Electronic and magnetic properties of the RuX3 (X = Cl, Br, I) family: two siblings—and a cousin?

Motivated by reports of metallic behavior in the recently synthesized RuI3, in contrast to the Mott-insulating nature of the actively discussed α-RuCl3, as well as RuBr3, we present a detailed comparative analysis of the electronic and magnetic properties of this family of trihalides. Using a combination of first-principles calculations and effective-model considerations, we conclude that RuI3, similarly to the other two members, is most probably on the verge of a Mott insulator, but with much smaller magnetic moments and strong magnetic frustration. We predict the ideal pristine crystal of RuI3 to have a nearly vanishing conventional nearest-neighbor Heisenberg interaction and to be a quantum spin liquid candidate of a possibly different kind than the Kitaev spin liquid. In order to understand the apparent contradiction to the reported resistivity ρ, we analyze the experimental evidence for all three compounds and propose a scenario for the observed metallicity in existing samples of RuI3. Furthermore, for the Mott insulator RuBr3, we obtain a magnetic Hamiltonian of a similar form to that in the much-discussed α-RuCl3 and show that this Hamiltonian is in agreement with experimental evidence in RuBr3.


INTRODUCTION
RuI 3 and RuBr 3 are recent additions to the RuX 3 family (X= Cl, Br, I) of layered Ru-based trihalides (Fig. 1a).The first member, α-RuCl 3 (in the following 'RuCl 3 ') has attracted considerable attention in recent years as a candidate material for the Kitaev honeycomb model [1].RuCl 3 is a spin-orbit assisted Mott insulator [2][3][4][5] whose magnetic low-energy degrees of freedom can be described in terms of j eff = 1/2 moments that interact through strongly anisotropic exchange [2,[6][7][8].While the material enters a so-called zigzag antiferromagnetic order (Fig. 1b) at low temperatures T N ≈ 7 K [4,9,10], various experiments at finite temperature [11][12][13][14] or at finite magnetic field [4,[15][16][17][18] have been interpreted as hallmarks of Kitaev physics, a subject which is presently under intensive debate [19][20][21][22][23][24].Recently, a sister compound with a heavier halogen, X = Br, was synthesized [25].Analogous to RuCl 3 , it is insulating and shows zigzag magnetic order, albeit with higher Néel temperature T N = 34 K [25].In contrast to RuCl 3 , the authors of Ref. 25 reported a Weiss constant with dominant antiferromagnetic interactions and a direction of the zigzag ordered moment different from RuCl 3 , and argued that this deviation suggests a closer proximity to the pure Kitaev model.To complete the RuX 3 family, two independent groups have now synthesized RuI 3 with the even heavier halogen iodine [26,27].In contrast to the two 'sibling' compounds, a quasi-metallic behavior was observed in RuI 3 , questioning the description in terms of localized j eff = 1/2 moments.Even though the dc resistivities measured in RuI 3 are orders of magnitude smaller than those of RuCl 3 or RuBr 3 , the reported values of 10 −3 to 10 −2 Ω cm [26] are uncharacteristically large for metals or even typical bad metals [28], and practically temperature-independent.While neither of the groups found clear signatures of magnetic ordering [26,27], they reported different behaviors of the magnetic susceptibility, which is either found to be temperature-independent [26], or with a strong upturn at low temperatures [27], suggesting that sample quality plays a crucial role.In order to understand the apparently distinct behavior of this family of trihalide materials, in this work we analyze the available experimental data and perform a detailed comparative study of the electronic and magnetic properties of the systems via first-principles calculations and extracted low-energy models.We find that: (i) The behavior of RuI 3 is not that far from RuCl 3 and RuBr 3 and the variations across the series are more quantitative than qualitative.(ii) Pristine samples of RuI 3 should be insulating with highly anisotropic magnetic exchange and nearly vanishing conventional Heisenberg interaction.We argue that the reported metallic behavior in RuI 3 could have its origin in sample quality.(iii) The magnetism in the Mott insulator RuBr 3 has predominantly ferromagnetic interactions, in contrast to what is suggested by the Curie-Weiss analysis of Ref. [25].We show that such interactions are consistent with experiment when taking into account spin-orbit coupling effects in the Curie-Weiss behavior.Our study derives model parameters and magnetic Hamiltonians for the whole RuX 3 family from ab-initio, that will be useful for future theoretical studies of these systems.In contrast to the usual model derivations that only include lo-  [10]), RuBr3 (Imai [25]), RuI3 (Ni [27], Nawa [26]).The shaded background depicts a typical range of resistivity for bad metals [28].
cal spin-orbit coupling (SOC) on the magnetic ion [2,6,8], our approach includes all SOC effects in the crystal.In fact, we show that SOC from the ligands leads to significant deviations from the Ruthenium-only SOC picture in the case of RuBr 3 and RuI 3 .

