Hopping induced ground-state magnetism in 6H perovskite iridates

TL;DR

This study reveals that in 6H perovskite iridates, substantial intra-dimer hopping modifies spin-orbit states, leading to unexpected magnetism despite strong SOC, and suggests tuning Ir-Ir distances could enable quantum criticality.

Contribution

It demonstrates that hopping significantly influences low-energy physics in iridates, challenging the assumption that strong SOC dominates and highlighting the importance of intra-dimer interactions.

Findings

Hopping reduces spin-orbit excitation energies in 6H iridates.

Magnetism observed contrary to expected non-magnetic ground state.

SOC is effectively scaled down to ~0.26 eV, enabling potential quantum criticality.

Abstract

Investigation of elementary excitations has advanced our understanding of many-body physics governing most physical properties of matter. Recently spin-orbit excitons have drawn much attention, whose condensates near phase transitions exhibit Higgs mode oscillations, a long-sought physical phenomenon [Nat. Phys. {\bf 13}, 633 (2017)]. These critical transition points resulting from competing spin-orbit coupling (SOC), local crystalline symmetry and exchange interactions, are not obvious in Iridium based materials, where SOC prevails in general. Here, we present results of resonant inelastic x-ray scattering on a spin-orbital liquid Ba$_3$ZnIr$_2$O$_9$ and three other 6H-hexagonal perovskite iridates which show magnetism, contrary to non-magnetic singlet ground state expected due to strong SOC. Our results show that substantial hopping between closely placed Ir$^{5+}$ ions within Ir$_2$O$_9$ dimers in these 6H-iridates, modifies spin-orbit coupled states and reduces spin-orbit excitation energies. Here, we are forced to use at least a two-site model, to match the excitation spectrum going in line with the strong intra-dimer hopping. Apart from SOC, low energy physics of iridates is thus critically dependent on hopping, and may not be ignored even for systems having moderate hopping, where the excitation spectra can be explained using an atomic model. SOC which is generally found to be 0.4-0.5~eV in iridates, is scaled in effect down to $\sim$0.26~eV for the 6H-systems, sustaining the hope to achieve quantum criticality by tuning Ir-Ir separation.

Tables

Table 1. Table 1: Effective values of hopping t eff subscript 𝑡 eff t_{\textup{eff}} estimated from DFT calculated values of bandwidth W = 2 ​ z ​ t eff 𝑊 2 𝑧 subscript 𝑡 eff W=2zt_{\textup{eff}} without SOC. z 𝑧 z =12 for Ba 2 YIrO 6 , and W 𝑊 W has no effect of distortions in IrO 6 as Δ CF NC superscript subscript Δ CF NC \Delta_{\textup{CF}}^{\textup{NC}} = 0. For the 6H-iridates, z 𝑧 z =2, and Δ CF NC superscript subscript Δ CF NC \Delta_{\textup{CF}}^{\textup{NC}} also contributes.

System	Bandwidth $W$ (eV)	$t_{\text{eff}}$ (eV)
Ba2YIrO6	1.49	0.06
Ba3MgIr2O9	1.46	0.36
Ba3ZnIr2O9	1.22	0.31
Ba3CaIr2O9	1.38	0.34
Ba3SrIr2O9	1.23	0.31

Table 2. Table 2: Ranges of physical parameters in the two-site model (equation (1) of main text and the matrices in Sections V and VI), using which the low-energy RIXS spectra could be fit for each system. The values given in italics are used for obtaining the energy-losses in Fig.  4 . We have kept U d subscript 𝑈 𝑑 U_{d} as 2.0 eV for all calculations.

