splittings of 69 and 72 meV when using the RuCl_{3} C2/m structure determined by Cao et al.^{28}, a minimal active orbital space of only three 4d orbitals and no SOC. A “trigonal” orbital basis is used in Table 1 to express the wave functions^{27}, in contrast to the Cartesian orbital basis employed for the Rh^{4+} states in ref. 32, better suited for Li_{2}RhO_{3} due to additional distortions of the ligand cages giving rise in the rhodate to one set of longer ligandmetalligand links with an angle of nearly 180°.
The corrections brought by the MRCI treatment are tiny, smaller than in the 4d oxide^{32} Li_{2}RhO_{3} due to less metald – ligandp covalency in the halide. The smaller effective ionic charge at the ligand sites in the halide — Cl^{−} in RuCl_{3} vs O^{2−} in Li_{2}RhO_{3}, in a fully ionic picture — further makes that the transitionmetal t_{2g} − e_{g} ligandfield splitting is substantially reduced in RuCl_{3}: by MRCI calculations without SOC but with all five Ru 4d orbitals active in the reference CASSCF, we see that the lowest states are at only 1.3 (1.5) eV above the lowlying component (see Table 2). Interestingly, for the “older” P3_{1}12 crystal structure proposed in ref. 29, we find that the sextet lies even below the lowest states, see Methods. The smaller effective ligand charge might also be the cause for the smaller t_{2g}shell splittings in the halide: ≈70 meV in RuCl_{3} (see caption of Tables 1 and 2) vs ≈90 meV in Li_{2}RhO_{3}^{32} at the MRCI level, in spite of having similar degree of trigonal compression in these two materials.
With regard to the split j_{eff} = 3/2like states that we compute at 195 and 234 meV by MRCI + SOC calculations involving all three , and configurations in the spinorbit treatment (see Table 2), clear excitations have been measured in that energy range in Raman scattering experiments with “crossed” polarization geometries^{22,26} and also in the optical response of αRuCl_{3}^{18,26}. The peak observed at 140–150 meV by Raman scattering^{26}, in particular, may find correspondence in the lowest j_{eff} = 3/2like component that we compute at 195 meV. It is interesting that in Sr_{2}IrO_{4} the situation seems reversed as there the Raman selection rules appear to favor the higherenergy splitoff 3/2 states^{33}, which are however shifted to somewhat lower energy as compared to resonant inelastic xray scattering (RIXS)^{34}. One should note however that in Sr_{2}IrO_{4} the crystalfield physics is rather subtle, as the local tetragonal distortion giving rise to elongated apical bonds is counteracted by interlayer cation charge imbalance effects^{35}.
The rather broad feature at 310 meV in the imaginary part of the dielectric function has been assigned to Ru^{3+} t_{2g}toe_{g} transitions^{26}. Our ab initio data do not support this interpretation, since the lowest t_{2g} → e_{g} excitations are computed at ≈1.3 eV, but rather favor a picture in which the 310 meV peak corresponds to the upper 3/2like component. The latter can become optically active through electronphonon coupling. The rather large width of that excitation has been indeed attributed to electronphonon interactions in ref. 26.
Comparison between our quantum chemistry results and the optical spectra^{18,26} further shows that the experimental features at 1.2 and 2 eV, assigned in ref. 26 to intersite d–d transitions, might very well imply onsite Ru 4dshell excitations. In particular, we find spinorbit states of essentially nature at 1.3–1.5 eV and of both and character at 1.7–2.2 eV relative energy, see Table 2. Experimentally the situation can be clarified by direct RIXS measurements on αRuCl_{3}, for instance at the Ru M_{3} edge.
We have also calculated the magnetic g factors in this framework. By spinorbit MRCI calculations with all five Ru 4d orbitals in the reference CASSCF, we obtain for the C2/m structure of ref. 28 g_{ab} = 2.51 and g_{c} = 1.09, where the crystallographic c axis is perpendicular to the (ab) Ru honeycomb plane. On the experimental side, conflicting results are reported for the g factors: while Majumder et al.^{19} derive from magnetic susceptibility data that both g_{ab} and g_{c} are 2, Kubota et al.^{20} estimate g_{ab} = 2.5 and g_{c} = 0.4. The latter g_{c} value implies a rather large t_{2g}shell splitting δ, with δ/λ > 0.75 (see the analysis in ref. 20). The quantum chemistry g factors are consistent with a ratio δ/λ ~ 0.5, i.e., t_{2g} splittings of ≈70 meV (see the data in Tables 1 and 2) for a 4d SOC in the range of 120–150 meV^{24,27,36}. Electron spin resonance measurements of the g factors might provide more detailed experimental information that can be directly compared to our calculations.
