Modified Curie-Weiss Law for j_{\rm eff} Magnets
Ying Li, Stephen M. Winter, David A. S. Kaib, Kira Riedl, Roser Valenti
MModified Curie-Weiss Law for j eff Magnets
Ying Li,
1, 2, ∗ Stephen M. Winter, † David A. S. Kaib, Kira Riedl, and Roser Valent´ı ‡ MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,School of Physics, Xi’an Jiaotong University, Xi’an 710049, China Institut f¨ur Theoretische Physik, Goethe-Universit¨at Frankfurt,Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany Department of Physics and Center for Functional Materials, Wake Forest University, NC 27109, USA (Dated: February 19, 2021)In spin-orbit-coupled magnetic materials, the usually applied Curie-Weiss law can break down.This is due to potentially sharp temperature-dependence of the local magnetic moments. We there-fore propose a modified Curie-Weiss formula suitable for analysis of experimental susceptibility. Weshow for octahedrally coordinated materials of d filling that the Weiss constant obtained from theimproved formula is in excellent agreement with the calculated Weiss constant from microscopic ex-change interactions. Reanalyzing the measured susceptibility of several Kitaev candidate materialswith the modified formula resolves apparent discrepancies between various experiments regardingthe magnitude and anisotropies of the underlying magnetic couplings. Great interest has been devoted towards searchingfor Kitaev spin liquid candidate materials with stronglyanisotropic Ising couplings on the honeycomb lattice [1–7]. Such interactions were proposed to be realizable in theedge-sharing octahedra of d transition metal ions, wherestrong spin-orbit coupling (SOC) splits the t g states intomultiplets with effective angular momentum j eff = 3/2and 1/2 [8–11]. While other proposals for realization ofthe Kitaev model also exist for materials with d filling[12, 13], as well as for complex magnetic interactions for d filling [14], we concentrate in this work on the well-studied d case. Promising candidate materials includeNa IrO [15–17], α -Li IrO [17–20], α -RuCl [21–25], aswell as H LiIr O [26–28].One of the persistent questions regarding all of thesespin-orbital coupled magnets is the specific details ofthe low-symmetry magnetic couplings, which are diffi-cult to extract from any single experiment. The overallscale and anisotropies are often first addressed via the(direction-dependent) Weiss constant Θ, appearing in thephenomenological Curie-Weiss (C-W) law describing thehigh-temperature magnetic susceptibility: χ = χ + N s ( µ eff ) k B ( T − Θ) , (1)where χ accounts for temperature-independent back-ground contributions, N s is the number of sites, and µ eff denotes the effective magnetic moment. While thermalfluctuations dominate for temperatures T (cid:38) Θ, quan-tum effects typically play a decisive role for T (cid:28) Θ.For this reason, quantum magnets nearby spin-liquidground states with finite but suppressed ordering tem-perature T N may still display a wide temperature range T N < T < Θ where responses resemble those of spin-liquid states. This occurs provided a large frustrationparameter f = Θ /T N can be defined. The excitationsin this temperature regime may even be interpreted interms of fractionalization [29–32]. Evidently, accurate estimation of Θ is an important first characterization ofa spin-liquid candidate and frustrated magnets in gen-eral. However, as we discuss in this work, standard C-Wfits are insufficient for j eff magnets with strong SOC.For the examples of Na IrO [17] and α -Li IrO [20],standard C-W fits suggest strongly anisotropic Θ-valuesas large as ∼ −
125 K despite antiferromagnetic orderingtemperatures of 15 K in α -Li IrO [17, 33] and 13 −