Comparative analysis of experiments
In the following, we analyze the reported electrical resistivity, specific heat and magnetic susceptibility data for RuX 3 (X=Cl, Br, I) [10,[25][26][27]29].In Fig. 1c we summarize the temperature dependence of the experimental resistivity data [10,[25][26][27] in all three compounds.In RuI 3 , the resistivity has a weak [27] or almost no [26] temperature dependence (Fig. 1c).Traditionally, metals are classified as materials where the resistivity ρ increases with temperature, distinguishing conventional metals (e.g., Cu) as those where in clean samples at temperatures roughly 300 to 600 K, ρ ∼ 10 −6 Ω • cm to ∼ 10 −5 Ω • cm, and bad metals as those with resistivities of ∼ 1−10 mΩ•cm This range is shown as a background shading in Fig. 1c.The reported resistivities for RuI 3 (ρ ∼ 40 mΩ•cm [27] and ρ ∼ 4 mΩ•cm [26]) are high even for bad metals, surpassing the Ioffe-Regel limit by more than an order of magnitude.Even more relevant, the lower-resistivity set of data [26] shows no discernible temperature dependence at all, while the data in Ref. 27 show a very weak positive derivative dρ/dT , but the absolute value is above anything traditionally considered metallic.Seemingly, as also pointed out in Ref. 27, electron transport in existing RuI 3 samples may be contaminated by grain boundaries.One possibility to interpret the measurements is that the pristine material is metallic, but insulating grain boundaries prevent percolation.Then, the in-grain resistivity can be neglected and what is measured is the resistivity of the insulating grain boundaries.In that case, however, thermal activation of carriers in the boundaries should give a positive temperature gradient of the resistivity, which is not observed.The opposite scenario is that of an insulating behavior in the bulk and (possibly bad) metallic one between the grains.In that case, the large resistivity reflects the small relative volume of metallic boundaries, where the transport is dominated by the residual resistivity.This scenario is compatible with the observations.Morphology of the grain boundaries can vary wildly depending on the growth conditions, including but not limited to vacancies, twins, dislocation and plain chemical dirt.Grain boundaries in semiconductors are often observed to be metallic.Apart from grain boundaries contaminating resistivity measurements, disorder (in form of vacancies, stacking faults, etc. [26,27]) could promote the bulk metallic phase over the Mott-insulating one, as has been shown for example for the Mott insulator κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl [30].Indeed, in our first-principles calculations discussed below, we find the ideal RuI 3 to already be quite close to a Mott-metal transition.Turning to the sibling compounds RuCl 3 and RuBr 3 , the resistivity (Fig. 1c) decreases with temperature, as expected for Mott insulators, and both systems show an approximate exponential activation gap behavior, E g,eff (T ) = −k B T 2 (d ln ρ/dT ), although with a significant blue-shift of the gap with increasing temperatures.Considering specific heat data in the compounds, the specific heat for RuCl 3 displays a well-defined peak at T N ≈ 7 K denoting the onset of the zigzag order, while the onset of long-range magnetic order in RuBr 3 is observed by a kink at T N = 34 K [25].None of this is observed for RuI 3 [26,27].
In Table 1 we summarize specific heat parameters reported experimentally [25][26][27]31], where γ (β) is the T -linear (T 3 ) contribution to C(T ).In RuI 3 , the T -linear contribution, even though contaminated by an extrinsic raise at small temperatures in Ref. [27] attributed to the nuclear quadrupole moment of Ru, yields γ ∼ 15 − 30 mJ•K −2 •mol −1 [26,27].From our electronic structure calculations of RuI 3 shown below, we ) contribution and given in units of mJ K −2 mol −1 (mJ K −4 mol −1 ).Debye temperature is given in Kelvin and TDM RuX 3 as a ratio to the value for RuCl3 (first row), where M RuX 3 is the harmonic average of the RuX3 mass.(*) Note that the values given for RuCl3 correspond to the asymptotic field-polarized limit extracted by Tanaka et al. [31], as otherwise at zero field the low-temperature specific heat behavior is dominated by vicinity to the Néel temperature of RuCl3, causing large magnetic contributions to β.
find that the unrenormalized metallic (i.e., nonmagnetic, not U -corrected) density of states corresponds to γ 0 ≈ 3 mJ•K −3 •mol −1 , suggesting a mass renormalization (if this γ is intrinsic) of a factor of 7-12.In the scenario where the metallic grain boundaries take up a sizeable fraction of the sample volume, this renormalization shall be even stronger, encroaching into the heavy fermions domain.This suggests that the origin of the anomalously large residual heat capacity may not be related to intrinsic metallicity.It is worth noting that the T 3 term β of C(T ), on the other hand, is rather reasonable for the three systems and scales roughly as the harmonic average M RuX3 of the atomic masses (last column in Table 1).We now turn our attention to magnetic susceptibility measurements.Figure 2a summarizes the powder-averaged measured magnetic susceptibilities χ(T ) as reported in Refs.[25][26][27]32].At low temperatures the RuCl 3 data [32] shows a clear signature of a transition to the ordered magnetic phase at 7 K.For RuBr 3 , the Néel transition T N ≈ 34 K is less apparent from the susceptibility, but the maximum in dχ/dT is consistent with the distinct transition seen in NMR relaxation measurements [25].The experimental report on powder samples of RuBr 3 utilized a standard Curie-Weiss (CW) fit, yielding an average Curie-Weiss temperature Θ avg std = −58 K [25], indicating predominantly AFM interactions.However, as we have recently shown [33], the Weiss constants obtained with such a standard CW fit may not anymore reflect the intrinsic exchange couplings in the case of significant SOC in the material, as it is the case for the Ru-based trihalides.With SOC, temperaturedependent van-Vleck contributions can arise, which can be effectively captured in a temperature-dependent magnetic moment µ eff (T, ∆) [33], as shown for ∆ = 0.018 eV in Fig. 2b, where ∆ can be directly associated to the crystal field splitting resulting from the distorted octahedral environment of Ru.In fact, for the sister compound RuCl 3 , a standard CW fit would lead to Θ avg std = −20 K, whereas an improved CW fit taking into account such van-Vleck-like contributions [33] provides CW constants Θ = +55 K for the magnetic field in the honeycomb plane and Θ ⊥ = +33 K   for the out-of-plane field, revealing an average CW constant, , of ≈ 48K.This indicates predominant ferromagnetic (FM) interactions, as they have become established for the magnetic Hamiltonian in RuCl 3 [34][35][36][37].Considering a similar strategy (see 'Methods' section), we fit the average susceptibility χ avg of RuBr 3 [25].However, since the crystal-field parameter ∆ primarily controls the in-plane vs out-of-plane anisotropy, and for RuBr 3 only powder-averaged data are available, we do not aim at extracting ∆ by fitting.Instead, we first fix ∆ using our first-principles calculations, enforcing µ eff /µ ⊥ eff (T = 0K) ∝ g /g ⊥ , where g /g ⊥ are taken from quantum chemistry calculations (see Fig. 5a, discussed below).This leads to ∆ = 0.018 eV.The best CW fit accounting for the implied µ eff (T, ∆ = 0.018 eV) (shown in Fig. 2c) yields Weiss constants Θ ≈ 5 K, Θ ⊥ ≈ 17 K and Θ avg ≈ 9 K, which are positive, indicating predominately ferromagnetic interactions for RuBr 3 , as seen before in RuCl 3 .In Fig. 2d we further analyze how the best-fit Weiss constants evolve for other choices of ∆.Indeed, for a wide range of reasonable ∆ around the first-principles value (indicated by the dashed vertical line), the average Weiss constant Θ avg remains positive.Importantly, for both materials, a standard residual 'background' term has to be included in the fitting, which in our case, depending on the material (RuBr 3 or RuCl 3 ) ranges from ∼ −3.5 × 10 −4 emu/mol to 1.5 × 10 −4 .This is of the same order of magnitude as the corresponding term in RuI 3 (∼ 3 to 8 × 10 −4 emu/mol) [26,27].Since in the former cases an intrinsic Pauli origin can be excluded, this observation also casts doubts on a metallic interpretation of this term in RuI 3 .Actually, the two available susceptibility measurements on RuI 3 display different behaviors, one nearly temperature-independent [26], and the other [27] showing a Curie-like rise at low temperatures, where a standard CW fit yields µ eff = 0.53µ B and Θ avg CW = −3 K [27].These differences are consistent with our hypothesis that the measured samples consist of magnetic insulating grains surrounded by metallic boundaries.Then, the samples with larger resistivity data [27] hint to larger insulating grains, hence less metallic boundaries are present, leading to the low-temperature Curie-like upturn in the susceptibility, compared to the samples in Ref. 26.