Intersite exchange for j ≈ 1/2 moments
NN exchange coupling constants were derived from MRCI + SOC calculations for embedded fragments having two edgesharing RuCl_{6} octahedra in the active region. As described in earlier work^{16,17,32,35}, the ab initio data for the lowest four spinorbit states describing the magnetic spectrum of two NN octahedra is mapped in our scheme onto an effective spin Hamiltonian including both isotropic Heisenberg exchange and symmetric anisotropic interactions. Yet the spinorbit calculations, CASSCF or MRCI, incorporate all nine triplet and nine singlet lowenergy states of predominant character. As in earlier studies^{16,17,32,35}, we account in the MRCI treatment for all single and double excitations out of the valence dmetal t_{2g} and bridgingligand p shells.
For onsite Kramersdoublet states, the effective spin Hamiltonian for a pair of NN ions at sites i and j reads
where and are 1/2pseudospin operators, J is the isotropic Heisenberg interaction, K the Kitaev coupling and the Γ_{αβ} coefficients are offdiagonal elements of the symmetric anisotropic exchange matrix with α, β ∈ {x, y, z}. Since the pointgroup symmetry of the Ru–Ru links is C_{2h} in the C/2m unit cell, the antisymmetric DzyaloshinskiiMoriya exchange is 0. Also, Γ_{zx} = −Γ_{yz} for C_{2h} bond symmetry. A local (Kitaev) reference frame is used here, such that for each RuRu link (see Table 3) the z axis is perpendicular to the Ru_{2}Cl_{2} plaquette (as also employed in refs 16, 17 and 32). Details of the mapping procedure, ab initio data to effective spin Hamiltonian, are described in ref. 35 and Methods.
From the quantum chemistry calculations, we obtain a FM Kitaev coupling K, for all three crystalline structures reported in the literature (see Table 3). Its strength is reduced as compared to the 4d^{5} honeycomb oxide Li_{2}RhO_{3}^{32}, with a maximum absolute value of 5.6 meV in the C2/m structure proposed by Cao et al.^{28}. We shall discuss and compare our finding of a FM Kitaev coupling to other theoretical and experimental findings in the next section. For the structure of Cao et al.^{28} the bond lengths and angles are very similar for the two types of pairs of NN octahedra. As a result, we find identical effective interactions up to the first digit. This is the reason we provide in Table 3 only one set of couplings for that particular crystal structure. Anisotropic interactions of similar size, i.e., both K and the offdiagonal couplings Γ_{αβ}, are computed for the C2/m configuration of ref. 30, characterized by bond lengths and bond angles rather close to the values derived by Cao et al.^{28}. The Heisenberg J, on the other hand, changes sign with decreasing RuClRu flexure but for the bond angles reported in refs 28, 29, 30 and explicitly given in Table 3 remains in absolute value significantly smaller than K.
The trends we find with changing the RuClRu bond angle, apparent from Table 3, and earlier results for the dependence of K and J on bond angles in oxide honeycomb compounds^{17,32} motivate a more detailed investigation over a broader range of RuClRu flexure. The outcome of these additional calculations is illustrated in Fig. 1. We fixed in these calculations the RuRu distance to 3.44 Å and varied the RuClRu angle by changing the amount of trigonal compression for each of the two NN RuCl_{6} octahedra. In contrast to the oxides, where K values in the range of 15–30 meV are computed for large angles of 98–100°, the Kitaev coupling is never as strong in RuCl_{3}. K shows a maximum of only ≈5 meV at 94° in Fig. 1 and its angle dependence is far from the nearly linear behavior in 4d^{5} and 5d^{5} oxides^{17,32}.
The Heisenberg J, on the other hand, displays a steep upsurge with increasing angle, more pronounced as compared to the honeycomb oxides. In other words, for large angles J dominates in RuCl_{3}, in contrast to the results found in 4d^{5} and 5d^{5} honeycomb oxides in the absence of bridgingligand displacements parallel to the metalmetal axis^{17,32}. These notable differences between the halide and the oxides suggest a somewhat different balance between the various superexchange processes in the two types of systems.
Magnetic phase diagram
To assess the consistency of our set of ab initio NN effective couplings with experimental observations, we carried out ED calculations for the honeycomb model described by (1) but including additionally the effect of second and thirdneighbor J_{2} and J_{3} isotropic exchange. Anisotropic longerrange interactions are however neglected since recent phenomenological investigations conclude those are not sizable^{37}. We first considered the case without external magnetic field, H = 0, and clusters of 24 sites with periodic boundary conditions (PBC’s). The static spinstructure factor was calculated as a function of variable J_{2} and J_{3} parameters while fixing the NN couplings to the MRCI results computed for the crystalline structure of ref. 28 and listed in Table 3.
