Spin–orbit effects in pentavalent iridates: models and materials

The primary ingredient for the formation of J = 0 state is the strong SOC, typical for elements with large atomic number Z, where the electrons experience a strong nuclear potential, leading to unavoidable relativistic effect. This results in stronger SOC effects towards the end members of the periodic table. For example, in the transition metal (TM) series, the strength of the SOC gradually increases with an added n quantum number as we go down in the periodic table, viz, λ_5d > λ_4d > λ_3d , as also schematically illustrated in figure 1(a).

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Interestingly, going from 3d to 5d TM elements, the d orbitals get spatially more extended (see figure 1(b)). As a result, the wide 5d orbitals hybridize strongly with the neighboring ligand atoms, leading to a larger crystal field splitting in 5d TM compounds compared to the systems with narrow 3d bands. This has some interesting consequences, which play crucial role in the formation of J = 0 state in a d⁴ electronic configuration. For a simple illustration, we consider TM-O₆ octahedral unit. Indeed such units are quite common in various TM oxides, particularly in iridates. We consider an ideal octahedral symmetry, which, then, splits the five-fold d states into a triplet t_2g and a higher lying doublet e_g orbitals. First, let us consider the case when the TM has 3d orbitals in the valence state. Due to weak crystal field splitting (Δ) in these systems, the Hund's coupling J_H that favors parallel spin alignment of the electrons in different d orbitals win, and 3d⁴ electronic configuration leads to a high spin S = 2 state (${t}_{2g}^{3}{e}_{g}^{1}$), as seen for example in LaMnO₃ [79]. The SOC effects being rather weak, the crystal field split states are only very weakly perturbed by SOC. In case of 4d TM elements, however, the 4d orbitals being wider than the 3d orbitals, Δ is relatively larger in magnitude. In this case, Δ wins over the Hund's coupling J_H, resulting in a low spin state S = 1, with four electrons in the t_2g orbitals (${t}_{2g}^{4}{e}_{g}^{0}$) [80, 81]. The formation of the low spin state is the first step towards the J = 0 state. However, due to its moderate value in 4d TMs, often SOC acts only as a perturbation to the t_2g states. It is interesting to point outhere that the recent report of a J = 0 ground state in 4d based Ru compound K₂RuCl₆within LS coupling scheme points to the importance of SOC (beyond perturbation) also in 4d based systems [82]. Nevertheless, the remaining criteria of strong SOC is, however, satisfied in 5d TMs, where strong SOC forms spin–orbital entangled states, destroying the individual orbital and spin identities of the constituents.

We pause here and discuss the formation of these new spin–orbit entangled states within a single particle formalism. Assuming the t_2g states are completely decoupled from the higher lying e_g orbitals, the spin–orbit entangled states can be immediately visualized by diagonalizing the SOC Hamiltonian in the t_2g basis, viz,

$$\begin{equation}{\mathcal{H}}_{\text{SOC}}=\left(\begin{matrix}\hfill {\mathcal{H}}_{+}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathcal{H}}_{-}\hfill \end{matrix}\right),\end{equation} \tag{ 1 }$$

where ${\mathcal{H}}_{+}$ (${\mathcal{H}}_{-}$) in the basis {|yz↓⟩, |zx↓⟩, |xy↑⟩} ({|yz↑⟩, |zx↑⟩, |xy↓⟩}) has the following form

$$\begin{equation}{\mathcal{H}}_{{\pm}}=\left(\begin{matrix}\hfill 0\hfill & \hfill \mp i\frac{\lambda }{2}\hfill & \hfill {\pm}\frac{\lambda }{2}\hfill \\ \hfill {\pm}i\frac{\lambda }{2}\hfill & \hfill 0\hfill & \hfill +i\frac{\lambda }{2}\hfill \\ \hfill {\pm}\frac{\lambda }{2}\hfill & \hfill -i\frac{\lambda }{2}\hfill & \hfill 0\hfill \end{matrix}\right).\end{equation} \tag{ 2 }$$

We have assumed, here, degenerate t_2g states in an ideal octahedral environment. The diagonalization of ${\mathcal{H}}_{\text{SOC}}$ yields four-fold degenerate Γ₈ (−$\frac{\lambda }{2}$) and two-fold degenerate Γ₇ (λ) states, where each of the eigenstates of ${\mathcal{H}}_{+}$ has its time-reversal counterpart in the corresponding eigenstate of ${\mathcal{H}}_{-}$, forming the Kramer's doublet. The explicit form of these eigenstates are given by:

$$\begin{align*}\vert 1\alpha \rangle & =-\frac{1}{\sqrt{2}}(\vert {d}_{yz{\uparrow}}\rangle +i\vert {d}_{zx{\uparrow}}\rangle ),\\ \vert 1\beta \rangle & =\frac{1}{\sqrt{2}}(\vert {d}_{yz{\downarrow}\rangle }-i\vert {d}_{zx{\downarrow}}\rangle ),\\ \vert 2\alpha \rangle & =\frac{1}{\sqrt{6}}(2\vert {d}_{xy{\uparrow}}\rangle -\vert {d}_{yz{\downarrow}\rangle }-i\vert {d}_{zx{\downarrow}}\rangle ),\\ \vert 2\beta \rangle & =\frac{1}{\sqrt{6}}(2\vert {d}_{xy{\downarrow}}\rangle +\vert {d}_{yz{\uparrow}}\rangle -i\vert {d}_{zx{\uparrow}}\rangle ),\end{align*}$$

and,

$$\begin{align*}\vert 3\alpha \rangle & =\frac{1}{\sqrt{3}}(\vert {d}_{xy{\uparrow}}\rangle +\vert {d}_{yz{\downarrow}}\rangle +i\vert {d}_{zx{\downarrow}}\rangle ),\\ \vert 3\beta \rangle & =\frac{1}{\sqrt{3}}(\vert {d}_{xy{\downarrow}}\rangle -\vert {d}_{yz{\uparrow}}\rangle +i\vert {d}_{zx{\uparrow}}\rangle ).\end{align*}$$

Note that the eigenstates above do not have pure spin or orbital characters any more, as they couple to form entangled states. In the literature, these quartet and doublet are commonly referred to as j_eff = 3/2 and j_eff = 1/2 states respectively. It is important to point out that these spin–orbit entangled effective states are obtained within a single particle formalism and they are different from the many-body states J = 1/2 or 3/2. To make this distinction clear, now on we represent the single particle states by j, while the many-body states are indicated as J.

If we now go ahead and fill up these spin–orbit coupled states with four electrons, present in the 5d shell, we can see that the lower lying j_eff = 3/2 state gets completely occupied while the j_eff = 1/2 state remains empty. As there is no unpaired electron left in the system, the system becomes non-magnetic. The effective total angular momentum, j_eff for this state may be determined from the effective orbital angular momentum l_eff = −1 (see appendix for the illustration of this point) and the spin angular momentum s = 1, which again give a non-magnetic j_eff = l_eff + s = 0 state. The evolution of the ground state of a d⁴ system in the different regimes of the competing energy scales is schematically shown in figure 1(c). Note that the formation of J = 0 state within the many-body framework can similarly be described assuming t_2g only model, which we discuss in detail later in the next section.