Electronic and magnetic calculations
In the following we present a comparison of the electronic and magnetic properties of the trihalide RuX 3 family obtained from a combination of density functional theory (DFT) and exact diagonalization of ab-initio-derived lowenergy models.Details of the calculations are given in the "Methods" section.Past experience with first-principles calculations for the Rubased trihalides [4,7,[38][39][40] indicates that the magnetic order and, to a considerably lesser extent, metallicity is very fragile, with several closely competing different magnetic phases.The ground states may vary depending on small changes in the crystal structure, on the way in which strong correlations are accounted for, and even on tiny details of the computational protocol.With this in mind, it is imperative to compare the calculated properties across the series, using the exact same computational setup.For the electronic structure calculations we consider the experimentally reported C2/m [4,9] and R 3 [41] structures for RuCl 3 , and the suggested R 3 structures for RuBr 3 [25] and RuI 3 [27].Structural details of the four models are summarized in the Supplementary Information.For RuCl 3 , the R 3 results are shown in Supplementary Information due to very similar results to the C2/m ones.Figure 3 shows the relativistic density of states (DOS) obtained within GGA+SOC+U as implemented in Wien2k, where a zigzag magnetic configuration with magnetic moments polarized perpendicular to the ab plane was considered.For the choice of U eff = U − J we take as a reference the ab initio estimates for the orbitally-averaged Hubbard on-site (U avg ) and Hund's coupling (J avg ) as obtained from constrained random-phase approximation (cRPA) calculations (see "Methods" section for calculation details).In contrast to previous cRPA estimates for RuCl 3 [42], our estimates incorporate all five d orbitals and extend to the complete Ru-based trihalide family.As shown in Fig. 4a, the effective Hubbard interaction parameters decrease with increasing ligand atomic number from Cl to I, which can be attributed to the more delocalized nature of the Ru d orbitals in RuI 3 compared to RuCl 3 when hybridizing with I instead of Cl.For RuCl 3 a U eff = 2.7 eV yields both the fundamental and direct gap to be ≈ 1 eV (Figure 3a) in agreement with the reported optical gap, apart from the presence of multiplets at 200 meV [43].We systematically reduced U eff to 2.1 eV for RuBr 3 and 1.4 eV for RuI 3 following the trend given by the cRPA results.With these values, RuBr 3 shows a gap of 0.56 eV (Figure 3b), while RuI 3 shows a small gap of 0.1 eV (Figure 3c).The gap closes in RuI 3 when U eff is further reduced to 1 eV.These results indicate a spin-orbit assisted Mott insulating state in disorder-free RuI 3 samples, which is on the verge of a metal-insulator transition.Possibly, as discussed above, disorder in the experimental samples could act as effective pressure, and bring the samples closer or over the Mott transition as seen in other Mott insulators [30].Note that these results hold regardless of the assumed magnetic pattern in the calculations.
In order to analyze the magnetic structure of the RuX 3 compounds, we first consider spin-polarized total energy calculations with VASP in the GGA+SOC+U approxima-   [37,44].Correspondingly, in our effective pseudospin model of RuCl 3 discussed below, classically, the energy of the ferromagnet is below that of zigzag, and only by including quantum fluctuations the zigzag ground state is recovered (as in, e.g., Ref. [37]).For RuBr 3 we find an energy minimum for the zigzag ordering in agreement with the experiment.Interestingly, for RuI 3 Néel and zigzag orders are energetically almost degenerate E Néel − E ZZ ≈ 1 meV/Ru, with the rest of magnetic orders we scanned being energetically rather close.All orders show very small and varying magnetic moments for Ru.These results hint to a magnetic frustration.
We proceed with the derivation of magnetic exchange models.In the first place, the magnetic Hamiltonian of RuBr 3 has been suggested to be closer to the pure Kitaev limit than in RuCl 3 [25], and, secondly, with our proposed scenario of a Mott insulating state for RuI 3 , the question of its magnetic properties is open.To investigate these issues from first principles, we derive via the ab-initio projED method [45] the pseudospin models Here, S stands for the relativistic pseudospin In the conventional parametrization of Kitaev materials, the exchange matrix J ij in R 3 symmetry on a nearest-neighbor Z 1 -bond (defined in Fig. 1) follows the form with the isotropic Heisenberg exchange J 1 , the bonddependent anisotropic Kitaev exchange K 1 , the bonddependent off-diagonal exchange terms Γ 1 and Γ Using U avg and J avg from cRPA (Fig. 4a), the complex hopping parameters extracted from full-relativistic DFT (magnitudes shown in Fig. 4b,c,d) and the projED method, we extracted the exchange constants shown in Fig. 5b,c,d.