System	${\Delta }_{\text{CF}}^{\text{NC}}$ (eV)	$t_{\text{dim}}$ (eV)	$J_{\text{H}}$ (eV)	$\lambda$ (eV)
Ba3MgIr2O9	$\Delta$=0.025	$t_1$=0.16; $t_2$=0.041	0.35-0.46	0.26
			0.44-0.51	0.27
			0.48-0.55	0.28
Ba3ZnIr2O9	$\Delta$=0.02	$t_1$=0.15; $t_2$=0.045	0.42-0.49	0.26
Ba3CaIr2O9	${\Delta }_2$=0.044, ${\Delta }_1$=0.01	$t_1$=0.01; $t_2$=0.001; $t_3$=0.007; $t_4$=0.0; $t_5$=$t_3$	0.45-0.5	0.26
Ba3SrIr2O9	${\Delta }_2$=0.028, ${\Delta }_1$=0.01	$t_1$=0.013; $t_2$=0.003; $t_3$=0.015; $t_4$=0.002; $t_5$=$t_3$	0.43-0.5	0.25
			0.54-0.58	0.26

Equations

$$H=\sum _{\begin{array}{c}i=1,2\end{array}}(H_i^{{\Delta }_{\mathrm{CF}}^{\mathrm{NC}}}+H_i^{\mathrm{int}}+H_i^{\mathrm{SO}})+H^t,$$

$$H$$

$$+$$

$$\mathrm{For}\mathrm{Mg}\mathrm{and}\mathrm{Zn},H^{{\Delta }_{\mathrm{CF}}^{\mathrm{NC}}}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& \Delta \end{array}\right],\mathrm{for}\mathrm{Ca},\left[\begin{array}{ccc}{\Delta }_1& 0& 0\\ 0& {\Delta }_2& 0\\ 0& 0& 0\end{array}\right],\mathrm{and}\mathrm{for}\mathrm{Sr},\left[\begin{array}{ccc}0& 0& 0\\ 0& {\Delta }_2& 0\\ 0& 0& {\Delta }_1\end{array}\right].$$

$$\mathrm{For}\mathrm{Mg}\mathrm{and}\mathrm{Zn},H^t=\left[\begin{array}{ccc}-t_1& 0& 0\\ 0& -t_1& 0\\ 0& 0& -t_2\end{array}\right],\mathrm{for}\mathrm{Ca}\mathrm{and}\mathrm{Sr},\left[\begin{array}{ccc}{t}_1& -t_2& t_4\\ t_2& -t_3& t_4\\ -t_4& t_4& t_5\end{array}\right]$$

Full text

Competition between lattice, spin and orbital degrees of freedom is often responsible for remarkable properties in condensed-matter systems rau2016arc . While atomic spin-orbit coupling is quenched in light transition metal oxides, it starts influencing physical properties in 4$d$ or 5$d$ systems kusch2018prb ; nag2018prbbyio ; agrestini2018prb . In these materials, spin-orbit coupling (SOC or $\lambda$) manifests itself by splitting wide $d$ bands and effectively enhancing electronic correlations resulting in Mott-insulating states akin to 3$d$ oxides kim2008prl ; imada1998rmp . A magnetic condensation of excitations across spin-orbit-coupled states (SOC-states) in some of these Mott-insulators ($d^{4}$ systems, whose ground states are expected be non-magnetic SOC $J$=0 ($J_{0}$) singlets, with $S$,$L$ nominally equal to one), was predicted by Khaliullin khaliullin2013prl , of which Ca2RuO4 has been an exemplary case jain2017np . With the free-ion SOC itself being small in the 4$d^{4}$ oxides like Ca2RuO4 jain2017np ; souliou2017prl , SOC-states are readily perturbed by non-cubic crystal-fields ($\Delta_{\textup{CF}}^{\textup{NC}}$) around the 4$d^{4}$ ions, allowing access to the magnetic $J$=1 ($J_{1}$) triplet states. In Ir5+ systems however, the prediction seems to have stayed experimentally unclear and debated on account of stronger SOC that raises the excitation energies (0.37 eV) compared to the inter-site exchange interactions kusch2018prb ; nag2018prbbyio . In addition, it has been shown recently that even a large $\Delta_{\textup{CF}}^{\textup{NC}}$ (0.325 eV), fails to compete with strong SOC to close the spin-orbit excitation gap comprehensively agrestini2018prb .