As an aside, it is interesting to point out that for five electrons in the 5d shell, the j_eff = 1/2 state gets partially occupied in addition to the completely filled j_eff = 3/2 state. This half-filled j_eff = 1/2 state is further split into empty upper and filled lower Hubbard bands in presence of Coulomb interaction, leading to the well known j_eff = 1/2 Mott insulating state [24]. In contrast to the case of four electrons, discussed above, here the system remains magnetically active. The role of SOC, here, is to just convert the exchange interactions between spin and orbital angular momentum into an effective total angular momentum. These interactions between the total angular momenta, however, may have interesting consequences, for example they could be bond dependent, inherited from the corresponding interactions between orbital angular momentum for the t_2g orbitals [83].

The non-magnetic J = 0 ground state, imposed by SOC, is very typical for Van-Vleck type paramagentism, where the admixture of the ground state |ψ₀⟩ with the magnetic excited states |ψ_n ⟩ gives rise to a magnetic response in presence of a magnetic field ($\overrightarrow {B}$). In the language of perturbation theory, this can be expressed as $\vert \psi \rangle =\vert {\psi }_{0}\rangle +{\sum }_{n}\;\frac{\langle {\psi }_{n}\vert ( \overrightarrow {B}\cdot \overrightarrow {J})\vert {\psi }_{0}\rangle }{{E}_{n}-{E}_{0}}$ [84]. The essence of the behavior of a singlet–triplet system in presence of a magnetic field is schematically illustrated in figure 2.

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2.2.1. Why iridates?

2.2.2. Singlet-triplet transition: excitonic magnetism

Following Khaliullin's idea [71] of singlet–triplet transition, here, we briefly illustrate this process considering 180°d–p–d bond geometry, which is relevant for the corner-shared octahedra. In the 180°d–p–d bond geometry, as shown in figure 3, only two of the three t_2g orbitals are active along a certain bond, while the third orbital, say c, remains inactive. For simplicity, we can call this bond c. Therefore, the nearest neighbor hopping matrix along this c bond is given by $t({a}_{i}^{{\dagger}}{a}_{j}+{b}_{i}^{{\dagger}}{b}_{j}+\mathrm{H}.\mathrm{c}.)$, indicating that the hopping is diagonal in the active orbital space. Here ${a}_{i}^{{\dagger}}({a}_{i}),{b}_{i}^{{\dagger}}({b}_{i})$ are the creation (annihilation) operators that create (annihilate) an electron in the respective active t_2g orbitals a and b at site i. In the illustration of figure 3, these active orbitals are d_xz and d_xy orbitals for the bond along x. We now introduce three different operators A, B, and C, which represent the three possible orbital configurations for the ${t}_{2g}^{4}$ shell, viz, A = {a² bc}, B = {ab² c}, and C = {abc²}. The one-to-one correspondence between the operators and the orbitals are apparent from this definition. With the help of these operators, we can also rewrite the components of the orbital angular momentum operator, e.g., L_x = −i(B^† C − C^† B), etc.

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In the strong coupling limit (U ≫ t, λ, J_H), this results in a spin–orbital Hamiltonian, having the form $H=({t}^{2}/U){\sum }_{\langle ij\rangle }[({ \overrightarrow {S}}_{i}\cdot { \overrightarrow {S}}_{j}+1){O}_{ij}^{(\gamma )}+{({L}_{i}^{\gamma })}^{2}+{({L}_{j}^{\gamma })}^{2}]$, where O^(γ) is the bond dependent orbital operator which can be written in terms of the A, B, C operators or directly expressed in terms of the orbital angular momentum operators L_x , L_y , and L_z . For example, the operator for the bond c can be written as ${O}^{(c)}={({L}_{i}^{x}{L}_{j}^{x})}^{2}+{({L}_{i}^{y}{L}_{j}^{y})}^{2}+{L}_{i}^{x}{L}_{i}^{y}{L}_{j}^{y}{L}_{j}^{x}+{L}_{i}^{y}{L}_{i}^{x}{L}_{j}^{x}{L}_{j}^{y}$. Similarly, the operators O^(a), O^(b) for other bonds a, and b can be obtained from cyclic permutations of L_x , L_y , and L_z . Clearly, the Hamiltonian H operates in the (M_L , M_S ) basis. We now want to project it onto the low-energy subspace of the singlet J = 0 ground state, $\vert 0\rangle =\frac{1}{\sqrt{3}}(\vert 1,-1\rangle -\vert 0,0\rangle +\vert -1,1\rangle )$ and the first excited state J = 1 triplet at an energy Δ_st determined by λ, $\vert {T}_{0}\rangle =\frac{1}{\sqrt{2}}(\vert 1,-1\rangle -\vert -1,1\rangle ),\;\vert {T}_{{\pm}1}\rangle ={\pm}\frac{1}{\sqrt{2}}(\vert {\pm}1,0\rangle -\vert 0,{\pm}1\rangle )$. Calculating the matrix elements of the orbital and spin angular momentum operators in this Hilbert space, we can write them in terms of the 'triplon' $\overrightarrow {T}$ with the Cartesian components ${T}_{x}=\frac{1}{i\sqrt{2}}({T}_{1}-{T}_{-1}),\enspace {T}_{y}=\frac{1}{\sqrt{2}}({T}_{1}+{T}_{-1})$, and T_z = iT₀, and the 'spin' $\overrightarrow {J}=-i({ \overrightarrow {T}}^{{\dagger}}{\times} \overrightarrow {T})$. For example, $\overrightarrow {S}=-i\sqrt{\frac{2}{3}}( \overrightarrow {T}-{ \overrightarrow {T}}^{{\dagger}})+\frac{1}{2} \overrightarrow {J}$, and $\overrightarrow {L}=i\sqrt{\frac{2}{3}}( \overrightarrow {T}-{ \overrightarrow {T}}^{{\dagger}})+\frac{1}{2} \overrightarrow {J}$. Using these relations we can immediately write down the magnetic moment for a ${t}_{2g}^{4}$ configuration in terms of $\overrightarrow {T}$ and $\overrightarrow {J}$, viz, $\overrightarrow {M}=2 \overrightarrow {S}- \overrightarrow {L}=-i\sqrt{6}( \overrightarrow {T}-{ \overrightarrow {T}}^{{\dagger}})+{g}_{J} \overrightarrow {J}$, where ${g}_{J}=\frac{1}{2}$. Clearly, the magnetism in a J = 0 state is driven solely by the $\overrightarrow {T}$ exciton as the second term in $\overrightarrow {M}$ does not contribute for a singlet state, J = 0, and hence, the magnetism is referred to as 'excitonic magnetism'.

The transformation $H(L,S)\to H( \overrightarrow {T}, \overrightarrow {J})$, as described above, gives us the desired effective singlet–triplet model [71],

$$\begin{equation}{H}_{\text{eff}}={{\Delta}}_{\mathrm{s}\mathrm{t}}\sum\limits _{i}{T}_{i}^{{\dagger}}{T\hspace{-2pt}}_{i}+\kappa \frac{2}{9}\sum\limits _{ij}\left[{ \overrightarrow {T}}_{i}^{{\dagger}}\cdot { \overrightarrow {T\hspace{-2pt}}}_{j}-\frac{7}{16}({ \overrightarrow {T}}_{i}\cdot { \overrightarrow {T\hspace{-2pt}}}_{j}+\mathrm{H}.\mathrm{c}.)\right].\end{equation} \tag{ 3 }$$

Here κ denotes the exchange interaction 4t²/U and terms up to quadratic in $\overrightarrow {T}$ bosons are considered. The three and four-boson interaction terms, that include the quadrupole operators and bi-quadratic Heisenberg exchange couplings respectively, have only small values near the phase transition for 180° bond geometry and are, therefore, not important for the qualitative understanding of magnetic phase transition driven by the condensation of $\overrightarrow {T}$ bosons. The diagonalization of this effective Hamiltonian gives the energy dispersion for the triplet excitation, which is useful to understand the condensation of the triplet state. For example, when κ < κ_c, where κ_c is some critical value of κ, the magnetic excitation has a finite gap $\lambda \sqrt{1-(\kappa /{\kappa }_{\mathrm{c}})}$, indicating a paramagnetic phase. At κ = κ_c this gap closes and a magnetic phase transition takes place due to condensation of the $\overrightarrow {T}$ bosons.