Evaluating the magnetic interactions of the complete RuX 3 family, we find a nearest-neighbor ferromagnetic Kitaev interaction K 1 to be the dominant in all three compounds.Additionally, a subdominant ferromagnetic nearest-neighbor Heisenberg exchange J 1 is present, which is, however, almost vanishing for the iodine case.The symmetric offdiagonal Γ 1 interaction is of similar magnitude as J 1 , changing sign going from Cl and Br to I. Γ 1 , often neglected in the RuCl 3 analysis, may become rather important, particularly for RuI 3 .Further-neighbor interactions are generally smaller than their nearest-neighbor counterparts for all three systems, but increase for larger ligand atomic number and may play, especially in RuI 3 , an important role.That the anisotropic interactions do not monotonically increase with stronger spin-orbit coupling of halogen elements can be related to the SOC source.In the original Jackeli-Khaliullin mechanism [2], the heavy magnetic ions are solely responsible for SOC effects, which can be well described within the SOC atomic limit.In the case of RuBr 3 and RuI 3 , however, ligand SOC starts to play an important role.
To evaluate the interplay of these two SOC sources, we extracted ab initio values for the RuX 3 materials and compared them to the SOC atomic limit (see Supplementary Information).We find that in these compounds SOC effects from magnetic ions and ligands do not enhance each other, but do compete.This leads to the observed inhomogeneous behavior of the magnetic anisotropic terms in Fig. 5 as a function of ligand atomic number.Another consequence of this breakdown of the SOC atomic limit is that the established analytic perturbation theory expressions [2,6,8] become unjustified in Kitaev materials where SOC arises from both the metal and the ligand elements.RuBr 3 and RuI 3 are therefore cases where more general approaches, like 3NN-Exchange (meV) 2NN-Exchange (meV) NN-Exchange (meV) ours, are indispensable.Another approach would be perturbation theory taking into account ligand orbitals, as recently derived for the S = 3/2 material CrI 3 [46].Along the halogen series Cl-Br-I we observe a decrease for nearest-neighbor couplings (Fig. 5b) and an overall increase in magnitude for second and third neighbors (Fig. 5c,d).This can be understood by consideration of the ligand-metal (p-d) hybridization.We quantify the hybridization strength by integrating the DFT(GGA) density of states (DOS) with Ru 4d orbital character in the energy window dominated by the ligand p orbitals (between -7 eV and -1.05 eV).In spite of respective larger Ru-Ru distances, this can be related to the magnetic exchange by consideration of the ab initio hopping parameters between Wannier d orbitals.As also pointed out in Ref. 39, in spite of the stronger hybridization the nearest-neighbor hopping parameters are reduced for heavier ligands, illustrated in Fig. 4b.This is reflected in the magnetic exchange parameters (Fig. 5b) in an overall reduced magnitude in the nearest-neighbor parameters.
In contrast, the second and third neighbors show a very different dependence on the halogen element.From the dominant further-neighbor hoppings (Fig. 4c,d), the hoppings show an overall tendency to increase, with few exceptions.Certain further-neighbor magnetic exchange parameters, depending on their relation to the individual hopping parameters, become therefore increasingly important for RuBr 3 and especially for the magnetic properties of RuI 3 .Finally, we also computed the gyromagnetic g-tensor for the RuX 3 family from first principles, in order to relate the pseudospin S of the effective Hamiltonian to the magnetic moment M = µ B G • S. The g-tensor can be approximately characterized by two components, the value parallel to the honeycomb plane, g , and the one perpendicular to it, g ⊥ , which are shown in Fig. 5a.We consistently find g > g ⊥ for the whole family, promoting a stronger Zeeman term for in-plane fields.
We now discuss the ramifications of the derived magnetic models for the magnetism in these materials.
For RuCl 3 , we can compare our result to a vast available literature of models that have been shown to reproduce various experimental observations.Indeed, the model presented Properties of derived pseudospin models.Θ avg CW is the powder-averaged Weiss temperature of each model.'GS' refers to the ground state computed by exact diagonalization, and the angles of the magnetic moment φ M , θ M are defined according to Fig. 1b.
here in Fig. 5, derived completely from first principles without adjustments or external parameters, is remarkably close to some well-benchmarked recent models [19,37,38], and is therefore expected to also describe the material quite well.
As we apply the same ab-initio setup for the new members of the RuX 3 family, we expect our models to be reliable for them too.The direction-averaged Weiss constant (Θ avg CW in Table 2) is predicted to be positive across the RuX 3 family, characteristic of ferromagnetic exchange interactions.This is in line with our analysis of the experimental magnetic susceptibilities of RuCl 3 and RuBr 3 above (Fig. 2).While in RuCl 3 and RuBr 3 a large FM contribution to the Weiss constant comes from a significant FM nearest-neighbor Heisenberg interaction J 1 , this interaction nearly vanishes for RuI 3 (Fig. 5b), leading to a smaller Weiss constant.Furthermore, the small J 1 in RuI 3 renders the nearest-neighbor interactions to be extremely anisotropic, with a dominant Kitaev interaction K 1 .While at first glance this might suggest a spin-liquid ground state in RuI 3 , the increased strength of the furtherneighbor interactions in RuI 3 (see, e.g., J 2 , K 2 in Fig. 5c) also needs to be considered [47].To find the magnetic ground state properties, we perform exact diagonalization (ED) calculations of the derived j eff = 1/2 models on the 24-site cluster shown in Fig. 6a.In Table 2 and Fig. 6b we summarize the encountered ground states, i.e. zigzag for RuCl 3 and RuBr 3 , and possibly a quantum spin liquid (QSL) in RuI 3 .This is discussed in detail