For a given set of J_{2} and J_{3} values, the dominant order is determined according to the propagation vector Q = Q_{max} providing a maximum value of S(Q). As shown in Fig. 2, the phase diagram contains seven different phases: four commensurate phases (FM, Néel, zigzag, stripy), three with incommensurate (IC) order (labelled as ICx1, ICx2, ICxy) and a SL phase. The ICx1 and ICx2 configurations have the same periodicities along the b direction as the stripy and zigzag states, respectively, and display IC wave numbers along a. The ICxy phase has IC propagation vectors along both a and b. The variety of IC phases in the computed phase diagram is related to the comparable strength of the NN J and the offdiagonal NN couplings Γ_{αβ}. For example, the system is in the ICxy state for J_{2} = J_{3} = 0. From the experimental observations, the lowtemperature magnetic structure of αRuCl_{3} is abplane zigzag AF order^{21,28,30}. We find indeed that the zigzag state is stabilized in a wide range of AF J_{2} and J_{3} values in our phase diagram.
To estimate the strength of J_{2} and J_{3} in αRuCl_{3}, we performed a fitting of the experimental magnetization curves^{30} by ED calculations. The PBC cluster we used is shown in Fig. 3(a). We find that different signs for J and K determine qualitatively different shapes for the magnetization curves. In particular, J > 0 and K < 0 values are required to reproduce the overall pattern of the measured magnetization, which exhibits a very slow saturation with increasing external field (see Methods). Additionally, AF values for J_{2} and J_{3} significantly shift the saturation to higher field and therefore these longerrange couplings must be small (1 meV, see Methods) to reproduce the experiment. The observed magnetization curves are set side by side to ED results in Fig. 3(b), for both and . We used MRCI g factors (g_{ab} = 2.51, g_{c} = 1.09) and MRCI NN interactions (see Table 3) and set J_{2} = J_{3} = 0.25 meV in Fig. 3(b). It is seen that the overall shapes of the experimental curves are well reproduced in these ED calculations. For comparison, additional ED results are provided in Fig. 3(c) with J = 1.0, K = −5.0, J_{2} = J_{3} = 0.3, g_{ab} = 2.4, g_{c} = 0.95 and vanishing offdiagonal NN couplings. It is difficult to extract information on the latter by using ED fits to the experimental data because the magnetization is not very sensitive to the strength of these offdiagonal NN exchange interactions. The magnetization is very sensitive, on the other hand, to the g factors — its strong directional dependence mainly comes from the strongly anisotropic g factors.
Most interestingly, a level crossing between the lowest two states is seen around H = 10 T for H ‖ [001] in all periodic clusters we considered. To better understand the nature of the changes at this level crossing, we analyzed the spinspin correlation functions . Results in the thermodynamic limit are presented in Fig. 3(d), while a detailed finitesize scaling analysis and further discussion on the spin correlations are provided in Methods. Below H ≈ 10 T the zigzag AF correlations are dominant; only the NN spinspin correlations being large for fields of 10–13 T (magnetization M/M_{s} ~ 0.25 − 0.45) is indicative of a Kitaevlike SL regime. The level crossing can be therefore associated to a transition between AF zigzag order and a SL. Static spinstructure factors for H = 0, 9.5 and 10.5 T are plotted in Fig. 3(e–g). A featureless static spinstructure factor is obtained for H > 10 T, consistent with the spinspin correlations shown in Fig. 3(d). In other words, we argue that the zigzag AF order is gradually weakened with increasing H, destroyed at H = 10 T and instead a SL ground state occurs for H > 10 T.
The MRCI calculations indicate K/J ratios in a range of 3 to 5 for the C/2m structures (see Table 3) while a commonly used criterion^{8} for identifying the Kitaev SL is having K/J > 7.8, so that the further frustration of magnetic interactions is relevant as well. One simple way to rationalize these findings is that an external field effectively weakens the effect of the AF NN J due to partial spin polarization and consequently K/J is effectively enhanced. Another way of qualitatively appreciating this point is that when one looks at the J_{2}–J_{3} phase diagram in Fig. 2, the main features of which are very similar to those^{16} found for Na_{2}IrO_{3}, a trajectory in the phase diagram from zigzag order (the low field state) to a saturated ferromagnet (the very high field state) is likely to pass through the SL phase. It is interesting that such a fieldinduced SL state due to frustration has been also predicted recently for the S = 1/2 AF kagomé lattice^{38}.
In the Kitaev limit, it is confirmed by earlier densitymatrix renormalization group (DMRG) calculations that topological phases can survive up to M/M_{s} ≈ 0.5^{39}, a critical value in agreement with our upper bound of the SL phase. It may be that due to the longerrange J_{2} and J_{3} couplings the topological phase in the lowfield regime of the Kitaev limit^{39} is replaced by the zigzag ground state in our model.