Interestingly, soon after this proposal of excitonic magnetism, Cao et al [74] experimentally showed the existence of magnetism in the d⁴ double perovskite iridate Sr₂YIrO₆. Indeed, in most of the pentavalent iridates, the Ir atom has a non-zero effective magnetic moment (see, for example, the list of pentavalent iridates with their magnetic and trasport properties in table 1), where the moment values are often attributed to the non-cubic crystal field, bandwidth effect, etc in the solid causing the deviation from the atomic like picture, described above for the formation of J = 0 state. Furthermore, since the strength of the SOC is renormalized in a solid as λ/2S [84, 85], where λ is the atomic SOC, and S is the total spin, one might anticipate the SOC strength would be weaker in d⁴ systems with S = 1 as compared to tetra-valent iridates with S = 1/2. Such weak SOC tends to favor excitonic magnetism in d⁴ iridates as the relative strength of the SOC constant and the superexchange energy scale determines the occurrence of excitonic magnetism. Interestingly, a recent theoretical study [86] based on density matrix renormalization also suggests spin–orbit induced transition to an excitonic anti-ferromagnetic condensate even at an intermediate Coulomb interaction regime.

Table 1. Structural, transport and magnetic properties of pentavalent iridates. The notations FM, AFM and SOL refer to ferromagnetic, anti-ferromagnetic and spin–orbital liquid states respectively. The connectivity of IrO₆ unit in each structural family is given within the parenthesis.

Crystal structure	Compound	Space group	Transport property	Magnetic ordering	μeff/Ir (μB )	Reference
Double perovskite (isolated)	Sr2YIrO6	P21/n	Insulator	AFM	0.91	[74, 75]
		$Fm\overline{3}m$	Insulator	None	0.21	[94]
	Ba2YIrO6	$Fm\overline{3}m$	Insulator	None	0.3	[75, 77]
			Insulator	None	0.44	[87–92]
		P21/n	—	—	—	[95]
			Insulator	AFM	1.44	[93]
	Sr2GdIrO6	Pn-3	Insulator	FM	—	[74, 75]
	Sr2ScIrO6	P21/n	Insulator	None	0.16	[96–98]
	Sr2LuIrO6	P21/n	—	None	—	[95]
	Ba2ScIrO6	$Fm\overline{3}m$	Insulator	None	0.39	[96, 97]
	Ba2LuIrO6	P21/n	—	None	—	[95]
	Ba2LaIrO6	P21/n	—	None	—	[95]
		$R\overline{3}$	—	—	—	[99]
	Ba2NdIrO6	Cubic	—	—	—	[100]
	Ba2SmIrO6	Cubic	—	—	—	[100]
	Ba2DyIrO6	Cubic	—	—	—	[100]
	Sr2InIrO6	P21/n	Insulator	None	1.608	[101]
	Sr2CoIrO6	I2/m	Insulator	AFM	0.47	[102–104]
	Sr2FeIrO6	I2/m	Insulator	AFM	—	[101]
		$I\overline{1}$	Insulator	AFM	—	[105]
	Ca2FeIrO6	$I\overline{1}$	Insulator	AFM	—	[105]
	La2LiIrO6	Pmm2	—	None	1.42	[106, 107]
	BaLaMgIrO6	Pseudocubic	—	None	—	[106]
	Bi2NaIrO6	P21/n	Insulator	None	0.19	[108]
	SrLaMgIrO6	P21/n	Insulator	None	0.61	[109]
	SrLaZnIrO6	P21/n	Insulator	None	0.46	[109]
	SrLaNiIrO6	P21/n	Insulator	AFM	0.19	[109]
Post perovskite (corner & edge shared)	NaIrO3	Cmcm	Insulator	NM	0	[75, 110, 111]
	Ba3ZnIr2O9	P63/mmc	Insulator	SOL	0.2	[76]
6H-perovskite (face shared)	Ba3MgIr2O9	P63/mmc	Insulator	None	0.5–0.6	[112, 113]
	Ba3CaIr2O9	C2/c	Insulator	None	—	[112, 113]
	Ba3SrIr2O9	C2/c	Insulator	None	—	[112, 113]
	Ba4LiIr3O12	P63/mmc	Insulator	Magnetic	—	[114]
	Ba4NaIr3O12	P63/mmc	Insulator	—	—	[114]
	Ba3CdIr2O9	C2/c	Insulator	None	0.3	[115]
Hexagonal (isolated)	Sr3NaIrO6	$R\overline{3}c$	Insulator	None	—	[116, 117]
	Sr3LiIrO6	$R\overline{3}c$	Insulator	None	—	[116–118]
Pyrochlore (corner shared)	Cd2Ir2O7	FCC	—	—	—	[119]
	Ca2Ir2O7	FCC	—	—	—	[119]
Others	KIrO3	Pn3	—	—	1.04	[120]
	Ba0.5IrO3	I23	Metal	—	—	[119]
	Sr0.5IrO3	BCC	—	—	—	[119]
	Na3Cd2IrO6	C2/m	—	—	—	[121]

The magnetism in pentavalent iridates are, however, not beyond doubt. The situation is, in particular, controversial and flooded with conflicting results for the double peroveskite iridate both in theory and experiments [75, 77, 87–93]. Nevertheless, the possibility of the breakdown of the anticipated J = 0 state, leading to the unconventional magnetism, have generated enough interest in pentavalent iridates, which are further accelerated by the observation of novel 'spin–orbital liquid' state in the 6H perovskite iridate [76]. The easy availability of the d⁴ electronic configuration in iridates, compared to other 5d TM oxides, have further boosted the search for the realization of J = 0 state and/or the excitonic magnetism in iridates.

For simplicity, we first consider a many-body atomic model and, then, eventually, we shall include the band structure effects by allowing hopping between the spin–orbit coupled atoms. Assuming the e_g orbitals are completely decoupled from the t_2g sectors owing to the strong octahedral splitting Δ, only the t_2g orbitals are considered at each atomic site. The corresponding atomic many-body Hamiltonian, spanned in the Hilbert space of dimension ⁶C₄ = 15, includes the electron–electron interactions as well as the SOC interaction, viz,

$$\begin{equation}{\mathcal{H}}_{\text{atom}}={\mathcal{H}}_{\text{int}}+{\mathcal{H}}_{\mathrm{S}\mathrm{O}}.\end{equation} \tag{ 4 }$$