18 Kin Na IrO [17, 34, 35]. While competition betweenanisotropic interactions of different signs may render f a poor measure of frustration [17], the scale of the cou-plings suggested by these Θ-values is much larger than ex-pected from ab-initio calculations [36–39]. Furthermore,recent analysis of RIXS data on Na IrO led to proposedmodels that account for neither the anisotropy nor themagnitude of the observed Weiss constants [40].Similarly, the magnetic susceptibility of α -RuCl ( T N ∼ ∼
130 K, corresponding to f > µ B are anomalously large comparedto the pure j eff = 1 / µ B ), indicating inade-quacy of the C-W form. Indeed, similar deviations ob-served in a wide range of Ru compounds support thisconclusion [49].The oversimplified use of the Curie-Weiss law can mis-judge frustrations [50], relative anisotropies, and signsof the underlying couplings. A key observation is thatthe Curie-Weiss law only represents an adequate high-temperature approximation for χ ( T ) if the quantum op-erators representing the magnetic moments and magnetic a r X i v : . [ c ond - m a t . s t r- e l ] F e b x yz ab ac c* M A ~~ ~~~~ X YZ ~ T (K/eV) µ e ff ( µ B ) (c) µ abeff ∆ = − ∆ = − ∆ =0 ∆ =0.1 ∆ =0.3 0 400 800 1200 T (K/eV)(d) µ c*eff T (K) µ e ff ( µ B ) (b) µ eff ∆ = 0 λ =0 λ =0.1 λ =0.2 λ =0.3 λ =0.4 (a) FIG. 1. (a) Kitaev candidate honeycomb layers ( M = { Ir , Ru } , A = { O, Cl } ) with view in the ab - and ac -plane.Arrows indicate local x, y, z directions and crystallographic a, b, c axes. c ∗ is the direction perpendicular to the ab -plane.Canonical X, Y, Z bonds are indicated. (b) Temperature-dependent effective magnetic moment µ eff ( T ) for differentSOC-strengths λ (in eV) and fixed trigonal splitting ∆ = 0 eV.(c) µ eff ( ˜ T ) with ˜ T = T /λ for different values of ˜∆ = ∆ /λ inthe ab -plane. (d) µ eff ( ˜ T ) perpendicular to the ab -plane. field commute. This holds only if the local moments areof pure spin composition, while strong SOC may inducesignificant deviations. For isolated paramagnetic metalcomplexes [49, 51–53], and dimers [54], this effect canbe modelled by temperature-dependent moments µ eff ( T )due to additional van Vleck contributions. Such effectsmust also be present in j eff quantum magnets, but areusually ignored in analysis of χ ( T ). In this work, wetherefore propose an improved formula accounting for µ eff ( T ). We then perform exact diagonalization of theone-site multi-orbital Hubbard model for d filling withinclusion of spin-orbit and crystal-field terms. This al-lows to compare the results of the improved formula ac-counting for µ eff ( T ) to standard Curie-Weiss fitting fora range of models where the underlying couplings areexactly computed. Finally, we apply the modified fittingformula to experimental susceptibilities to yield correctedWeiss constants, and discuss corresponding implications.The electronic Hamiltonian for the d filling of octahe-drally coordinated transition metal ions in edge-sharinggeometries (see Fig. 1(a)) is given by: H tot = H hop + H CF + H SO + H U , (2)which is the sum of, respectively, the kinetic hopping term, crystal field splitting, spin-orbit coupling, andCoulomb interaction. The explicit expression for eachterm is given in the Supplemental Material [55]. Locally,SOC splits the t g levels into j / and j / states, witha single hole in the j / level in the ground state. Thelow-energy states are thus spanned by j / doublet de-grees of freedom, which can be described by an effectivespin model with j eff = 1 /