RuBr3
< l a t e x i t s h a 1 _ b a s e 6 4 = " g R Q H N H 9 / R g P B p P x r P x 8 p l q J O K a d f q 2 j N c P a s + f T g = = < / l a t e x i t > below.For RuCl 3 , the model in Fig. 5 (as well as the R 3 model discussed in the Supplementary Information) yields zigzag AFM order, identifiable by a maximum at k = M in the static spin structure factor, shown in Fig. 6b.The computed ordered magnetic moment direction (see 'Methods' section), parametrized by θ and φ in Table 2 (compare Fig. 1b), is found to be tilted by θ M ≈ 34 • out of the plane, in excellent agreement with the recent experiment, where θ M = 32 ± 3 • [36] was reported.Interestingly, on the classical level, the ferromagnetic state is lower in energy than the zigzag state, meaning that the latter only becomes the ground state through quantum fluctuations, as discussed also in Ref. [37].We will now focus on the recently synthesized compounds, starting with RuBr 3 .The static spin structure factor for the RuBr 3 model of Fig. 5 is shown in Fig. 6b, indicating also a zigzag AFM order (k = M and C 6 -rotated vectors), in agreement with experiment [25].However, the calculated tilt angle of the magnetic moment, θ M = 32 • , is more in line with RuCl 3 than with the reported measured θ M = 64 • of RuBr 3 [25].The authors of Ref. 25 argued that this anomalously large tilt angle indicates an exceptionally strong relative Kitaev coupling, i.e., larger |K 1 /J 1 | and |K 1 /Γ 1 | compared to RuCl 3 .To investigate to what extent a closer proximity to the pure Kitaev model could produce such high tilt angles, we take our RuBr 3 Hamiltonian of Fig. 5 as a starting point and tune towards the pure Kitaev model, where K 1 is the only non-zero coupling.This is done by multiplying every exchange coupling except K 1 by (1 − f ) and sweeping f from 0 to 1.As shown in Fig. 6c, the moment indeed rotates further away from the honeycomb plane upon moving towards the pure Kitaev model, however even right before the transition to the Kitaev spin liquid (indicated by the vertical dashed line), θ S does not , which is to be compared to the neutron diffraction experiment, is even smaller due to the anisotropy g > g ⊥ in our calcu-lated g-tensor (Fig. 5a).A reconciliation with the reported θ M = 64 • would therefore require quite drastic changes to the g-tensor anisotropy and/or the exchange parameters.While in the whole J 1 -K 1 -Γ 1 parameter space with Γ 1 > 0, no angles of θ S beyond ∼ 40 • are expected in the zigzag phase [48], significant negative Γ 1 < 0 can in principle lead to θ S beyond 60 • [49].However such terms seem incompatible with the ab-initio results and would likely need strong distortions from the present considered RuBr 3 crystal structure to be realized.More distinct from the other two compounds are our results for RuI 3 .As discussed above, in our GGA+SOC+U calculations we find a very flat energy landscape of competitive magnetic configurations, indicative of strong magnetic frustration.Fittingly, the ground state from exact diagonalization of the present exchange model does not show a dominant ordering wave vector in the spin structure factor, see Fig. 6b.Although this is a signature generally associated with quantum spin liquid (QSL) states, we note that in the present model, the Kitaev Z 2 flux operator yields W p = 2 6 S x 1 S y 2 S z 3 S x 4 S y 5 S z 6 ≈ 0.29 (site indices refer to Fig. 6a).While this is clearly elevated compared to classical collinear states, where W p is restricted to | W p | ≤ 1 27 < 0.04, it is still significantly below the value of the pure unperturbed Kitaev spin liquid, where W p = 1 [1].Hence, if the ground state constitutes a QSL state, it is presumably not the Z 2 Kitaev spin liquid.The precise nature of the encountered magnetically disordered state might be interesting for future studies.It appears to be stabilized by the further-neighbor interactions, as we find a clear ferromagnetic ground state when omitting the second-and third-neighbor interactions in the present model.While a QSL scenario for our full RuI 3 model is compelling, we note that finite-size effects in our calculation could play a role.In particular, the finite-size cluster could be incompatible with the supposed correct ordering wave vector of the model, e.g. in case of an incommensurate ordering vector.