Here the Kanamori Hamiltonian ${\mathcal{H}}_{\text{int}}$ contains the intra-orbital and the inter-orbital Coulomb interactions U and U', which are related to each other via the Hund's coupling J_H, viz, U = U' + 2J_H. The schematic illustration of the various interaction terms are given in figure 4. The explicit form of the Hamiltonian at an atomic site i is given by [98, 123–125]

$$\begin{align*}{\mathcal{H}}_{\text{int}}^{i}& =U\sum\limits _{l}{n}_{i,l{\uparrow}}{n}_{i,l{\downarrow}}+\frac{{U}^{\prime }-{J}_{\mathrm{H}}}{2}\sum\limits _{l\ne m,\sigma }{n}_{i,l\sigma }{n}_{i,m\sigma }\\ & \quad +\frac{{U}^{\prime }}{2}\sum\limits _{\sigma \ne {\sigma }^{\prime }}\sum\limits _{l\ne m}{n}_{i,l\sigma }{n}_{i,m{\sigma }^{\prime }}\\ & \quad -\frac{{J}_{\mathrm{H}}}{2}\sum\limits _{l\ne m}\left({d}_{i,m{\uparrow}}^{{\dagger}}{d}_{i,m{\downarrow}}{d}_{i,l{\downarrow}}^{{\dagger}}{d}_{i,l{\uparrow}}\right.\\ & \quad \left.+{d}_{i,m{\uparrow}}^{{\dagger}}{d}_{i,m{\downarrow}}^{{\dagger}}{d}_{i,l{\uparrow}}{d}_{i,l{\downarrow}}+\mathrm{h}.\mathrm{c}.\right)\end{align*}$$

and the spin–orbit interaction is

$$\begin{equation}{\mathcal{H}}_{\mathrm{S}\mathrm{O}}^{i}=\frac{\dot {\iota }\lambda }{2}\sum\limits _{lmn}{{\epsilon}}_{lmn}\sum\limits _{\sigma {\sigma }^{\prime }}{\sigma }_{\sigma {\sigma }^{\prime }}^{n}{d}_{i,l\sigma }^{{\dagger}}{d}_{i,m{\sigma }^{\prime }}.\end{equation} \tag{ 5 }$$

Here, i-l-σ denotes the site-orbital-spin index and d_i,lσ (${d}_{i,l\sigma }^{{\dagger}}$) is the annihilation (creation) operator.

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The exact diagonalization of the atomic model, equation (4), gives the spin–orbit coupled eigenstates that are characterized by the total angular momentum quantum number J: J = 0 (1), J = 1 (3), J = 2 (5), J = 2 (5), and J = 0 (1) with increasing order in energy (see figure 5(a)). Here, the number within the parenthesis denotes the degeneracy of the states. As anticipated, J = 0 singlet forms the ground state (similar to the single particle picture, described in the previous section) which is separated from the excited triplet J = 1 state. The energies of these states are dictated by the interaction parameters such as Hund's coupling J_H, Coulomb interaction U and SOC constant λ, while their energy difference is independent of U. The exact energy eigenvalues of the atomic states J = 0 and J = 2 depend on the relative strength of J_H and λ (In the limit J_H ≫ λ, E_J=0 = E₀ − 4U + 7J_H − λ and E_J=2 = E₀ − 4U + 7J_H + λ/2; for J_H ≪ λ, E_J=0 = E₀ − 4U + 7J_H − 2λ and E_J=2 = E₀ − 4U + 7J_H − λ/2) but their energy difference is always 3λ/2. The energy eigen value of the J = 1 state, on the other hand, always have the same analytical form E_J=1 = E₀ − 4U + 7J_H − λ/2 [126].

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It is noteworthy that in the simple atomic model in equation (4), we have assumed an ideal cubic crystal field with degenerate t_2g states. In most of the real materials, however, the TM-O₆ octahedral unit undergoes distortions, tilt, and rotation, removing the degeneracy of the t_2g states. Indeed, the resulting non-cubic crystal field is important to destabilize the atomic J = 0 state, leading to magnetism even within an atomic model. This is straight forward to see by adding a non-cubic crystal field term, ${\mathcal{H}}_{\text{NCF}}^{i}={\sum }_{lm\sigma }{\varepsilon }_{lm}{d}_{i,l\sigma }^{{\dagger}}{d}_{i,m\sigma }$ in the Hamiltonian, equation (4), where the onsite energies of the different orbitals are depicted by ɛ_lm and the difference in the onsite energies is the measure of non-cubic crystal field Δ_NCF. The phase diagram, figure 5(b), for the ground state shows the emergence of magnetism (J ≠ 0) in the d⁴ system due to non-cubic crystal field. The competition between Δ_NCF and SOC λ is evident from this phase diagram. While in the strong SOC regimes (λ ≫Δ_NCF) the non-magnetic J = 0 state remains intact, magnetism appears in the energy regime Δ_NCF ≫ λ (see figure 5(b)).

The atomic model, described above, misses the important ingredient of a solid which is the hopping between different atoms. The simplest model that incorporates the hopping effect is a two-site model where in addition to the atomic interactions, described in equation (4), hopping between two atomic sites are also allowed. The two-site many body model Hamiltonian in a Hilbert space of ¹²C₈ = 495 basis states reads as

$$\begin{equation}\mathcal{H}={\mathcal{H}}_{t}+\sum\limits _{i}({\mathcal{H}}_{\text{int}}^{i}+{\mathcal{H}}_{\mathrm{S}\mathrm{O}}^{i}),\end{equation} \tag{ 6 }$$

where the first term represents the inter-site hopping between different orbitals, designated by ${t}_{ij}^{l\sigma ,m{\sigma }^{\prime }}$

$$\begin{equation}{\mathcal{H}}_{t}=\sum\limits _{\text{i}\;\text{j}}\sum\limits _{l,m}\sum\limits _{\sigma ,{\sigma }^{\prime }}({t}_{ij}^{l\sigma ,m{\sigma }^{\prime }}{d}_{il\sigma }^{{\dagger}}{d}_{jm{\sigma }^{\prime }}+\mathrm{H}.\mathrm{C}.)\end{equation} \tag{ 7 }$$

and the second and the third terms in equation (6) represent the atomic interactions. While the second atomic term, spin–orbit interaction, is the same as described before, the first atomic term, the interaction term, may be rewritten in terms of the total occupation number operator ${\hat{N}}_{i}$, total spin momentum operator ${\hat{S}}_{i}$, and total orbital angular momentum operator ${\hat{L}}_{i}$ at a site i as [127]

$$\begin{equation}{\mathcal{H}}_{\text{int}}^{i}=\frac{U-3{J}_{\mathrm{H}}}{2}{\hat{N}}_{i}({\hat{N}}_{i}-1)+\frac{5}{2}{J}_{\mathrm{H}}{\hat{N}}_{i}-2{J}_{\mathrm{H}}{\hat{S}}_{i}^{2}-\frac{1}{2}{J}_{\mathrm{H}}{\hat{L}}_{i}^{2}.\end{equation} \tag{ 8 }$$

A diagonal hopping in the orbital and spin space i.e., ${t}_{ij}^{l\sigma ,m{\sigma }^{\prime }}\to t{\delta }_{lm}{\delta }_{\sigma {\sigma }^{\prime }}$ is assumed, where all three t_2g orbitals actively participate in the hopping. The exact diagonalization results of the above Hamiltonian, as discussed in reference [127], are more reliable in the Mott limit (U/t ≫ 1). In this regime, a magnetic phase transition occurs from the non-magnetic J = 0 state to a magnetic J = 1 state. The phase diagram in the parameter space of U, λ and t for a chosen value of J_H = 0.2U is shown in figure 6(a). Interestingly, the phase transition J = 0 → J = 1 at a critical value of λ/t is associated with abrupt changes in both the expectation values of individual J_i and total J quantum numbers across the transition. The local L_i and S_i expectation values, however, remain unchanged throughout (see figure 6(b)). In the magnetic state the local moments are produced by hopping (superexchange) driven mixing of the onsite singlet state with the higher energy triplet states. It is important to note, in contrast to standard 3d Mott insulators, here the local moments are not robust and deviate from their atomic values.