2. The effective Hamiltonian iswritten as H eff ≡ P ( H tot + H Z ) P , where [56]: P H tot P = (cid:88) iαjβ J αβij S αi S βj , (3) P H Z P = − (cid:88) iαβ h α g αβi S βi . (4)Here, P is a projection operator onto the low-energy sub-space, J αβij describe interactions between j / pseudospincomponents S αi ( α ∈ { x, y, z } ), g αβi are effective g -values,and h α respective magnetic field components. The conju-gate high-energy subspace contains states with finite den-sity of local j / → j / spin-orbital excitons, and inter-site particle-hole excitations. In reality, the Zeeman op-erator H Z mixes the j / and j / states, generating con-tributions to the magnetic susceptibility that are not cap-tured within this low-energy theory. Such van Vleck -likecontributions may modify the high-temperature suscepti-bility significantly. We therefore consider a regime wherethe temperature k B T is large compared to the magneticinteractions between j / moments ( k B T (cid:29) J αβij ∼
10– 100 K), but small compared to the splitting betweenthe j / and j / levels ( k B T (cid:28) λ ∼ . − . ∼ χ α ( T ) ≈ χ α + C α ( T ) T − Θ α , (5) C α ( T ) = N s k B [ µ α eff ( T )] , (6)Θ α = − S ( S + 1)3 k B (cid:80) iγjδ g αγi J γδij g δαj (cid:80) iγ g αγi g γαi . (7)In this approximation, the effective temperature depen-dence of Θ α is neglected, which is adequate for thepresent cases (see [55]). The most important observationis that the temperature dependence of µ α eff ( T ) severelycomplicates the extraction of Θ α from experimental sus-ceptibility data. It is often possible to fit such data toa conventional Curie-Weiss form χ = χ + C/ ( T − Θ);however, the values of C and Θ obtained from such fitsare not directly relatable to the exchange constants ofthe low-energy spin model. The way to proceed in or-der to extract reliable C-W constants is to first obtainthe effective moment µ α eff ( T ) of a single magnetic site,which can be computed exactly by diagonalizing the local ~ ~~ ~ − − − −
200 0 200 400 Θ z0 (K/eV) Θ z f i t ( K / e V ) Θ zfit,2 with constant µ fit J > 0, K < 0J > 0, K > 0J < 0, K < 0J = 0, K < 0 − − − −
200 0 200 400 Θ z0 (K/eV) Θ z f i t ( K / e V ) Θ zfit,2 with constant µ fit Θ zfit,1 with calculated µ effz (T)J > 0, K < 0J > 0, K > 0J < 0, K < 0J = 0, K < 0 FIG. 2. Comparison of fitted renormalized Weiss constant˜Θ fit = Θ fit /λ for fitting functions ˜Θ z fit,1 with temperature-dependent µ eff ( T ) (Eqs. 5 and 6) and ˜Θ z fit,2 with constant µ fit (Eq. 1) vs. the intrinsic renormalized Weiss constant ˜Θ =Θ /λ (Eq. 7), over a wide range of parameters. The fitted˜Θ z fit,1 agree much better with the intrinsic ˜Θ . Hamiltonian H CF + H SO + H U . For specific cases of trigo-nal and tetragonal distortions, analytical expressions arealso available [52, 57]. The C-W constants can then beextracted by fitting Eq. (5) to the measured χ α ( T ).In what follows we demonstrate this procedure for thecase of octahedral transition metal ions with trigonalsymmetry, where the t g electron level is split into an a g singlet and an e g doublet with a splitting equal to3∆ = E ( a g ) − E ( e g ). Fig. 1(b) illustrates the temper-ature dependence of µ eff ( T ) for ∆ = 0 and in Figs. 1(c)and (d) we show µ eff ( T ) as a function of ˜ T = T /λ fordifferent values of ˜∆ = ∆ /λ considering only the t g or-bitals. For Ru , we take λ = 0 .
15 eV, while for Ir , wetake λ = 0 . d compounds, for all ori-entations of the magnetic field. This implies that C ( T )and χ ( T ) are anomalously enhanced with increasing tem-perature entirely due to local van Vleck contributions.As we show next, if such data is fitted with a conven-tional Curie-Weiss form, it leads to large Curie constants C > g S ( S + 1) / (3 k B ) and anomalously antiferromag-netic Weiss temperatures Θ compared to Eq. (7).In order to benchmark the standard C-W function ver-sus the improved Eq. (5), we analyze two-site t g -onlyHubbard models for edge-sharing octahedra with the fieldoriented perpendicular to the plane of the bond [ i.e. par-allel to the cubic z -direction, for the canonical Z-bonddefined in Fig. 1(a)]. We then consider a range of pa-rameters with ∆ /λ ∼ − . . t /λ ∼ t /λ ∼ − . U/λ ∼ . J H /λ ∼ .