We also analyzed the gap Δ in the SL state. Due to discrete effects in clusters with PBC’s , this analysis was performed using a setup with open boundary conditions (OBC’s). To remove artifacts related to individual motions around the open edges, we calculated the excitation spectrum for a spin flip at a site in the central region of the cluster,
where is the spinflip operator at site i and ψ_{ν}〉 and E_{ν} are eigenstates and eigenvalues of the system, respectively (ν = 0 corresponds to the ground state). The position of site i and the computed spectrum E_{ν} − E_{0} are shown in Fig. 4(a). Obviously, a gap linear in H (Δ ∝ H − H_{s}) in high fields (H > H_{s} ≈ 15 T) indicates a fully polarized FM state. But a sizable gap opens as well for fields in the range of 8–15 T, in spite of having no longrange magnetic order. This is another result that indicates a gapped SL state.
Furthermore, to check the topological properties of the gapped SL state, we considered the hexagonal plaquette operator^{5}
where the labeling of links and sites is illustrated in the inset of Fig. 4(b). The expectation value of O_{h} was calculated using the 24site PBC cluster. Results for αRuCl_{3} are provided as a function of H in Fig. 4(b). For comparison, a plot of 〈O_{h}〉 for the plain KitaevHeisenberg model is shown in Fig. 4(c) (the definition of the KitaevHeisenberg Hamiltonian is given in Methods). In the Kitaev limit, the operator (3) commutes with the Hamiltonian and the expectation value 〈O_{h}〉 is exactly ±1. On the other hand, it rapidly drops to 〈O_{h}〉 ~ 0 when moving away from the Kitaev SL regime. The expectation value we compute for αRuCl_{3} is 〈O_{h}〉 ≈ −0.13 at H = 0. It monotonously decreases in absolute value with increasing H and displays a steep enhancement at H = 12.3 T. The absolute value of 〈O_{h}〉 in the SL regime (fields of 12.3–16.1 T) is significantly lower than the limit 〈O_{h}〉 = ±1 for the pure Kitaev model, pointing again to the important role of longerrange AF interactions. The secondneighbor couplings, in particular, frame a triangular AF Heisenberg net — a wellknown playground for frustrationinduced SL physics.
Finally, for insights into the topological properties of the system, we investigated by DMRG methods the field dependence of the entanglement spectrum (ES)^{40}. Using Schmidt decomposition, the ground state can be expressed as
where the states correspond to an orthonormal basis for the subsystem S (either A or B). We studied a cylindrical cluster with 44 sites whose subdomains A and B are sketched in Fig. 4(d). In our calculations, the ES {ξ_{i}} is simply obtained as ξ_{i} = −log λ_{i}, where {λ_{i}} are the eigenvalues of the reduced density matrices after the bipartite splitting. The lowlying ES levels are plotted as function of magnetic field in Fig. 4(e). A relatively large “gap” is seen below H = 11.5, since the AF zigzag state is topologically trivial. With increasing H, a crossover is clearly seen around H = 11.5 T. Interestingly, there exist many (nearly) degenerate lowlying levels for fields in the interval 11.5–14 T. This is the window for which a SL ground state is suggested by the behaviour of other quantities and parameters discussed above. The lowlying levels are distributed in a rather broad range of the partition spin sectors: for example, at H = 13.2 T, ξ_{1} = 0 , ξ_{2} = 0.0097 , ξ_{3} = 0.1697 , ξ_{4} = 0.4243 , ξ_{5} = 0.4823 , ξ_{6} = 0.5327 , ξ_{7} = 0.7968 etc., where is the total S^{z} of subsystem A. This also supports the appearance of the SL state. For higher fields, the ξ_{2} − ξ_{1} gap increases linearly, reflecting the fieldinduced FM state.
Discussion
Our finding of a FM Kitaev interaction can be first compared with the conclusions of other theoretical investigations. In fact, the analysis of effective superexchange models using hopping matrix elements and effective HubbardU interactions derived from densityfunctional (DF) electronicstructure calculations lead to contradictory results: an AF NN Kitaev coupling has been earlier predicted by Kim et al.^{24} and a FM K has been more recently found by Winter et al.^{37}. Our result is qualitatively consistent with the latter. Relevant in this regard is further the trends we observe for the effective K by running spinorbit calculations at different levels of approximation: restricted openshell HartreeFock (ROHF), CASSCF and MRCI. The respective K values are 1.2, −2.5 and −5.6 meV, for the C2/m structure of ref. 28. It is seen that accounting for intersite t_{2g} − t_{2g} hopping by CASSCF changes the sign of K from AF to FM and that by additionally taking into account superexchange paths involving the bridgingligand 3p and metal e_{g} levels by MRCI calculations with single and double excitations only pushes K more on the FM side. It is unlikely that additional excitations, “triple” etc., would change the sign of K back to the AF ROHF.