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The magnetic transition, discussed above for orbitally symmetric hopping, assumes full rotation symmetry which is usually absent in a real material. For example, in simple cubic systems where the hopping t between TM atoms occur from the superexchange via the oxygen atoms, only two orbitals participate in the hopping (see figure 3), so that ${t}_{ij}^{l\sigma ,m{\sigma }^{\prime }}\to t{\delta }_{lm}(1-{\delta }_{km}){\delta }_{\sigma {\sigma }^{\prime }}$. Here the direction of the line, connecting the two sites i and j, determines the blocked orbital k. On the other hand, for the hopping between nearest neighbor atoms on a face-centred cubic lattice may be approximated by only one active orbital k, viz, ${t}_{ij}^{l\sigma ,m{\sigma }^{\prime }}\to t{\delta }_{lm}{\delta }_{km}{\delta }_{\sigma {\sigma }^{\prime }}$. Here, the active orbital is determined by the common plane, shared by the two sites. Interestingly, the magnetic phase transition occurs for both these cases, emphasizing the crucial role of hopping in generating the magnetism in d⁴ spin–orbit coupled systems [128]. A next question that naturally occurs is the nature of the magnetism. Meetei et al [127] showed that the generated local moments favor ferromagnetism and a phase transition from the AFM superexchange to the FM superexchange takes place at a critical value of J_H/U. An intuitive picture for the Hund's coupling driven ferromagnetism, may be followed from reference [127], based on the simple perturbative analysis. The atomic ground state in the d⁴-d⁴ configuration with L_i = 1 and S_i = 1 has a total energy, E₀ = 12U − 26J_H, computed using equation (8). The hopping in equation (7) acts as a perturbation and the ground state gains energy via virtual hopping. For FM superexchange, the intermediate d³–d⁵ state (see figure 6(d)) with S₁ = 3/2, L₁ = 0, S₂ = 1/2, and L₂ = 1 has the total energy E_FM = 13U − 29J_H. This results in an energy gain by the FM state ${\Delta}{E}_{\text{FM}}=-\frac{2{t}^{2}}{U-3{J}_{\mathrm{H}}}$. Similarly, the intermediate paths for the AFM superexchange, as schematically illustrated in figure 6(e), gains an energy ${\Delta}{E}_{\text{AFM}}=-\frac{2}{3}\frac{{t}^{2}}{U-3{J}_{\mathrm{H}}}-\frac{14}{6}\frac{{t}^{2}}{U}-\frac{{t}^{2}}{U+2{J}_{\mathrm{H}}}$. Comparing the two energy scales ΔE_FM and ΔE_AFM, it is easy to argue that the superexchange gradually changes from the AFM to FM state with increasing value of J_H/U. It is to be noted that in the above analysis, SOC is ignored and inclusion of SOC tends to compete against the ferromagnetism.

An independent exact diagonalization study [98] with the similar two-site model including also the effect of non-cubic crystal field, also suggests the competition between ferromagnetism and AFM with SOC in d⁴ systems. While ferromagnetism is stabilized in absence of SOC (λ = 0), with increase in λ ferromagnetic interaction become weaker, and eventually at a critical value λ_c, an AFM interaction is favored. The critical value of λ_c depends on the strength of the other parameters of the Hamiltonian as shown in figure 6(c). In particular, a larger Hund's coupling tends to increase the λ_c, thereby favoring the ferromagnetism. Dynamical mean-field theory (DMFT) study combined with the continuous-time quantum Monte Carlo [129] also supports these findings.

Inspired by the results of the two site exact diagonalization, the lattice problem of magnetism in d⁴ systems has also been analysed by constructing an effective spin–orbital lattice model [127, 128, 130].

The effective model was employed by Svoboda et al [128] for the analysis of magnetism for the three-different cases of active t_2g orbitals, mentioned before, viz, three active orbitals (N_orb = 3), two active orbitals (N_orb = 2) and single active orbital (N_orb = 1). While the detail discussion of the effective magnetic models, developed from the full Hamiltonian in equation (6), can be found in reference [128] for each of the three mentioned cases, here we focus on the key finding of this analysis. According to this study, while ferromagnetism is stabilized at large J_H/U limit for N_orb = 2 and 3 cases, for the single active orbital case, ferromagnetic interaction is not supported for any reasonable value of J_H/U. The authors argue that the absence of ferromagnetism in N_orb = 1 is due to the fewer available exchange paths. As illustrated in figure 6, the number of superexchange paths supporting a FM state is less than the corresponding number of such paths available for the AFM state. However, there are more possible active paths in N_orb = 2 and 3 to reduce the energy of the FM state so that Hund's coupling takes over, favoring ferromagnetism. This is, however, not the case for N_orb = 1, where the energy of the FM state is lowered by only a single factor of −t²/U, failing to favor ferromagnetism.

In addition, reference [128] also predicts an intriguing orbital frustration for the cases of N_orb = 1 and 2 with broken rotational symmetry in contrast to the orbital symmetric N_orb = 3 case. The orbital frustration is demonstrated on a non-frustrated crystal geometry, as illustrated in figure 7, considering the case of N_orb = 2. In a d⁴ configuration, each site has at least one doubly occupied t_2g orbital. For an AFM interaction between two sites, each with two active orbitals (N_orb = 2), the doubly occupied orbitals on both sites remain inactive (see figure 6(e)). As a result, inter-site AFM interaction in the N_orb = 2 model favors the orbital, perpendicular to the line joining these two sites, to be doubly occupied, e.g., d_xz orbital in figure 7(a) is doubly occupied for the bond 1, along y-direction. These doubly occupied orbitals are, however, not inactive for bond 2, say along z-direction, simply because d_xz orbital is not perpendicular to this second bond along z. As a result, AFM interaction is no longer favorable and, therefore, the next energetically favorable FM interaction occurs along bond 2. This FM interaction, in turn, leads to a different doubly occupied orbital (d_xy orbital in figure 7) for the bond 3 along y-direction, which neither favors energetically lowest FM interaction nor the energetically lowest AFM interaction, stabilizing rather an energetically higher AFM interaction along bond 3. Similarly, starting from the FM interaction between two sites along a particular bond (see bond 1 in figure 7(b)), it is possible to show that a higher energy configuration is taken by one of the four bonds on a plaquette. This leads to a frustration due to orbital degrees of freedom even on a non-frustrated lattice, which may result in an orbital liquid state in the regime λ ≪ zJ_SE (z being the coordination number) with strong orbital effects in absence of strong octahedral distortions.

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We now turn to the transport properties of d⁴ spin–orbit coupled systems. The transport properties in d⁴ systems, as studied in reference [131] using the rotational invariant Gutzwiller approximation method as well as the DMFT. The calculations reveal metal–insulator transition as a function of SOC constant λ and Coulomb interaction U. A Hubbard model for the t_2g orbitals with full Hund's rule coupling and spin–orbit interaction, similar to the many-body model, discussed above, is considered in this work. The computed phase-diagram in the parameter space of λ and U (see figure 8) depicts the three different states, viz, metallic state, Mott insulating state, and band insulating state. The two insulating states can be visualized by considering the two limiting cases. First, let us consider the strong SOC limit (U = 0). In this limit, the degenerate t_2g bands split into j_eff = 3/2 and j_eff = 1/2 bands, separated from each other with an energy gap of ≈3λ/2. With increase in λ, the higher lying j_eff = 1/2 bands tend to get empty as the electrons move to the j_eff = 3/2 bands. Consequently, at a critical value of λ/D, D being the half-bandwidth, the j_eff = 3/2 bands are completely occupied, resulting in a transition to the band insulating state from a metallic state. Moving to the other limit where λ = 0, the degenerate t_2g orbitals have four electrons at each site. With increasing U, the system undergoes an interaction driven Mott transition at some critical value of U/D with each band having 4/3 electrons.