Conclusions and outlook
To summarize, we have presented a comparative analysis of the electronic and magnetic properties of the Ru-based trihalide family, including the recently synthesized RuBr 3 and RuI 3 , by combining state-of-the-art ab initio microscopic modelling with analysis of reported resistivity, specific heat and magnetic susceptibility data.The evolution of the magnetic order and Mott-Hubbard correlations along the halogen series, as well as possible role of disorder, have been a central part of our study.We conclude that: 1.All three ideal compounds are spin-orbit-assisted Mott insulators, but their fundamental gap decreases with higher ligand atomic number, Cl→Br→I, with RuI 3 coming rather close to a metal-insulator transition.
2. From DFT total-energy calculations, in ideal, pristine crystals the zigzag magnetic order is even more stable in RuBr 3 than in RuCl 3 , while RuI 3 shows significant magnetic frustration.Our ab-initio extracted low-energy models predict RuI 3 to feature either an incommensurate magnetic ordered state or a quantum spin liquid, which, interestingly, is possibly of a different kind to the Z 2 Kitaev spin liquid.
3. A number of reported experimental observations seem to be adversely affected by the sample quality, in particular by dirty grain boundaries.In fact, most of the observations in RuI 3 can be reconciled with theory by assuming insulating grains surrounded by (bad) metallic boundaries.The experimental evidence is consistent with a 'dirty' insulator, or a bad metal.Disorder would favor either of these.
4. In all three systems the dominant nearest-neighbor interaction is FM Kitaev K 1 , with a subdominant FM Heisenberg Interaction J 1 , that nearly vanishes for RuI 3 .We observe a non-monotonous behavior of the magnetic anisotropic terms as a function of ligand atomic number that we trace back to a competition of the SOC effects from magnetic ions and ligands.
5. RuBr 3 has predominantly ferromagnetic interactions, in contrast to what is suggested by standard Curie-Weiss analysis [25].Such interactions are consistent with the experimental susceptibility when taking into account high-temperature SOC effects.Our ab-initio magnetic model predicts zigzag order in agreement with experiment, with a tilting angle of θ M = 32 • for the magnetic moments, similar to RuCl 3 , but in contradiction to the reported θ M = 64