Interestingly, in these systems either the Ir5+ ions are far apart intervened by closed shell ions (Fig. 1) kusch2018prb ; nag2018prbbyio , or are within corner/edge-shared octahedral geometries du2013epl . In contrast, significant magnetic responses are obtained from 6H-hexagonal perovskite iridates Ba3$M$Ir2O9, ($M$ = Mg, Zn, Ca, and Sr), where the Ir5+ ions form face-sharing octahedral dimeric units (Fig. 1(a)) nag2016prl ; sakamoto2006jssc ; nag2018prbbmio . Since each Ir5+ ion in such a situation ‘sees’ another Ir5+ ion as its nearest neighbour, hopping between them via O2- ions becomes important nag2018prbbmio ; kugel2015prb . In addition to these three Ir-O-Ir superexchange pathways, the small intra-dimer Ir-Ir distances (about 2.7Å) nag2018prbbmio provide for direct exchanges between the Ir*5+*s. For example, bandwidths ($W=2zt_{\textup{eff}}$) obtained from density functional theory (DFT) without the influence of SOC, give the effective hopping $t_{\textup{eff}}$ to be 0.06 eV in double perovskite (DP) Ba2YIrO6 while in 6H Ba3ZnIr2O9 it is around 0.31 eV (see Supplementary Material (SM)) SM .

Resonant Inelastic X-Ray Scattering (RIXS), a second order “photon-in, photon-out” spectroscopic technique has emerged to be particularly suited to study the hierarchy of these SOC-states in iridates ament2011prbr ; sala2014prl ; kim2017prb ; paremkanti2018prb . The process is sensitive to the changes in energy, momentum and polarization of inelastically scattered x-ray photons representing elementary excitations in a material. With the development of high-resolution spectrometers sala2018jsr , it is now possible to resolve low-energy excitations to such SOC-states in a RIXS experiment kusch2018prb ; yuan2017prb . The low-energy spin-orbit excitonic process for an atomic scenario is depicted in the inset of Fig. 1(a) and explained below. Strictly within an atomic model with strong octahedral crystal-field and SOC, 5$d^{4}$ electrons in $t_{2g}$ orbitals have their spins $S$=1 coupled to orbital angular momenta $L$=1, forming nominally non-magnetic SOC-states of $J$=0 ($J_{0}$). The incoming photons with energy $\omega_{i}$ excite the system from $|i\rangle=|2p^{6}t_{2g}^{4}\rangle$ initial state to an intermediate $|n\rangle=|2p^{5}t_{2g}^{5}\rangle$ state with core-holes. This excited state then decays into a final state $|f\rangle=|2p^{6}t_{2g}^{4\ast}\rangle$ different from the initial state in terms of the arrangement of the SOC electrons ($\frac{6!}{4!2!}-1$ possibilities), and in the process emits photons of energy $\omega_{1}$,$\omega_{2}$ and so on. The energy losses $\Delta\omega=\omega_{\rm i}-\omega_{1},\omega_{2}$, then correspond to the energy costs for the different arrangements of the electrons within the SOC $t_{2g}$ orbitals. Even though RIXS has already been employed to study spin-orbit excitations in DP Ir5+ systems, marginal inter-site hopping has had an undetectable effect on the RIXS spectra which could be interpreted by a bare atomic model nag2018prbbyio ; kusch2018prb ; yuan2017prb ; SM . In contrast, we here show using RIXS that enhancement in inter-site hopping due to close placement of Ir5+ ions drives the 6H-iridates (Ba3$M$Ir2O9 ($M$ = Mg, Zn, Ca, and Sr)) into a regime where the effective SOC strength is weakened and the ground state deviates from non-magnetic singlets (Fig. 1). This is the first observation of spin-orbit excitons having such low energies in Ir5+ systems kusch2018prb ; nag2018prbbyio . Since an atomic model fails SM , particularly to explain a low-energy peak around 0.2 eV and a two-peak feature around 0.4 eV as shown in Fig. 1 (b), we implement exact diagonalisation of a minimal two-site model including the influences of physical parameters: $\lambda$, $\Delta_{\textup{CF}}^{\textup{NC}}$, Hund’s coupling $J_{\textup{H}}$, and intra-dimer hopping ($t_{\textup{dim}}$), to map the low-energy excitations in RIXS spectra. We find that the observed features are characteristic of a two-site model with a finite inter-site hopping (see Fig. 1). This is not surprising given the fact that the Ir-ions in these 6H-iridates interact strongly with their nearest neighbours forming Ir-Ir dimers nag2016prl ; nag2018prbbmio , in comparison to DP iridates where the Ir-ions have little interaction. Strangely however, we find that $\lambda$, usually considered to be 0.4-0.5 eV for iridates, is scaled down effectively to $\sim$0.26 eV in the presence of strong $t_{\textup{dim}}$. Moderate hopping in other systems like Ba2YIrO6, may similarly be suspected to rescale the atomic $\lambda$, even though imperceptible in RIXS nag2018prbbyio .