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Coming back to the discussion of the phase diagram in figure 8, we can see that the phase diagram may be subdivided into three different regions: (I) the regime 0 < λ/D < 0.24, where increase in U leads to a transition from metal to Mott insulator. The decrease in the critical value of U for this metal–insulator transition with increasing SOC, suggests the enhancement of the correlation effects by the SOC. Intuitively, this may be attributed to the narrow bands, resulting from the SOC driven band splitting. It is interesting to point out that similar cooperation between SOC and correlation has been observed for d⁵ filling as well [23]. In this Mott phase, the local moment at each site vanishes, forming a spin–orbital singlet (SOS) state and the ground state is simply the product state of the local singlets at each site. (II) The next regime is 0.24 < λ/D < 1.33. Two back to back transitions occur in this regime as we increase U. The first transition is from metal to band insulator, followed by the second transition from band to Mott insulator. For the intermediate U, the effective bandwidth of the system is reduced by the correlation effect first, which upon relatively small band splitting due to intermediate λ results in a band insulating state. Increasing U further pushes the system into Mott regime. (III) Finally, in the regime λ/D > 1.33, strong SOC leads to completely empty j_eff = 1/2 bands even at U = 0, suggesting a band insulating state in the non-interacting case. Increasing U to a large value induces again a transition from band insulating state to Mott insulating state.

The general formula for the double perovskite iridates, that we discuss here, is A₂ BIrO₆, where the A site is occupied by the divalent cations, e.g., Sr, Ba. In this case, the trivalent ions such as Y³⁺, Sc³⁺, Lu³⁺, La³⁺, etc, at the B site renders the desired 5+ charge state at the Ir site. The basic structure of the double perovskite iridates consists of corner shared BO₆ and IrO₆ octahedra that alternate along each of the crystallographic axes. These octahedra are either ideal or undergo specific types of distortions, depending on the particular crystal symmetry of the structure. In general, the crystal structure accompanies strong geometric frustration due to face-centered cubic lattice formed by the neighboring Ir atoms (see figure 9).

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Owing to the compositional and structural flexibility, double perovskite iridates have particularly received enormous attention. The interest in the d⁴ double perovskite iridates began with the observation of the onset of magnetism in double perovskite iridate Sr₂YIrO₆ in the year 2014, although, the magnetism in iridates with d⁴ electronic configuration was reported even earlier, e.g., in another double perovskite system Sr₂CoIrO₆ [102] (this is even before the theoretical prediction by Khaliullin [71]). In this work Narayanan et al [102] studied the structural, electronic and magnetic properties of La_2−x Sr_x CoIrO₆ for 0 ⩽ x ⩽ 2, the end member of which is Sr₂CoIrO₆ with high spin Co³⁺ and low spin Ir⁵⁺ ions. Sr₂CoIrO₆ undergoes successive structural transitions starting from a two-phase region upon heating, viz, $P{2}_{1}/n+I2/m\to I2/m\to I4/m\to Fm\overline{3}m$, and exhibits AFM ordering as revealed from magnetization measurements. Interestingly, the estimated effective magnetic moment for the system μ_eff = 5.1(1)μ_B /f.u. suggests for magnetic contribution from the Ir⁵⁺ ion, as the contribution (μ_eff = 4.9μ_B /Co) for the Co³⁺ ion alone cannot describe the estimated moment value. This small moment, however, remains undetected in the neutron powder diffraction measurements. Nevertheless, the density functional calculations, presented in the same work, also suggest significant spin and orbital moments at the Ir site. However, the material is reported to have significant anti-site disorder and, also, the presence of strongly magnetic Co³⁺ ions brings up the question if the Co ions have any role in the magnetism at the Ir site. Later, it was also pointed out that in addition to Ir⁵⁺ ion, the system also has Ir⁶⁺ ions, which have rather strong magnetic signal [104]. In view of the above, the double perovskite iridates where the magnetism can appear only via Ir (B site is non-magnetic) are the best candidates for the investigation of magnetism in the Ir⁵⁺ ion.

A series of such double perovskite iridates A₂ R³⁺Ir⁵⁺O₆ (A = Sr, Ba and R = La, Y, Sc, Lu) were synthesized long ago in 1999 by Wakeshima et al [95] and all of these compounds were reported to have paramagnetic behavior down to 4.5 K. With the new insight into the SOC effect in iridates, later, Cao et al [74] revisited the compound Sr₂YIrO₆ and showed that the system undergoes a novel magnetic transition below T = 1.3 K. The observed magnetism was surprising as both the presence of Ir⁵⁺ ion as well as strong geometric frustration in the double perovskite structure in general should forbid any long range magnetic order. The reported moment μ_eff at the Ir site is 0.91μ_B and the Curie–Weiss temperature θ_CW is as large as −229 K, indicating the presence of AFM interaction as well as strong frustration in the system. Sr₂YIrO₆ crystallizes in monoclinic P2₁/n symmetry, where the IrO₆ octahedron undergo distortion. The authors argued the non-cubic crystal field, resulting from this distortion, as the possible origin of deviation from the singlet J = 0 state, leading to magnetism in Sr₂YIrO₆.

Interestingly, the parallel study of another d⁴ double perovskite iridate Sr₂GdIrO₆ in the same work [74] showed that unlike the Y-compound, the Gd-compound orders ferro-magnetically at T_c = 2.3 K with a comparable Curie Weiss temperature θ_CW = 1.2 K. The saturation moment was found to be M_s ∼ 7μ_B /fu consistent with the Gd³⁺ (4f⁷) charge state, present in the system. The authors proposed Ir atoms in Sr₂GdIrO₆ to be non-magnetic, however the presence of a strong magnetic signal from Gd³⁺ ion makes the confirmation difficult. According to this study, the latter iridate crystallizes in cubic space group Pn-3 and, therefore, is free from any non-cubic distortion. In absence of any non-cubic crystal field, SOC dominates, resulting in a J = 0 state at the Ir site in Sr₂GdIrO₆. Both double perovskite iridates were reported to be insulating with an energy gap of 490 meV and 272 meV respectively.

The magnetic properties of these two double-perovskite iridates were later studied within the framework of density functional theory [75]. The electronic structure calculations presented in this work reveal that in addition to non-cubic crystal field the band structure effects are, more crucial for the breakdown of the anticipated J = 0 state leading to the formation of moments. It is interesting to point out that the crucial role of the band structure effects has also been discussed in the context of tetravalent iridates [132]. The conclusions of reference [75] is based on the estimated non-cubic crystal field and the magnetic moments at the Ir site in the two double perovskite iridates, mentioned above, along with another double perovskite, Ba analogue of Sr₂YIrO₆, Ba₂YIrO₆. The authors showed that even though Sr₂GdIrO₆ crystallizes in the cubic structure, the Pn-3 symmetry allows for the rotation of the IrO₆ octahedra, that leads to a strong trigonal splitting of the Ir-t_2g orbitals, which is about an order of magnitude larger than that in Sr₂YIrO₆. Indeed, the electronic structure calculation showed the presence of finite moment at the Ir site in both these systems. The situation became even more interesting with the cubic double perovskite iridate Ba₂YIrO₆, where the $Fm\overline{3}m$ symmetry does not allow any tilt or rotation of the IrO₆ octahedra, resulting in a triply degenerate t_2g orbitals. Even in this case, finite moment at the Ir site persists. This emphasizes the band structure effects to be crucial in driving magnetism in these series of double perovskite iridates (see scenario I in figure 9(b)). Interestingly, despite the different non-cubic crystal field, the bandwidths in this series of double perovskites are comparable and large in magnitude (∼1.2 eV) that leads to significant mixing between j_eff = 1/2 and j_eff = 3/2 states (see inset of figure 10(b)), resulting in deviation from the ideal atomic limit. The details of the evolution of the electronic structure of Ba₂YIrO₆, obtained in this work, is summarized in figure 10.