75, corresponding to Ir. The conclusionsbelow are also valid for parameter values correspondingto Ru. For each set of hoppings, we first compute theprecise low-energy couplings via the projection indicatedin Eqs. (3) and (4). In terms of the cubic ( x, y, z ) coordi- nates [Fig. 1(a)], the exchange couplings J ij are conven-tionally parametrized [10, 38] as: J ij = J Γ Γ (cid:48) Γ J Γ (cid:48) Γ (cid:48) Γ (cid:48) J + K . (8)From these, we obtain via Eq. (7) the intrinsic Weissconstant Θ z = [ g ab (8Γ (cid:48) − J +2 K )) − g ab g c ∗ ( − Γ+Γ (cid:48) + K ) − g c ∗ (2Γ + 4Γ (cid:48) + 3 J + K )] / [12 k B (2 g ab + g c ∗ )]where g ab and g c ∗ are the g -tensor components in the ab -plane and along c ∗ . We then compute χ z ( ˜ T ) via fulldiagonalization of H tot (Eq. 2) on the cluster, and fitit within the region from ˜ T = 800 K/eV to 1500 K/eV,which corresponds to 300 ∼
600 K for iridates and 120 ∼
220 K for α -RuCl . The results are shown in Fig. 2, wherewe compare two fitting procedures. The first fit to χ z ( ˜ T ),yielding ˜Θ z fit , (Θ z fit , /λ ), uses the improved Eq. (5) thatincludes the temperature-dependent µ eff ( T ) (determinedas described in the previous paragraph). The second fitfunction, yielding ˜Θ z fit , (Θ z fit , /λ ), is the standard Curie-Weiss law, with µ eff being a temperature-independentfitting constant. In all cases, we set χ α = 0. We findthat ˜Θ z fit , < ˜Θ z over the entire range of parameters, withdeviations from the intrinsic Θ z as large as ∼ −
120 K forIr and ∼ −
50 K for Ru. In comparison, ˜Θ z fit , does notdeviate nearly as strong from the intrinsic Θ z .Having validated the use of Eq. (5) for a model system,we now turn to the experimental susceptibilities of the d Kitaev candidate materials A IrO ( A = { Na, Li } ) and α -RuCl . In each case, we make a global fit to data in the c ∗ axis and ab -plane [defined in Fig. 1(a)] using Eq. (5)with five fitting parameters: χ c ∗ , χ ab , Θ c ∗ , Θ ab , and ∆.Note that standard Curie-Weiss fits for these materialsemployed six free parameters. The effective moments µ α eff ( T, ∆ , λ ) were computed via exact diagonalization of H CF + H SO + H U on a single site in each case [as shownpreviously in Fig. 1(c) and (d)]. For practical applica-tions, approximative analytical expressions [52, 57] for µ eff may be alternatively used.The fitting results are presented in Fig. 3. For eachcompound, we show fitted Θ c ∗ and Θ ab as a function ofcrystal field ∆, together with (1 − R ) to indicate thequality of the fit. Below, we discuss the fitted Weissconstants for each compound and their implications forthe microscopic couplings by recalling:Θ ab = − k B [ J + 13 K −
13 (Γ + 2Γ (cid:48) )] , (9)Θ c ∗ = − k B [ J + 13 K + 23 (Γ + 2Γ (cid:48) )] . (10)For Na IrO , we refit the susceptibility data fromRef. 17 over the range 150 – 300 K [see Figs. 3(a) and (d)].A standard Curie-Weiss fit yields Θ ab fit , = −
259 K andΘ c ∗ fit , = −
90 K, which are unlikely to be accurate. Micro-scopic considerations [10, 11, 38] suggest that Γ >