In order to characterise the spin-orbit excitons in presence of intra-dimer hopping (beyond capacity of single-particle theories), we have employed many-body multiplet formalism incorporated within the following two-site model Hamiltonian for the Ir-$t_{2g}$ orbitals matsuura2013jpsj ; bhowal2018prb :

[TABLE]

where $H_{i}^{\Delta_{\textup{CF}}^{\textup{NC}}}$, $H_{i}^{\textup{int}}$ and $H_{i}^{\textup{SO}}$ are the respective Hamiltonians representing the non-cubic crystal-field, Coulomb interactions and SOC for the $i^{\rm th}$ site. $H_{i}^{\textup{int}}$, known as Kanamori Hamiltonian, includes intra- ($U_{d}$) and inter-orbital ($U_{d}^{\prime}$) Coulomb interactions alongside Hund’s coupling ($J_{\textup{H}}$), and are related by $U_{d}=U_{d}^{\prime}+2J_{\textup{H}}$ (see SM for detailed expressions) SM . In addition to these on-site terms, we introduce $H^{t}$ to account for inter-site hopping $t_{ij}^{l\sigma,m\sigma^{\prime}}$ between the three $t_{2g}$ orbitals at the two Ir5+ sites $i$ and $j$. While within the atomic model SM , SOC gives rise to multiplet states $J_{0}$, $J_{1}$, $J_{2}$ … (see Fig. 1 (a)), in presence of strong $t_{\rm dim}$ these SOC-states are influenced by neighbouring atomic multiplet states forming two-site SOC-states (shown schematically in Fig. 1 (a)). The spin-orbit exciton energies can be obtained from the difference in energy eigenvalues calculated using exact diagonalization of the two-site model. Due to the inclusion of inter-site hopping in the two-site model, new sets of SOC-states appear out of the interaction between the atomic SOC-states at the two sites: $J_{0}$-$J_{0}$ (1), $J_{0}$-$J_{1}$ (6), $J_{1}$-$J_{1}$ (9), $J_{0}$-$J_{2}$ (10),$J_{1}$-$J_{2}$ (30),$J_{2}$-$J_{2}$ (25) etc. The new sets of two-site SOC-states are denoted according to their parent atomic SOC-states ($J_{0}$-$J_{0}$, $J_{0}$-$J_{1}$, etc.) and the numbers within the parentheses indicate the number of SOC-states having the same origin. Excitations from the $J_{0}$-$J_{0}$ SOC-states to the sets originating from the interaction between atomic SOC-states $J_{1}$-$J_{1}$ or $J_{0}$-$J_{2}$, form the inelastic features in the RIXS spectra. The fact that strong intra-dimer hopping plays a key role in modifying low-energy SOC excitation spectrum, is directly verified from the experimental results (Fig. 1 (b)).