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In complete contrast to references [74, 75], DMFT calculations [88] suggest absence of any long range magnetic order in both the compounds Sr₂YIrO₆ and Ba₂YIrO₆. In the above work, taking density functional theory as a starting point, within the constrained random phase approximation, a multi band Hubbard model is constructed which is further analyzed within static and dynamic mean field theories and strong coupling expansion. The authors showed that a large U (U = 4 eV) is required to get an insulating solution, reducing significantly the mixing between j_eff = 1/2 and j_eff = 3/2 states that was present within GGA + SOC. Using effective strong coupling model, the authors concluded that the reduced magnitude of hopping due to large Ir–Ir distance in the double perovskite structure is too small to overcome the singlet–triplet energy gap, which was predicted to be in the range of 200 meV. This results in a non-magnetic J = 0 singlet ground state in both the iridates.

Around the same time, Dey et al [87] revisited and studied in detail the structural, magnetic, and thermodynamic properties of Ba₂YIrO₆, which was originally synthesized more than 45 years ago [100]. This study confirmed the ideal cubic structure of Ba₂YIrO₆, in contrast to the monoclinic symmetry reported earlier [95], and showed that the system does not order down to 0.4 K. However, the Curie–Weiss fit yields a Curie Weiss temperature θ_CW ∼ −8.9 K and an effective moment of 0.44μ_B /Ir, unexpected for a J = 0 state. The authors exclude the possibility of any anti-site disorder due to different sizes of Ir⁵⁺ and Y³⁺ ions or oxygen vacancy from their magnetization measurements on the oxygen-annealed crystals. On the contrary, the DFT calculations presented in the same work show a non-magnetic ground state in Ba₂YIrO₆ with significant mixing between j_3/2 and j_5/2 states (the effective j_eff = 1/2 state is part of this full j_5/2 manifold [58]). While in absence of Hubbard U, the system remains metallic, the insulating state is obtained with a rather small value of U, viz, U = 1.4 eV and J_H = 0.5 eV in contrast to reference [88]. The origin of the moment at the Ir site remains elusive from this work.

In the meantime, independent experiments [133, 134] probed the effect of structural distortions on the magnetism of the series of double perovskites Ba_2−x Sr_x YIrO₆ (0 ⩽ x ⩽ 1). Both studies show no evidence for magnetism down to 2 K. The effective moment, reported in reference [133], is rather small, 0.16μ_B /Ir and was speculated as a result of local disorder in the structure which was not captured in the diffraction measurements. These results were, further, disputed by Terzic et al [93], emphasizing that the realization of the magnetic ground state requires investigation below 2 K. They confirmed the presence of magnetic order in the single-crystal, double-perovskite Ba₂YIrO₆ and Sr-doped Ba₂YIrO₆ below 1.7 K using dc magnetization, ac magnetic susceptibility, as well as heat-capacity measurements. From a fitting over a wide temperature range 50 < T < 300 K in this work, the authors estimated an effective moment of 1.44μ_B /Ir in Ba₂YIrO₆ and a large Curie–Weiss temperature of −149 K. The authors questioned the large singlet–triplet gap predicted earlier [88], claiming that such a large gap does not explain the significant temperature dependence of the magnetic susceptibility at low temperature.

The debate continued as, in the same year, Corredor et al [94] even questioned the previously reported monoclinic symmetry of Sr₂YIrO₆ [74, 133] and proposed instead the same cubic structure as its Ba analogue. According to this work Sr₂YIrO₆ does not undergo any magnetic transition down to 430 mK. They pointed out the low-temperature anomaly in the specific heat as a Schottky anomaly due to paramagnetic impurities, and is, therefore, not related to the onset of magnetic order. The small effective magnetic moment μ_eff = 0.21μ_B /Ir and the small correlation were attributed to the J = 1/2 impurities in the sample. Along the same line, the observed moment in Ba₂YIrO₆ was ascribed to extrinsic origin, such as paramagnetic impurity [89], anti-site disorder [90], Ir⁴⁺/Ir⁶⁺ magnetic defects [91], excluding the possibility of any excitonic magnetism (see scenario II in figure 9(b)).

On the contrary, Nag et al [77] pointed out that the temperature dependence of the magnetic response below 10 K for Ba₂YIrO₆ as revealed from their μSR studies is rather unusual. In particular, the temperature dependence of the stretched exponent parameter β of the fitted muon polarization P(t) = $\mathrm{exp}-{\left(\alpha t\right)}^{\beta }$ cannot be corroborated with a J = 0 state, for which no such temperature dependence of β should exist. The observed behavior is explained as follows: while a small iridium moment exists ubiquitously in Ba₂YIrO₆, at higher temperature the system behaves as a paramagnet. When the temperature is lowered below 10 K, most of these Ir moments start to interact with each other anti-ferromagnetically and pair up to form SOSs, leaving behind a few Ir ions due to Y/Ir disorder and with a very low residual interaction between them. This is in sharp contrast with the previous proposal where the impurity in the J = 0 background has been suggested for Ba₂YIrO₆. The proposed scenario of Nag et al is quite similar to its 4d analogue Ba₂YMoO₆ [135], supporting the scenario I in figure 9(b).

Clearly the ground states of Sr₂YIrO₆ and Ba₂YIrO₆ are quite controversial. The situation becomes even more puzzling with the report of resonant inelastic x-ray scattering (RIXS) measurements in these double perovskite systems [77, 136] that facilitate a quantitative estimation of the parameters λ and J_H in these systems. We shall discuss the RIXS technique in some detail later in the context of 6H perovskite iridates. Interestingly, regardless of the presence of the non-cubic crystal field (Sr vs Ba compound) or its strength (Y vs Gd compound), the RIXS spectra in all these double perovskite iridates are quite similar and can be described by an atomic model with λ ≈ 0.39–0.42 and J_H ≈ 0.24–0.26 eV [77, 136]. While theoretical studies [90, 92, 137] suggest excitonic magnetism is unlikely in Ba₂YIrO₆ due to large estimated singlet–triplet gap in RIXS measurement, reference [93] indicates the comparable energy scales of Hund's coupling and SOC, as a possible factor to drive magnetism in these double perovskite iridates. Furthermore, recent DFT calculations [138] show the presence of type-I AFM state in Ba₂YIrO₆, similar to the initial DFT results [75], confirming the possibility of interesting physics beyond the atomic limit. The insulating state in reference [138], however, occurs at a higher U than in reference [75] (U = 4 vs U = 2 eV). The magnetism in reference [138] is attributed to the 'intracluster excitons' that arise from the Ir–O interaction within an IrO₆ cluster.

The formation of moments at the iridium site was also discussed in the Sc counterparts of Sr₂YIrO₆ and Ba₂YIrO₆, viz, Sr₂ScIrO₆ and Ba₂ScIrO₆ [96–98]. Similar to the Y compounds, Ba–Sc compound crystallizes in the cubic $Fm\overline{3}m$ structure while Sr–Sc compound undergoes monoclinic distortion of the P2₁/n space group. Experimentally, the systems do not exhibit long range magnetic order while an effective moment of 0.16μ_B /Ir for the Sr compound and even larger moment of 0.39μ_B /Ir for the Ba compound was estimated. The generated magnetic moment was attributed to the hopping induced de-localization of holes in these systems [97, 98].