To make direct contact with experimental observations, one can compare the measured fielddependent magnetization with the theoretical results, as we did above, finding that only J > 0 and K < 0 are consistent with the measurements^{30}. This however contradicts the interpretation of recent inelastic neutron scattering data on the magnetic excitation spectrum^{23}, according to which K is very similar in magnitude to our finding but AF.
This point remains to be clarified but a possible explanation is related to modeling the experimental magnetic excitation spectra in the zigzag ordered state in terms of a pure KitaevHeisenberg Hamiltonian without longerrange couplings. In such a restricted model, zigzag order can only occur when J < 0 and K > 0, i. e., using the zigzag ordered ground state as input for the pure KitaevHeisenberg model fixes K > 0 from the beginning and a description of the magnetic excitations on top of this ground state in terms of linear spinwave theory is necessarily confined to this boundary condition. We find however that αRuCl_{3} is in a parameter regime where without longerrange, secondneighbor and thirdneighbor, interactions the ordering pattern would be an incommensurate AF state (see Fig. 2) which is close to the stripelike AF phase. This is the consequence of having J > 0 and K < 0. A weak AF thirdneighbor exchange J_{3} is essential to stabilize the zigzag order that is experimentally observed — this zigzag ground state is driven by the geometric frustration induced by J_{3} and consistent with K being dominant and FM.
For an interpretation of the magnon features in the neutron spectrum ref. 23 employs linear spinwave theory while for resolving the signatures of the fractionalized excitations — the actual fingerprint of the system being proximate to a Kitaev SL state — relies on a comparison to a Kitaevonly Hamiltonian. This should provide a full quantum description of the relevant physics on energy scales larger than weak interlayer magnetic couplings. The Kitaev point is particularly interesting because exact statements can be made^{5,41,42}. In the honeycomb Kitaev model the excitations are exactly fractionalized into localized fluxes and delocalized Majorana modes. Its dynamic spinstructure factor, which determines the inelastic neutron scattering response, is dominated by a spin excitation creating two fluxes. As the fluxes are localized, the spinstructure factor is rather dispersionless and only a weak momentum dependence arises from screening of the fluxes by gapless Majorana modes^{41}. The sign of K sets the sign for the dispersion of these Majorana modes that screen the fluxes^{5}. The upshot is that the dynamic structure factor in the Kitaev model strongly depends on the magnitude of K (which sets the energy threshold for flux creation) but only very weakly on its sign — fits to the data with K and −K then provide very similar results.
Conclusions
In sum, quantum chemistry calculations show that in αRuCl_{3} there is sizable trigonal splitting of the Ru 4d^{5} levels. This results in splitting of the spinorbit excitation energies, which can be accurately measured by e.g. resonant inelastic xray scattering, and in admixture of the j_{eff} = 1/2 and j_{eff} = 3/2 states. The resulting anisotropy of the magnetic g factors that we compute is consistent with experimental observations^{20}.
The nearestneighbor Heisenberg interaction J is found to be weak and antiferromagnetic in the ab initio computations while the Kitaev K is 3–5 times larger and ferromagnetic. Using these magnetic couplings as a basis for effectivemodel exactdiagonalization calculations of the magnetic phase diagram, we show that J > 0 and K < 0 values are required to reproduce the shape of the observed magnetization. The latter exhibits a very slow saturation with increasing the external field. As residual longerrange magnetic interactions would significantly shift the saturation to higher field, these couplings must be small. At the same time, however, we find the longerrange couplings are essential in producing the experimentally observed zigzag magnetic order in αRuCl_{3}.
We also determine by quantum chemistry calculations the dependence of the NN K and J interactions on the angle defined by two adjacent metal sites and a bridging ligand. Along with similar curves we have computed for the “213” honeycomb compounds^{17,32} — Na_{2}IrO_{3}, Li_{2}IrO_{3} and Li_{2}RhO_{3} — these results provide theoretical benchmarks for strain and pressure experiments on 4d^{5}/5d^{5} honeycomb halides and oxides.
In our numerical investigations, a level crossing between the lowest two states is seen for field along the [001] direction around H = 10 T, i. e., a transition from AF zigzag order to a gapped spinliquid state. We note that qualitatively similar features are also found for other field directions. Our calculations suggest that not only αRuCl_{3} but also Na_{2}IrO_{3} is a candidate material to observe such a transition, either at lowtemperature ambient conditions or under external pressure.