In our calculation, we have assumed the hopping to be diagonal in the spin-space i.e., $t_{\rm dim}=t_{ij}^{l\sigma,m\sigma}$. Also we have supposed that the hopping parameters are diagonal and equal in the orbital space to illustrate the effect of different parameters on the energies of the spin-orbit excitons (Fig. 2). A trigonal crystal-field is also assumed in this case. However, more realistic hopping and non-cubic matrices guided by the first principles calculations are considered to fit the RIXS spectra of the 6H-iridates. It is important to note that the non-cubic crystal-field and the hopping terms take care of the non-local effects within the solid. Such a simple $t_{2g}$-only two-site model captures the essential features of these 6H-iridates owing to the dimeric interaction and large $t_{2g}$-$e_{g}$ splitting (allowing us to neglect the $e_{g}$ orbitals (see SM Fig. 2)) SM . Taking into account the $\Delta_{\textup{CF}}^{\textup{NC}}$ and $t_{\textup{dim}}$ (= $t_{ij}^{l\sigma,m\sigma^{\prime}}$), which are both set-up in part by O2- ions, also indirectly includes their effects in the model. Fig. 2 shows the influence of each of these physical parameters on the excitonic gaps within the two-site model calculation. In contrast to the atomic states, spin-orbit excitation energies obtained from the two-site model also depend on Coulomb repulsion $U_{d}$ since the superexchange energy varies as $\sim t_{\rm dim}^{2}/U_{d}$ in the latter case. It is clear from Fig. 2(a-d) that $\Delta_{\textup{CF}}^{\textup{NC}}$ and $t_{\textup{dim}}$ control spread of each set of states, while their mean energies are primarily dictated by $J_{\textup{H}}$ and $\lambda$. An interesting feature in the spin-orbit exciton energy spectrum is crossover between $J_{0}$-$J_{2}$ and $J_{1}$-$J_{1}$ states as we go from $J_{\textup{H}}<<\lambda$ to the $J_{\textup{H}}>>\lambda$ limit SM . Even though this two-site model is a simplistic representation of processes in a real solid, it captures the salient features of the low-energy RIXS spectra of the 6H-perovskite iridates as shown ahead. Most importantly, as seen from Fig. 2(e) that for $\lambda$ = 0.25 eV (relevant for the 6H-iridates), the two-site ground state deviates from $J~{}=~{}0$ state by about 10% and as a consequence expectation value $<J^{2}>$ of the ground state attains a non-zero value and hence small moments.

High-resolution (0.029 eV at Ir-$L_{3}$-edge) RIXS spectra of the 6H-iridates (Ba3$M$Ir2O9, $M$ = Mg, Zn, Ca, and Sr) collected at 20 K at the ID20 beamline of ESRF – The European Synchrotron at 20 K sala2018jsr , are shown in Fig. 3(a-d), all of which show similar low-energy inelastic features unlike the RIXS from DPs like Ba2YIrO6 (See Fig. 1(b)) exhibiting the strong influence of intra-dimer hopping in these systems. The inelastic features of each spectrum, are fitted with four Gaussian line-shapes representing excitations from the $J_{0}-J_{0}$ state to higher lying two-site SOC states (see SM for details) SM . Since excitations to the set of SOC-states $J_{1}-J_{1}$ and $J_{2}-J_{0}$ are not clearly resolved (strongest splitting is seen for Ba3SrIr2O9, Fig. 3(d)), the fits to these two sets of peaks are done by limiting their Gaussian variances to 0.05 eV. A Gaussian tail from the high-energy $t_{2g}$-$e_{g}$ excitation peak is also included in the fitting, although its effect on the 4-SOC peaks is minimal (Fig. 3(e)). Although, excitations within the higher-energy states cannot be ruled out given their strong intermixing, these excitations would have comparatively weak intensities and are not resolved in the present experiments. A broad continuum of excitations can be seen (Fig. 3(e)) above 0.7 eV in all the systems which may be related to low-intensity transitions to $J_{2}-J_{2}$ states besides multiparticle excitations kim2017prb ; paremkanti2018prb . We note that the spectral weights of the $J_{2}-J_{2}$ excitation features ($\sim 0.75$ eV) are substantially suppressed in spite of large number of states for all the 4 systems again showing significant hopping induced deviation of these states from the pure atomic limit (see SM Fig. 5)) SM .