0, so ( − R ) x ∆ (eV) −160−80 0 80 (c) RuCl Θ α ( K ) ( − R ) x ∆ (eV) −160−80 0 80 (b) α −Li IrO Θ α ( K ) ( − R ) x ∆ (eV) −150−100−50 0 50 100 (a) Na IrO Θ α ( K ) Θ ab Θ c* (f) RuCl T (K) χ ( − c m / m o l ) (e) α −Li IrO T (K) χ ( − c m / m o l ) (d) Na IrO T (K) χ ( − c m / m o l ) χ abexp χ c*exp χ abfit χ c*fit FIG. 3. Fits to experimental χ ( T ) using Eq. (5). (a-c):Best fit Weiss constants as a function of crystal field ∆, with1 − R shown in green indicating fit quality. Vertical dashedlines mark the best overall fits. (d-f): Experimental datafrom Ref. [17, 20, 59] together with best overall fit. For eachmaterial, | χ α | < . · − cm /mol, thus influencing the fitsnegligibly. The Weiss constants obtained with Eq. (5) differsignificantly from conventional Curie-Weiss analysis neglect-ing the T -dependent µ eff ( T ). the finding of Θ ab < Θ c ∗ would require very large Γ (cid:48) < ab-initio calculations[36, 38, 60] and RIXS experiments [61, 62]. Using theimproved Eq. (5), we instead obtain Θ ab fit , = −
71 Kand Θ c ∗ fit , = −
75 K. The global best fit corresponds to3∆ = −
156 meV [indicated in Fig. 3(a) by a dashed line],which is compatible with the estimate of | | ∼
170 meVfrom RIXS [18]. The revised Weiss constants are re-duced in magnitude and nearly isotropic, indicating thatthe anomalous susceptibility anisotropy in this tempera-ture range likely results from µ eff , i.e. from the g -tensoranisotropy due to the local trigonal distortion. Assumingthe largest nearest-neighbor coupling to be ferromagnetic K <
0, the antiferromagnetic sign of the Weiss constantsmay be explained by further-neighbor antiferromagnetic(Heisenberg) couplings, as previously anticipated for thiscompound [38, 63]. Along this line, we note that therevised Weiss constants are in better agreement with arecently proposed model featuring such couplings fromRef. 40 that was inspired by analysis of RIXS measure-ments (for which Θ ab = −
73 K, Θ c ∗ = −
116 K).Turning to α -Li IrO , a standard Curie-Weiss fit ofthe reported susceptibility data [20] yields Θ ab fit , = −
52 Kand Θ c ∗ fit , = −
459 K. In contrast, for the modified Eq. (5),the revised Weiss constants are Θ ab fit , = +6 K and Θ c ∗ fit , = −
21 K, which are significantly reduced. The global bestfit over a temperature range 150 – 300 K correspondsto 3∆ = +96 meV [see Figs. 3(b) and (e)]. Consider-ing Eq. (9) and (10), this relatively small magnitude ofthe Θ-values may be related to a competition betweendifferent couplings, i.e. a ferromagnetic Kitaev coupling
K <
0, and competitive antiferromagnetic Heisenbergterms ( i.e. K ∼ − J ). The enhanced anisotropy com-pared to Na IrO may indicate relatively larger Γ, Γ (cid:48) couplings. All of these suggestions are consistent withprevious ab-initio estimates [38], and place α -Li IrO in a region of the J - K -Γ-Γ (cid:48) phase diagram [10] consis-tent with the experimentally observed incommensurateordered state [64].For α -RuCl , single crystal susceptibility data from[59] is fitted over the temperature range 175 – 400 K.A standard Curie-Weiss fit with constant µ eff yieldsΘ ab fit , = +35 K, Θ c ∗ fit , = −
129 K, in line with previousreports [41–45]. For the modified Eq. (5), the global bestfit corresponds to 3∆ = +51 meV [see Figs. 3(c), and(f)], which agrees surprisingly well with recent analysisof RIXS data in Ref. 65, and Raman scattering and in-frared absorption data in Ref. 66. For this case, the fittedWeiss constants are Θ ab fit , = +55 K and Θ c ∗ fit , = +33 K.These values differ significantly in terms of both mag-nitude and anisotropy from most previous reports (ex-cluding Ref. [65]). However, they are compatible withthe suggested ranges of parameters estimated from ab-initio approaches [38, 67–71], employing Eq. (9) and (10).The overall scale of the couplings also accords with thesaturation of nearest-neighbor spin correlations around T ∼ Θ ∼