Methods
Ru^{3+} 4dshell electronic structure
Ab initio manybody quantum chemistry calculations were first carried out to establish the nature of the Ru^{3+} 4d^{5} ground state and lowest Ru 4dshell excitations in RuCl_{3}. An embedded cluster having as central region one [RuCl_{6}]^{3−} octahedron was used. To describe the finite charge distribution in the immediate neighborhood, the three adjacent RuCl_{6} octahedra were also explicitly included in the quantum chemistry computations while the remaining part of the extended solidstate matrix was modeled as a finite array of point charges fitted to reproduce the ionic Madelung field in the cluster region^{43}. Energyconsistent relativistic pseudopotentials were used for the central Ru ion, along with valence basis sets of quadruplezeta quality augmented with two f polarization functions^{44}. For the Cl ligands of the central RuCl_{6} octahedron, we employed allelectron valence triplezeta basis sets^{45}. For straightforward and transparent analysis of the onsite multiplet physics (see Table 2 in main text and Table 4 in this section), the adjacent Ru^{3+} sites were described as closedshell Rh^{3+} ions, using relativistic pseudopotentials and valence triplezeta basis functions^{44}. Ligands of these adjacent octahedra that are not shared with the central octahedron were modeled with allelectron minimal atomicnaturalorbital basis sets^{46}. Results in excellent agreement with the experiment were found by using such a procedure in, e.g., Sr_{2}IrO_{4}^{35} and CaIrO_{3}^{47}.
All computations were performed with the Molpro quantum chemistry package^{48}. To access the Ru onsite excitations, we used active spaces of either three (see Table 1 in main text) or five (Table 2 in main text and Table 4 in this section) orbitals in CASSCF. In the subsequent MRCI^{49,50}, the Ru t_{2g} and Cl 3p electrons at the central octahedron were correlated. The PipekMezey localization module^{51} available in Molpro was employed for separating the metal 4d and Cl 3p valence orbitals into different groups, i. e., centered at sites of either the central octahedron or of the adjacent octahedra. The spinorbit treatment was carried out as described in ref. 52.
One important finding in our quantum chemistry investigation is that compared to the 4d and 5d oxide honeycomb systems — Li_{2}RhO_{3}, Li_{2}IrO_{3}, Na_{2}IrO_{3} — the smaller ligand ionic charge in the halide gives rise to significantly weaker t_{2g} − e_{g} splittings. This is apparent in Table 2 in the main text: for the C2/m crystalline structure of Cao et al.^{28}, we compute excitation energies of only ≈1.3 eV for the lowest states. Even more suggestive in this regard is the energylevel diagram we compute for the P3_{1}12 crystalline structure of ref. 29. For the latter, the sequence of Ru^{3+} levels is shown in Table 4: it is seen that the ^{6}A_{1} state is even lower in energy than ^{4}T_{1} . Such lowlying excited states may obviously play a more important role than in the oxides in intersite superexchange.
Ru 4d^{5} g factors were computed following the procedure described in ref. 35. The values provided in the main text, g_{ab} = 2.51 and g_{c} = 1.09, were obtained by including the ^{2}T_{2} , ^{4}T_{1} , ^{4}T_{2} , and ^{6}A_{1} states in the spinorbit treatment. The orbitals were optimized for an average of all these states. The strength of the coupling to external magnetic field can also be extracted from more involved calculations as described in the next subsection.
The effect of onsite mixing on the computed g factors appears to be modest — by spinorbit CASSCF calculations based on a minimal orbital active space (three Ru t_{2g} orbitals), g_{ab} = 2.63 and g_{c} = 1.03; if states are also considered by CASSCF (as described above), g_{ab} = 2.59 and g_{c} = 1.18.
Intersite exchange
NN magnetic coupling constants were derived from CASSCF + MRCI spinorbit calculations on units of two edgesharing [RuCl_{6}]^{3−} octahedra. Similar to the computations for the onsite excitations, the four octahedra adjacent to the reference [Ru_{2}Cl_{10}]^{4−} entity were also included in the actual (embedded) cluster. We used energyconsistent relativistic pseudopotentials along with valence basis sets of quadruplezeta quality for the two Ru cations in the reference unit^{44}. Allelectron basis sets of quintuplezeta quality were employed for the bridging ligands and triplezeta basis functions for the remaining chlorine anions of the reference octahedra^{45}. We further utilized two f polarization functions^{44} for each Ru ion of the central, reference unit and four d polarization functions^{45} at each of the two bridging ligand sites. Ru^{3+} ions of the four adjacent octahedra were modeled as closedshell Rh^{3+} species, following a strategy similar to the calculations for the onsite 4dshell transitions. The same computational scheme yields magnetic coupling constants in very good agreement with experimental estimates in CaIrO_{3}^{47}, Ba_{2}IrO_{4}^{53}, and Sr_{2}IrO_{4}^{35,54}.