The peak positions and their widths extracted from the fitting of the energy-loss features are shown in Fig. 3(f). The slight variation in the peak positions and widths are related to the details of the local geometry of these four iridates. This becomes evident when we try to place the difference in eigenvalues obtained from the exact diagonalization of the two-site model Hamiltonian given in equation (1) with different parameters, within the FWHMs of these peaks (SM Fig. 4) SM .

For the two smaller cations Mg2+ (0.72 Å) and Zn2+, (0.74 Å) the compounds Ba3MgIr2O9 and Ba3ZnIr2O9 stabilize in a $P{\rm 6}$3/$mmc$ space group, while Ca2+ (1.00 Å) and Sr2+ (1.18 Å) allow formation of Ba3CaIr2O9 and Ba3SrIr2O9 only within a lower symmetry space group $C2/c$ nag2016prl ; nag2018prbbmio ; sakamoto2006jssc ; shannon1976ac ; SM . Each of these 6H-iridates therefore possesses slightly modified IrO6 octahedral environment resulting in different $t_{\rm dim}$ and $\Delta_{\textup{CF}}^{\textup{NC}}$ values. We find that RIXS spectra for the 6H-iridates can be fitted over a small range of values for all the parameters (SM Table II) SM . A realistic estimation of hopping and non-cubic crystal-field energies for the different local geometries of the Ir5+ ions in the 6H-iridates is obtained by extracting the low-energy tight-binding model retaining only Ir-$t_{2g}$ in the basis from the muffin-tin orbital (MTO) based Nth order MTO (NMTO) method  NMTO1 ; NMTO2 ; NMTO3 as implemented in the Stuttgart code. While $P{\rm 6}$3/$mmc$ symmetry of Ba3MgIr2O9 and Ba3ZnIr2O9 leads to trigonal distortion of the IrO6 octahedra so that Ir-$t_{2g}$ orbitals split into singly degenerate $a_{1g}$ and $e_{g}^{\pi}$ states, monoclinic distortions in Ba3CaIr2O9 and Ba3SrIr2O9 remove completely the degeneracy of the $t_{2g}$ states and is reflected accordingly in the non-cubic crystal field matrices (see SM Section V). This results in diagonal hopping matrices for Ba3MgIr2O9 and Ba3ZnIr2O9, while Ba3CaIr2O9 and Ba3SrIr2O9 have relatively complicated forms with non-zero off-diagonal elements (see SM Section VI). The values obtained from the first-principles calculations are taken as initial guesses and are then renormalized to fit the RIXS spectra, while keeping the ratios between different elements of the matrices for hopping and non-cubic crystal-field fixed to that obtained from DFT. Parameters of the Hamiltonian which give energy-loss values within the FWHMs of the Gaussian fits to the RIXS peaks are then extracted as shown in SM Fig. 4 (a)-(d). In our calculations, we keep the value of $U_{d}$ fixed at 2 eV kusch2018prb .