35 K, as measured via optical spectral weightfor spin-dependent transitions [72].Assuming that the revised Θ-values are more accurate,we consider their full implications for α -RuCl , as it isthe most intensively studied compound. For this mate-rial, a broad inelastic neutron scattering response rem-iniscent of the Kitaev spin-liquid ground state was re-ported for 40 K < T <
100 K in Ref. 30. This was dis-cussed in terms of T H ∼ Θ ∼
100 K, where T H is anenergy scale associated with the Majorana spinon band-width. However, if the true interaction scale is muchsmaller than these estimates, then this range would in-stead correspond to the thermal paramagnet ( T >
Θ),where a relatively wide range of couplings can producea response similar to the experiment [73]. Similarly, inRef. 29 the temperature dependence of the Raman scat-tering intensity for 25 K < T <
300 K was shown to becompatible with fermionic statistics of the Majorana cita-tions of the Kitaev model. However, the data was mod-elled with K ∼
10 meV, corresponding to Θ ∼
30 K.Evidently, the majority of the data falls in the thermalparamagnet regime, where coherent magnetic quasipar-ticles with well-defined statistics are unlikely to persist.In summary, we have investigated the failure of thestandard Curie-Weiss law for several Kitaev candidatematerials with strong spin-orbit coupling. For suchmaterials, additional temperature-dependent van Vleck-like contributions always appear, with the lowest-ordercontribution providing an anisotropic and temperature-dependent effective moment. Failure to account for thiseffect in fitting of experimental susceptibility yields Weissconstants that are not representative of the underlyingmagnetic couplings. We therefore proposed and validateda modified formula that accounts for µ eff ( T ). The latterquantity may be estimated either via exact diagonaliza-tion of a local model Hamiltonian, or from analyticalexpressions [52, 57] when available. This was appliedto various j / honeycomb materials with d filling, andshown to resolve several previous apparent discrepanciesbetween χ ( T ) and other experiments. We conclude thatsome previous reports likely overestimated the scale ofthe magnetic couplings and possibly the degree of mag-netic frustration. For other classes of materials, and otherfillings, different deviations may be expected and must beconsidered. This work should aid in the improved anal-ysis of experimental χ ( T ), as a first characterization ofnovel quantum magnets. Acknowledgement .— We thank A. Loidl, A. Tsirlinand P. Gegenwart for discussions and for providingdata for α -RuCl and A IrO . We also thank I.I.Mazin for useful comments. RV, DAK and KR ac-knowledge support by the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) for fundingthrough Project No. 411289067 (VA117/15-1) and TRR288 — 422213477 (project A05). 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Electronic Hamiltonian
The Coulomb terms of H tot (Eq. 1 in the main text)are given by [38]: H U = U (cid:88) i,a n a, ↑ n i,a, ↓ + ( U (cid:48) − J H ) (cid:88) i,a