The mapping of the ab initio quantum chemistry data onto the effective spin model defined by (1) implies the lowest four spinorbit states associated with the different possible couplings of two NN 1/2 pseudospins. The other 32 spinorbit states within the manifold^{16,32} involve j_{eff} ≈ 3/2 to j_{eff} ≈ 1/2 charge excitations^{7,32} and lie at 150 meV higher energy (see Tables 1 and 2 and refs 32 and 36), an energy scale much larger than the strength of intersite exchange. To derive numerical values for all effective spin interactions allowed by symmetry in (1), we additionally consider the Zeeman coupling
where L_{q} and S_{q} are angularmomentum and spin operators at a given Ru site while g_{e} and μ_{B} stand for the freeelectron Landé factor and Bohr magneton, respectively (see also ref. 35). Each of the resulting matrix elements computed at the quantum chemistry level, see Table 5, is assimilated to the corresponding matrix element of the effective spin Hamiltonian, see Table 6. This onetoone correspondence between ab initio and effectivemodel matrix elements enable an assessment of all coupling constants in (1).
For C_{2h} symmetry of the [Ru_{2}Cl_{10}] unit^{28}, it is convenient to choose a reference frame with one of the axes along the RuRu link. The data collected in Tables 5 and 6 are expressed by using such a coordinate system, with the x axis along the RuRu segment and z perpendicular to the Ru_{2}Cl_{2} plaquette. The tensor reads then
where and the “prime” notation refers to this particular coordinate system. The Kitaevlike reference frame within which the data in Table 3 are expressed implies a rotation by 45° about the z axis^{16,17,32}. The connection between the parameters of Table 3, corresponding to the Kitaevlike axes, and the “prime” quantities in Tables 5 and 6 is given by the following relations^{16,17,32}:
The terms Δ_{n} and Ω_{n} in Table 6 (where n ∈ {y, z}) stand for:
The g factors are here expressed in the local coordinate frame related to the Ru_{2}Cl_{2} plaquette, different from the values provided in Sec. 2.
Magnetization curves for the KitaevHeisenberg model
Magnetization curves of the pure KitaevHeisenberg model, calculated by ED on a 24site cluster, are plotted in Fig. 5. The overall shapes are qualitatively well determined once the signs of J and K are fixed. For J > 0 and K > 0 [Fig. 5(b)], the magnetization increases linearly at low field and more steeply at higher field. This behavior is similar to that of the twodimensional (2D) bipartite Heisenberg systems; the main difference is the existence of a kink near the saturation, due to local AF interactions and the mixing of different S^{z}sectors. Below the kink, the NN spin correlations remain AF. For J < 0 and K < 0 [Fig. 5(c)], the magnetization “jumps” to finite values at H = 0^{+} and gradually saturates with increasing field. This gradual saturation is the result of local FM interactions S^{x}S^{x} and S^{y}S^{y}. For J < 0 and K > 0 [Fig. 5(a)], the magnetization increases linearly at low field, reflecting the AF J, smoothly connects to the higherfield curve and then saturates gradually with increasing field, similar to the case of J < 0 and K < 0. This qualitative behavior is basically the result of competing FM J and AF K. When K is small, the magnetization saturates rapidly with increasing field due to the FM J; as the AF K increases, the saturation is shifted to higher field. The shape of the magnetization curve itself is almost unchanged with changing K and the magnetic field can be simply rescaled by K · H. Typically, the effect of a FM K on the magnetization curve is small but the saturation becomes slower for larger K. A linear increase in weak fields and very slow saturation at higher fields was experimentally observed for αRuCl_{3}. Such behavior is found in the calculations only for J > 0 and K < 0 [Fig. 5(d)].
Generally, the magnetization curve of the Heisenberg model is a step function in calculations on finitesize systems, due to discrete effects. However, in the KitaevHeisenberg model, the total S^{z} is no longer conserved due to terms such as S^{+}S^{+} and S^{−}S^{−}. The magnetization curve can be then a smooth function. In our results, small steps are still visible in the magnetization curve for the case of J > 0 and K > 0. There, since the Néel (or zigzag) fluctuations are strong, the mixing of different S^{z}sectors is not sufficient to mask discrete effects.
Magnetization curves with longerrange interactions
We find that J > 0 and K < 0 values are required to reproduce the experimental magnetization curves. Looking in more detail to the dependence on longerrange interactions J_{2} and J_{3} is also instructive. Magnetization curves at J = 1, K = −5 and J = 1, K = −8 are shown in Fig. 6 for several J_{2} = J_{3} values. The effect of longerrange interactions seems to be even quantitatively similar for the two different K values. As long as J_{2} and J_{3} are much smaller than J (J_{2}, J_{3} < 0.2 J), the saturation is simply shifted to higher field but the overall shape of the magnetization curve is conserved. On the other hand, for J_{2}, J_{3} > 0.3 J, the overall shape changes somewhat, approaching that for the case of J > 0 and K > 0. We thus infer that J_{2} and J_{3} must be smaller than 0.3 J to reproduce the experimental magnetization curves. Only results for the case of J_{2} = J_{3} are shown here for simplicity, since we find that J_{2} and J_{3} have similar effect on the magnetization curves and affect those almost independently.