The best matched energy-loss values of the two-site SOC-states calculated using a set of values for the parameters $\Delta_{\textup{CF}}^{\textup{NC}}$, $t_{\textup{dim}}$, $J_{\textup{H}}$ and $\lambda$ for each of the four systems, are shown as vertical bars in Fig. 3(a-d). The maximum components of $\Delta_{\textup{CF}}^{\textup{NC}}$ and $t_{\textup{dim}}$ matrices used for the fits are then compared to the corresponding values obtained using NMTO method on these systems in Fig. 4(a-d). We find that the parameters extracted from RIXS are scaled down w.r.t to the ones obtained from first principles. However, as can be seen from panels of Fig. 4(a-d), the reduction in values is not arbitrary and is qualitatively similar for the four 6H-iridates. It should be mentioned here, that overestimation of parameters from first principles is known and reported for honeycomb iridates with $d^{5}$ electronic configuration kim2016prl . The Hund’s coupling estimated from RIXS ($\sim 0.45$ eV) is similar for all four (see Fig. 4(e)), comparable to other 5$d^{5}$ systems but higher than the 5$d^{4}$ DP-iridates yuan2017prb ; kusch2018prb . However, most striking is the substantially suppressed value of SOC ($\sim 0.26$ eV) compared to any other iridate reported as yet, proving the decisive role of intra-dimer hopping in these systems. In this context, reduced covalency has recently been ascribed to increased SOC in iridium fluorides rossi2017prb . RIXS of these 6H-iridates provide first clear experimental evidences of atomic non-magnetic SOC $J$ = 0 state breakdown, in presence of solid state effects. The two-site model employed may thus be appropriate to describe the low-energy physics of other systems where the Ir-ions are placed close-by in the crystal structure takayama2018 ; streltsov2018prb . The polycrystalline nature of our samples and two-site theoretical modelling forbid estimation of the momentum dependence of the spin-orbit excitons presently and can be a future direction of work. The identification of inter-site hopping as a critical parameter which can drastically change the effective strength of SOC even so, may be utilised to tune magnetism in iridates.

Acknowledgements

SR and IDG thank Technical Research Center of IACS. SR also thanks Department of Science and Technology (DST) [Project No. WTI/2K15/74], UGC-DAE Consortium for Scientific Research, Mumbai, India [Project No. CRS-M-286] for support, and Jawaharlal Nehru Centre for Advanced Scientific Research from DST-Synchrotron-Neutron project, for performing experiments at ESRF (Proposal No. HC-2872).

I Sample synthesis:

Polycrystalline samples of Ba3$M$Ir2O9 ($M$ = Mg, Zn, Ca, and Sr) were synthesized by standard solid state reaction using stoichiometric amounts of BaCO3, MgO, ZnO, CaCO3, SrCO3, and IrO2 as starting materials SI_sakamoto2006jssc ; SI_nag2016prl ; SI_nag2018prbbmio . The structural parameters used for the calculations are given in earlier reports SI_nag2018prbbmio .

II Resonant Inelastic X-ray Scattering

Ir-$L_{3}$-edge RIXS on polycrystalline 6H-iridates were done at the I20 beamline of ESRF – The European Synchrotron at 20 K SI_sala2018jsr . Polarisation of the incident x-ray beam was kept parallel to the horizontal scattering plane and the scattered beam was collected at 90∘ w.r.t the incident beam. Energy of the incident x-ray was varied across Ir-$L_{3}$ absorption edge and the corresponding energy-loss spectrum was recorded (see Fig. 1). Low-energy features adjacent to the elastic peaks arising due to excitations across the SOC-states were found to be enhanced at an incident energy of 11.216 keV and was fixed for subsequent high resolution measurements. Low-resolution RIXS (0.280 eV) focussing on high-energy loss features for all the four systems are shown in Fig. 2. The high-energy loss features identified as $t_{2g}$-$e_{g}$ transitions and charge-transfer excitations from O 2$p$ to vacant Ir-orbitals were found to be similar to DP Ba2YIrO6 SI_sala2014prl ; SI_ishii2011prb . To distinguish the low-energy SOC-states, we collected RIXS spectrum on each sample with a resolution of 0.029 eV.