For the theoretical derivation of the modified Curie-Weiss law we set the independent background χ α = 0.To first determine an expression for the generalized sus-ceptiblity χ η , consider a Hamiltonian H = H + η H η ,where H is independent of η . The susceptibility of anobservable O with respect to η is defined as χ η = − ∂ (cid:104)O(cid:105) ∂η ,where (cid:104)O(cid:105) = Tr[ e − β H O ] / Tr[ e − β H ] is the thermodynamicexpectation value. We assume that O itself has no ex-plicit dependence on η . In general, the susceptibility canbe computed from: χ η = − β (cid:104)H η (cid:105)(cid:104)O(cid:105) + (cid:90) + ∞−∞ dω (cid:18) − e − βω ω (cid:19) C H η , O ( β, ω ) . (S8) C H η , O ( β, ω ) are temperature-dependent dynamical cor-relation functions: C H η , O ( β, ω ) ≡ Z (cid:88) n,m e − βE n (cid:104) n |H η | m (cid:105)(cid:104) m |O| n (cid:105)×× δ [ ω − ( E m − E n )] . (S9)where | n (cid:105) , | m (cid:105) are eigenstates of H with energies E n , E m ,and Z is the partition function.In the case where | n (cid:105) , | m (cid:105) are also eigenstates of either O or H η (i.e. [ H , O ] = 0 and/or [ H , H η ] = 0), then thecorrelation function is finite only at ω = 0. In this case,the susceptibility reduces to: χ η = β ( (cid:104)H η O(cid:105) − (cid:104)H η (cid:105)(cid:104)O(cid:105) ) . (S10)However, for general operators O and H , this formuladoes not hold. Finite-frequency corrections to Eq. (S10)include e.g. van Vleck paramagnetic contributions to themagnetic susceptibility of materials with significant spin-orbit coupling, which we discuss in more detail below.In general, the Zeeman operator is given by: H Z = − h · M ; M = (cid:88) i g s ˜ S i + g L L i . (S11)Here ˜ S i denotes the pure spin angular momentum, in con-trast to the pseudospin S i . The magnetic susceptibilitytensor is then: χ αβh = − lim h → ∂ (cid:104) M α (cid:105) ∂h β , (S12)where α, β ∈ { x, y, z } . Since g s (cid:54) = g L , eigenstates ofthe total angular momentum ( J = L + ˜ S i ) are gener-ally not eigenstates of the Zeeman operator for systemswith unquenched orbital angular momentum. It is use-ful to divide the states into (i) low-energy states, withenergies ω ≈
0, which are described by the low-energyspin Hamiltonian, and (ii) high-energy states, with ener-gies ω (cid:38) λ . Let P be the projection operator onto thelow-energy space, and let Q = 1 − P project onto thehigh-energy space. The dynamical correlation functionscan then be divided into two contributions: C M α ,M β ( β, ω ) = C (+) α,β ( β, ω (cid:38) λ ) + C ( − ) α,β ( β, ω ≈ . (S13)The contribution from low-frequency correlations can becomputed within the low-energy theory; in the high-temperature limit, it is: (cid:90) ∞−∞ dω (cid:18) − e − βω ω (cid:19) C ( − ) α,β ( ω ) ≈ β Tr[ e − β H eff P M α P M β P ] , (S14)with P M α P = (cid:80) i,β g αβi S βi and H eff as defined in themain text. Expanding the exponential for large temper-atures gives: lim h → β Tr[ e − β H eff P M α P M β P ] ≈ β S ( S + 1)3 (cid:88) i,γ g αγi g γβi − β (cid:18) S ( S + 1)3 (cid:19) (cid:88) iγjδ g αγi J γδij g δβj + O ( β ) . (S15)If this were the only contribution to the susceptibility, itwould be conventional to match these lowest order termswith the expansion of the Curie-Weiss law: CT − Θ ≈ Ck B β + Ck B Θ β + O ( β ) (S16)from which one would identify: C αβ = S ( S + 1)3 k B (cid:88) i,γ g αγi g γβi (S17)Θ αβ = − S ( S + 1)3 k B (cid:80) iγjδ g αγi J γδij g δβj (cid:80) i,γ g αγi g γβi . (S18)However, this does not account for the high frequencycontributions, i.e. van Vleck-like terms that arise frommixing of the low-energy states with high-energy states: C (+) α,β ( ω ) = (cid:80) n,m e − βE n (cid:104) n | P M µ Q | m (cid:105)(cid:104) m | Q M ν P | n (cid:105) Tr[ P e − β H P ] ×× δ [ ω − ( E m − E n )] . (S19)This represents two contributions. The first contributionis a temperature-dependent modification of the effective − − − −