Spinspin correlations in the spinliquid phase
To describe in more detail the Kitaev SL phase in the intermediatefield region, we calculated the fielddependent spinspin correlation functions and compared them to those of the zerofield Kitaev SL phase of the 2D KitaevHeisenberg model on the honeycomb lattice^{8}. The NN interactions of the KitaevHeisenberg model can be written as
where γ(=x, y, z) labels the three distinct types of NN bonds in the “regular” honeycomb plane. Following the notation of ref. 8, we define the effective parameter and an angle φ via and . In Fig. 7(a), spinspin correlations near the FM Kitaev limit (φ = 1.5) of the KitaevHeisenberg model are plotted, for a 24site cluster with PBC’s. The Kitaev SL state is characterized by a rapid decay of the spinspin correlations: in the Kitaev limit, only the NN correlations are finite and longerrange ones are zero; that is faithfully reproduced by the 24site calculations. Even away from the Kitaev limit, the longerrange (not NN) spinspin correlations fall within a narrow range in the Kitaev SL phase (1.40 < φ < 1.58). As seen in Fig. 7(b), using the same 24site cluster, our fieldinduced SL state exhibits similar features; the values of longerrange correlations are distributed within a narrow range in the SL phase (10.8 T < H < 14.2 T). In other words, a rapid decay of the spinspin correlations is seen in our fieldinduced SL state, at the same level as in the FM Kitaev SL phase of the 2D KitaevHeisenberg model. We also found that the zigzagSL level crossing and the associated rapid decay of the spinspin correlations occur for any cluster which can stabilize a zigzagordered ground state at low or no field [see Fig. 7(c–e)]. On the other hand, there is no level crossing for clusters geometrically inconsistent with zigzag order, e.g., the clusters depicted in Fig. 8. Finitesize scaling analysis of the NN, secondneighbor and thirdneighbor spinspin correlation functions within the SL phase (H = 13.2 T) is shown in Fig. 7(f). The rather small dependence on cluster size is a natural consequence of having no finitesize effects in the Kitaev limit.
Additional Information
How to cite this article: Yadav, R. et al. Kitaev exchange and fieldinduced quantum spinliquid states in honeycomb αRuCl_{3}. Sci. Rep. 6, 37925; doi: 10.1038/srep37925 (2016).
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Acknowledgements
We thank A. Tsirlin and S.E. Nagler for fruitful discussions. We also thank S.H. Kim and B. Büchner for discussions and for sharing unpublished experimental data. S. N. and L. H. acknowledge financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG — SFB1143 and HO4427/2). J.v.d.B. acknowledges support from the HarvardMIT CUA. Part of the computations were carried out at the High Performance Computing Center (ZIH) of the Technical University Dresden.
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Author notes
 Nikolay A. Bogdanov
Present address: Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany.
 Vamshi M. Katukuri
Present address: Institute of Theoretical Physics, Ecole Polytechnique Féderale de Lausanne (EPFL), CH1015 Lausanne, Switzerland.
Affiliations
Institute for Theoretical Solid State Physics, IFW Dresden, Helmholtzstrasse 20, 01069 Dresden, Germany
 Ravi Yadav
 , Nikolay A. Bogdanov
 , Vamshi M. Katukuri
 , Satoshi Nishimoto
 , Jeroen van den Brink
 & Liviu Hozoi
Department of Physics, Technical University Dresden, Helmholtzstrasse 10, 01069 Dresden, Germany
 Satoshi Nishimoto
 & Jeroen van den Brink
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
 Jeroen van den Brink
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Contributions
R.Y. carried out the ab initio quantum chemistry calculations and subsequent mapping of the ab initio data onto the effective spin Hamiltonian, with assistance from N.A.B., V.M.K. and L.H. S.N. performed the ED and DMRG calculations and subsequent analysis, with assistance from J.v.d.B. L.H. and J.v.d.B. designed the project. R.Y., S.N., L.H. and J.v.d.B. wrote the paper, with contributions from all other coauthors.
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The authors declare no competing financial interests.
Corresponding authors
Correspondence to Ravi Yadav or Satoshi Nishimoto or Liviu Hozoi.
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Further reading

1.
Ordered states in the KitaevHeisenberg model: From 1D chains to 2D honeycomb
Scientific Reports (2018)

2.
Majorana fermions in the Kitaev quantum spin system αRuCl3
Nature Physics (2017)

3.
Atomicscale observation of structural and electronic orders in the layered compound αRuCl3
Nature Communications (2016)
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