III Atomic model Hamiltonian:

An atomic many-body Hamiltonian for the $d^{4}$ configuration nag2016prl in presence of strong SOC gives rise to spin-orbit coupled states: $J_{0}$ (1), $J_{1}$ (3), $J_{2}$ (5), $J_{2}$ (5), and $J_{0}$ (1), where numbers within parentheses denote the number of SOC-states having same total angular momentum (see Fig. 1(a) of main text). The energy eigenvalues of these spin-orbit coupled-states are dictated by the interaction parameters such as Hund’s coupling $J_{\textup{H}}$, Coulomb interaction $U_{d}$ and SOC constant $\lambda$. The energy difference between these states are however independent of Coulomb interaction $U_{d}$. It is interesting to note that the explicit energies of the $J_{0}$ and $J_{2}$ states depend on the relative strength of the parameters $J_{\textup{H}}$ and $\lambda$ keeping the energy difference always fixed at $3\lambda/2$. In the limit $J_{\textup{H}}>>\lambda$, $E_{J_{0}}=E_{0}-4U+7J_{\textup{H}}-\lambda$ and $E_{J_{2}}=E_{0}-4U+7J_{\textup{H}}+\lambda/2$; for $J_{\textup{H}}<<\lambda$, $E_{J_{0}}=E_{0}-4U+7J_{\textup{H}}-2\lambda$ and $E_{J_{2}}=E_{0}-4U+7J_{\textup{H}}-\lambda/2$ SI_kim2016prl .

Such an atomic many-body Hamiltonian is sufficient to describe the RIXS spectra of the DP iridates SI_yuan2017prb ; SI_nag2018prbbyio ; SI_kusch2018prb ; SI_paremkanti2018prb with triplet ($J_{0}\rightarrow J_{1}$) and quintet ($J_{0}\rightarrow J_{2}$) spin-orbit excitons, however it fails to interpret the RIXS spectra for the present 6H-iridates (Ba${}_{3}M$Ir2O9, $M$ = Zn, Mg, Ca and Sr).

IV Two-site model Hamiltonian:

In order to decipher the low-energy RIXS spectra of these 6H-iridates we consider the following two-site many-body Hamiltonian for the Ir-$t_{2g}$ orbitals  SI_matsuura2013jpsj ; SI_bhowal2018prb ,

[TABLE]

where $i$,$j$ are site indices, $l$, $m$ are orbital indices and $\sigma,\sigma^{\prime}$ are spin indices. $H_{i}^{\Delta_{\textup{CF}}^{\textup{NC}}}$, $H_{i}^{\textup{int}}$ and $H_{i}^{\textup{SO}}$ are the respective Hamiltonians representing the non-cubic crystal-field, Coulomb interactions and SOC for the $i^{\rm th}$ site. The last term represents the hopping between two different sites $i$ and $j$. The parameters $\epsilon_{lm}$, $U_{d}$, $U_{d}^{\prime}$, $J_{\textup{H}}$, $\lambda$ and $t_{ij}^{l\sigma,m\sigma^{\prime}}$ are respectively the on-site energies of the $t_{2g}$ orbitals, intra and inter-orbital Coulomb interaction, Hund’s coupling, SOC and the hopping between nearest neighbour sites $i$ and $j$.

A crossover in energy between $J_{1}$-$J_{1}$ and $J_{0}$-$J_{2}$ SOC-states can be seen depending on the relative strength of the $J_{\textup{H}}$ and $\lambda$ (marked by arrows in Fig. 2(b,d) of the main paper). This is due to the fact that while $E_{J_{1}}$ has always the same analytical form, the energies of $J_{0}$ and $J_{2}$ SOC-states depend on relative strengths of the $J_{\textup{H}}$ and $\lambda$ as discussed earlier in Ref. SI_kim2016prl .

V Form of non-cubic crystal field matrices for the 6H-iridates

[TABLE]

VI Form of hopping matrices for the 6H-iridates

[TABLE]