200 0 200 400 Θ z0 (K/eV) Θ z f i t ( K / e V ) Θ A Θ B ~ ~ ~~ FIG. S1. Comparison of ˜Θ z fit to intrinsic Weiss constant ˜Θ z for two-site Z-bond model, with blue indicating fits employingthe temperature-independent ˜Θ A (Θ A /λ ), given in Eq. (S25),and orange indicating ˜Θ B (Θ B /λ ), given in Eq. (S26). Theformer function performs slightly better. magnetic moment at each site, due to field-induced mix-ing of different spin-orbital states. The second contri-bution is a similar modification to the effective intersiteinteractions. To distinguish these, we further subdividethe high-frequency correlations into single and multi-sitecorrelations: (cid:90) ∞−∞ dω (cid:18) − e − βω ω (cid:19) C (+) α,β ( ω ) ≡ (S20) β S ( S + 1)3 (cid:88) i Λ α,βi ( β ) − β (cid:18) S ( S + 1)3 (cid:19) (cid:88) ij Ω α,βij ( β ) + ... Here, Λ α,βi ( β ) gives the single-site contribution that re-mains in the limit where all intersite interactions ( e.g. hopping) are taken to zero. In contrast, β Ω α,βij ( β ) con-tains all corrections that result from two-site correlations, e.g. intersite interactions between excited j / and j / levels. The factors of β are introduced for convenience.With these contributions included, the Curie andWeiss terms are modified: C αβ ( T ) = S ( S + 1)3 k B (cid:88) i (cid:32) Λ αβi ( T ) + (cid:88) γ g αγi g γβi (cid:33) (S21)Θ αβ ( T ) = − S ( S + 1)3 k B (cid:80) ij (cid:16) Ω αβij ( T ) + (cid:80) γδ g αγi J γδij g δβj (cid:17)(cid:80) i (cid:16) Λ αβi ( T ) + (cid:80) γ g αγi g γβi (cid:17) . (S22)Evidently, the van Vleck corrections render both theCurie and Weiss constants temperature dependent, whichmay complicate the estimation of the low-energy inter-actions J γδij from temperature-dependent susceptibilities.Restricting now to diagonal susceptibilities (i.e. α = β ),the modified Curie-Weiss law is written: χ αh = C α ( T ) T − Θ α ( T ) (S23) C α ( T ) = N s [ µ α eff ( T )] k B . (S24)As discussed in the main text, µ α eff ( T ) can be estimatedfor single sites. However, Ω αβij ( T ) is unknown a-priori,so we have considered two approximations for the Weissterm:Θ αA ( T ) = Θ α = − S ( S + 1)3 k B (cid:80) iγjδ g αγi J γδij g δαj (cid:80) iγ g αγi g γαi , (S25)Θ αB ( T ) = (cid:18) µ eff (0) µ eff ( T ) (cid:19) Θ α . (S26)In Fig. S1, we compare the performance of these ap-proximations for a two-site model of the Z-bond of edge-sharing octahedra with d filling and a j eff = 1 / J αβij were extracted by numerical projection tothe low-energy space, and used to compute the intrin-sic Θ z . The susceptibility was then computed exactly,and fit with Eq. (S23), using the two approximations forΘ z ( T ). For this case, we find that both approximationsyield similar values, with Θ A performing better over theparameter range. This suggests that the major devia-tions are due to the temperature-dependence of the Curieconstant, rather than the Weiss constant.Note that in Eq. (7) of the main text and Eqs. (S18),(S22) and (S25), the expressions for Θ may be signif-icantly simplified for field directions that are principalaxes of the g -tensor. In particular, if the g -tensor is di-agonal in the basis, the Weiss constant becomes indepen-dent of the g -tensor,Θ α = S ( S + 1)3 k B (cid:88) ij J ααij . (S27)In the present case of Kitaev materials, two coordinatesystem were used: The cubic axes x , y , z and the crys-tallographic axes ˆ a = ( x + y − z ) / √
6, ˆ b = ( y − x ) / √ c ∗ = ( x + y + z ) / √
3. The g -tensor is approximately di-agonal in the a, b, c ∗ basis, while the couplings J ij areusually expressed in the x, y, zx, y, z