METHODS
Modified Curie-Weiss fit of RuBr 3 We fit the experimental average susceptibility of Ref. 25 with four fitting parameters χ ⊥ 0 , χ 0 , Θ ⊥ , Θ using the modified Curie-Weiss formula where Θ , Θ ⊥ are the Weiss constants and C α (T ) ∝ [µ α eff (T, ∆)] 2 is determined through ∆ as described in Ref. 33.Superscripts and ⊥ indicate the in-and out-ofhoneycomb-plane direction respectively.The susceptibility is fitted over the temperature range 150 -300 K and SOC strength λ = 0.15 eV is taken.

DFT calculations
To make sure that the calculated features within density functional theory are robust with respect to the choice of the basis set, we have tested the results using two different methods: the projector augmented wave method [50,51] as implemented in the VASP code [52,53], and the full potential linearized augmented plane-wave (LAPW) basis as implemented in Wien2k [54].Throughout the paper we have used the Generalized Gradient Approximation (GGA [55]) to the exchange-correlation functional.Hubbard correlation effects were included on a mean field level in the rotationally invariant implementation of the GGA+U method [56].All calculations included spin-orbit coupling (SOC) effects.For VASP we used the Ru pv pseudopotential, treating Ru p states as valence, and the standard pseudopotentials for the halogens.The Γ-centered 8×8×8 mesh in the nonmagnetic rhombohedral Brillouin zone was used, or the correspondingly scaled meshes for other structures.The energy cut-off was 350 eV, and the energy convergence criterion 1×10 −08 eV.For each type of magnetic order a number of collinear starting configurations with randomly selected Néel vectors were used, and the lowest-energy result was selected as the ground state.Individual results can be found in the Supplementary Information.For Wien2k we chose the plane-wave cutoff K max corresponding to RK max = 8 and a k mesh of 8 × 8 × 2 for the R 3 structure in the hexagonal Brillouin zone and 8 × 4 × 6 in the first Brillouin zone of the conventional unit cell for the C/2m structure.The density of states are calculated using a k mesh of 12×12×3 for the R 3 structure and 12 × 6 × 9 for the C/2m structure.The zigzag configurations are constructed using a conventional cell of the C/2m structure for RuCl 3 while a 1 × 2 × 1 supercell of the R 3 structures for RuBr 3 and RuCl 3 .

cRPA calculations
In order to obtain ab-initio estimates for the effective Coulomb interaction for the Ru-trihalide family, we employed the constrained random-phase approximation (cRPA) [57,58], as implemented in the FHI-gap code [59], based on the Wien2K electronic structure.The low-energy limit of the screened interaction was projected on the five Ru d orbitals, where screening processes in the same window were excluded.Convergence with respect to the discretization of the Brillouin zone and energy cutoff was ensured.

DFT-based derivation of magnetic models
To derive bilinear exchange parameters for each material, we employed the projED method [45], which consists of two steps.First, complex ab-initio hopping parameters between the ruthenium ions are estimated with projective Wannier functions [60] applied on full relativistic FPLO [61] calculations on a 12 × 12 × 12 k mesh.This allows to construct an effective electronic model H tot = H hop + H U , where the complex ab-initio hopping parameters enter the kinetic term H hop = ijαβ σσ t σσ iα,jβ c † iασ c jβσ and the cRPA effective Coulomb interaction parameters enter the two-particle term H U = iαβγδ σσ U σσ iαβγδ c † iασ c † iβσ c iδσ c iγσ .Second, the effective spin Hamiltonian H eff is extracted from the electronic model via exact diagonalization (ED) and projection of the resulting energy spectrum onto the lowenergy subspace, mapped onto pseudo-spin operator representation in the j eff picture with the projection operator P: H eff = PH tot P = ij S i J ij S j .Note that for RuCl 3 the exchange constants slightly differ from previously calculated values by some of the authors [8,38].The reason for this lies in the following details of the calculation setup: (i) first principles input parameters U avg and J avg from cRPA in contrast to previous choices, (ii) consideration of all five 4d ruthenium orbitals with the cost of restriction onto two-site clusters, (iii) SOC effects from both Ru 3+ and ligands considered through complex hopping parameters in contrast to the atomic limit, and (iv) consideration of the experimental crystal structure in contrast to relaxed ambient pressure structure as it was done in Ref. 38.For the calculation of the gyromagnetic g-tensor, we considered [RuX 6 ] 3− molecules within the quantum chemistry ORCA 3.03 package [62,63] with the functional TPSSh, basis set def2-TZVP and complete active space for the d orbitals CAS (5,5).

Exact diagonalization
Exact diagonalization calculations of the j eff = 1/2 models were performed on the 24-site cluster shown in Fig. 6a.To identify possible magnetic ordering, we analyze the static spin structure factor µ=x,y,z S µ −k S µ k .For the ordered moment direction, we compute the eigenvector with maximal eigenvalue of the correlation matrix ( S µ −k S ν k ) µ,ν (µ, ν ∈ {x, y, z}) at the ordering wave vector k = Q.This eigenvector then represents the ordered pseudospin direction S [49], which relates to the magnetic moment direction M ∝ G • S, as measured by neutron diffraction, via the anisotropic g-tensor G.

FIG. 1 .
FIG. 1. RuX 3 (X=Cl, Br, I) crystal structure, magnetic structure and resistivity.a Honeycomb layer in the RuX3 (X=Cl, Br, I) trihalides with bond definitions, cubic axes (xyz) and crystallographic axes (abc) in the R 3 structure, b Zigzag magnetic order in a honeycomb layer from two perspectives, with definitions of in-plane-angle φ and out-of-plane-angle θ. c Comparison of experimental dc resistivities as a function of temperature.Data was extracted from plots in the following references and labelled by respective first-author names: RuCl3 (Banerjee[10]), RuBr3 (Imai[25]), RuI3 (Ni[27], Nawa[26]).The shaded background depicts a typical range of resistivity for bad metals[28].

c 3 EFIG. 3 .
FIG.3.Density of states for RuX 3 (X=Cl, Br, I) Density of states (DOS) for the experimental structures of RuCl3,RuBr3 and RuI3, obtained from GGA+SO+U calculations with Wien2k, considering antiferromagnetic zigzag magnetic configurations.For RuCl3 we employed U eff = 2.7 eV, for RuBr3 U eff = 2.1 eV and for RuI3 U eff = 1.4 eV.Shown is also the contribution of Ru and halogen states to the DOS.

FIG. 5 .
FIG.5.Ab-initio computed pseudospin models across the RuX 3 family.a Quantum chemistry results for local gyromagnetic g -tensor components g (in-plane) and g ⊥ (out-of-plane).b-d projED results for the magnetic exchange couplings on nearestneighbor (NN), second-neighbor (2NN) and third-neighbor (3NN) bonds.Tabular form of all values is given in the Supplementary Information.

FIG. 6 .S y 2 S z 3 S x 4 S y 5 S z 6 .
FIG. 6. Exact diagonalization of RuX 3 pseudospin models.a Employed periodic cluster.Labeled sites 1, . . ., 6 define the Kitaev plaquette operator Wp = 2 6 S x 1 S y 2 S z 3 S x 4 S y 5 S z 6 .b Static spin structure factor in reciprocal space.Inner (outer) hexagon mark the edge of the first (third) Brilluoin Zone.High-symmetry k-points Γ, M, Γ are labelled.Color scale is the same for all three plots.c Out-of-plane angle θ S (θ M ) of the pseudospin (magnetic moment) within the zigzag phase when tuning from the RuBr3 model (f = 0) towards the pure Kitaev model (f = 1).Dashed vertical line indicates phase transition to the Kitaev spin liquid, identified by a peak in −∂ 2 E/∂f 2 .

TABLE 1 .
Overview of reported specific heat parameters.γ (β) is the coefficient of the T -linear (T 3 Ab-initio-computed multi-orbital Hubbard model parameters across the RuX 3 family.acRPA results for the orbitallyaveraged on-site Hubbard interaction (Uavg), Hund's coupling (Javg), and the nearest-neighbour Vavg coupling.bAbsolute magnitude of hopping parameters t1 = t (yz,yz) , t2 = t (xz,yz) , t3 = t (xy,xy) , and t = t (xy,z 2 ) on nearest-neighbor (NN), second-neighbor (2NN) and third-neighbor (3NN) Z-bonds.tion(see also 'Methods').Detailed results are listed in the Supplementary Information.For RuCl 3 , the calculated energy of the ferromagnetic state E FM is very competitive with the energy of the experimentally observed zigzag ordered state E ZZ : E ZZ − E FM ≈ 2 meV/Ru.This observation is consistent with the evidence for a metastable ferromagnetic state in RuCl 3 Y-bonds follow by respective C 3 rotations about the out-of-plane axis([111]in pseudospin coordinates).Analogously follow the definitions for second and third neighbor exchange terms (or see, e.g., Ref. 8).
1and correction terms η 1 and ν 1 .The latter correction terms are found to be small in our calculated Hamiltonians, and are neglected in what follows.The exchange matrices on X-and [25]].We showed that such a large angle cannot be simply explained by proximity to the pure Kitaev model, but would require quite drastic changes to the exchange parameters, such as sizeable negative Γ 1 < 0. Those would necessitate strong distortions on the reported RuBr 3 crystal structures.Answering the question posed in the title, our results and analysis strongly suggest that the ideal RuCl 3 , RuBr 3 and RuI 3 compounds constitute a family of three Mott-insulating siblings.The challenging task of getting better samples will hopefully help resolve the open issues.