The compositional flexibility within the general formula of A₂ BIrO₆ leads to a large number of pentavalent double perovskite iridates, that includes but not limited to the listed materials in table 1. Further flexibility is achieved by doping at the A-site or/and using a magnetic atom at the B site. While doping offers the possibility to explore the effect of the resulting structural distortion on the magnetic properties of the system, magnetic B-site offers the direct study of 3d–5d interaction, combining the strong correlation of 3d TM ion to the strong SOC of 5d TM ion. While this offers a broad area of research spanned by some excellent works [101, 103–105], we do not discuss them in the present review.

The first post-perovskite oxide with pentavalent atom NaIrO₃, containing Ir in the d⁴ configuration, was synthesized using high-pressure solid state method [110]. The post perovskite NaIrO₃ crystallizes in the layered orthorhombic Cmcm space group. In contrast to the corner shared IrO₆ octahedra in the double perovskite iridates, discussed above, here the structure consists of distorted IrO₆ octahedra which not only share their corners along the c-axis but also share the edges along the a-axis and, thereby, form IrO₃ sheets. As shown in figures 11(a) and (b), these sheets are, further, stacked along the b direction, separated by the Na atoms.

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Interestingly, the magnetic measurement as well as DFT calculations [110] confirm the nonmagnetic state of Ir in NaIrO₃. This was, thus, considered an ideal example for the realization of the J = 0 nonmagnetic state [110], in marked contrast with the series of compounds, discussed in the previous section. The electrical resistivity in this work showed that the system is non-metallic, however the DFT calculation fails to describe the insulating state observed in experiments, leaving the origin of the resistivity unclear. A variable range hopping model was used to describe the low temperature resistivity of NaIrO₃. This put forward two important questions for this post-perovskite iridate: firstly, whether NaIrO₃ is a true realization of the J = 0 ground state? Second, whether the material is closer to a 'renormalized band insulator', where the correlation reduces the effective bandwidth which upon SOC becomes gapped out, or is it closer to 'Mott insulator', driven by strong Coulomb interaction. The DFT calculations in references [75, 111] addressed these issues, providing insights to the answer to these questions, which we now proceed to discuss.

4.2.1.  S = 0 instead of J = 0 state in NaIrO₃

4.2.2. Insulating state

We now discuss the d⁴ 6H-perovskite iridates with the general formula Ba₃ MIr₂O₉. The divalent cation M, e.g., Ca, Mg, Zn, Sr renders 5+ charge state to Ir. These 6-H perovskite iridates were synthesized by Sakamoto et al [113] back in 2006 and, then, further revisited with an idea to understand the SOC effects [76, 78, 112, 115]. It is interesting to point out here that similar to the double perovskite structure, 6H pervoskite structure also offers the composition flexibility, viz, the varying charge state of the M cation results in different valency of the Ir ion, which includes the rather unusual fractional charge states in addition to the common 4+ charge state of Ir atom [100, 113, 139–142]. For example, Li ion at the M site or trivalent M ions such as, Y, Sc, In, give rise to fractional charge states 5.5+(d^3.5), and 4.5+(d^4.5) at the Ir site respectively, while tetravalent M ions, e.g., Ti and Zr result in a 4+ charge state of Ir ion (d⁵). Here we address the two key issues, viz, the impact of the local distortion, assisted by the M cation, on the magnetic properties of the 6H perovskite iridates and, also, discuss the hopping induced formation of the magnetic moment at the Ir site.

4.3.1. Structural distortion and its impact on the magnetic properties of Ba₃ MIr₂O₉

The basic crystal structure of Ba₃ MIr₂O₉ is quite intriguing. In contrast to the previously discussed structures, here the IrO₆ octahedra share their faces with each other forming thereby Ir₂O₉ bi-octahedra, which in turn leads to the formation of structural Ir–Ir dimer. Such dimers are again arranged on a geometrically frustrated triangular network, as illustrated in figure 12. Furthermore, the size of the cation M, has a significant impact on the crystal geometry, which, in turn, affect the magnetic properties in different members within the same 6H perovskite family. For example, the Rietveld analyses of the x-ray diffraction data [76, 112, 113] show that while Ba₃ZnIr₂O₉ and Ba₃MgIr₂O₉ with the smaller cations Zn and Mg crystallize in the hexagonal P6₃/mmc space group, having less distorted IrO₆ octahedra, rest of the compounds, where the cation M is larger in size, adopt the crystal structure with monoclinically distorted C2/c space group, having significantly distorted IrO₆ octahedra. While in the Zn and Mg compounds, the trigonal distortion of the face-shared IrO₆ octahedra in the hexagonal symmetry splits the Ir-t_2g levels into a low-lying doublet (${e}_{g}^{\pi }$) and a singlet (a_1g ) with a non-cubic crystal field splitting Δ_NCF, the monoclinic symmetry of the Ca and Sr compounds leads to complete removal of the degeneracy of the t_2g orbitals, denoted by t₁, t₂, and t₃ in figure 12(d), with the corresponding energy differences Δ₁ and Δ₂ respectively (see figure 12(d) for pictorial representation). The stronger distortion in the latter compounds also leads to stronger non-cubic crystal field splitting in these systems. The comparison of the crystal geometry and the resulting non-cubic crystal field splitting (Δ_NCF or Δ₁, Δ₂) are listed in table 3.

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Table 3. The comparison of the structural parameters and the resulting non-cubic crystal field splitting in Ba₃ MIr₂O₉, M = Zn, Mg, Ca and Sr.

	Ba3ZnIr2O9	Ba3MgIr2O9	Ba3CaIr2O9	Ba3SrIr2O9
a (Å)	5.770 34	5.7705	5.9091	5.9979
b (Å)	5.770 34	5.7705	10.2370	10.3278
c (Å)	14.3441	14.3216	14.7624	15.1621
β (°)			91.14	92.78
Ir–O1 (Å)	2.021	2.058	2.010	2.041
Ir–O2 (Å)	1.948	1.932	2.079	2.026, 2.181
Ir–O3 (Å)			1.849	1.991
Ir–O4 (Å)			1.865	1.839
Ir–O5 (Å)			2.092	2.079
${d}_{\text{Ir}}^{\text{intra}}$ (Å)	2.75	2.76	2.735	2.69
${d}_{\text{Ir}}^{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r},c}$ (Å)	5.54	5.52	5.52, 5.78, 5.86	5.86, 5.99, 6.11
${d}_{\text{Ir}}^{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r},ab}$ (Å)I	5.78	5.77	5.909, 5.91, 5.91	5.97, 5.97, 5.99
∠Ir–O1–Ir (°)	85.92	85.96	85.75	82.55
∠Ir–O2–Ir (°)			80.06	79.50
ΔNCF (eV)	0.04	0.06	0.07, 0.24	0.09, 0.16

The impact of the crystal geometry on the magnetic properties have been studied both theoretically and experimentally. While Sakamoto et al showed that all the systems behave paramagnetically with no long range magnetic order down to 1.8 K and described the magnetic properties of these systems by Kotani's theory [143], later a combined experimental and theoretical analysis [76] of Ba₃ZnIr₂O₉ showed that the compound exhibits a fascinating spin–orbital liquid state down to at least 100 mK. In contrast to the favorable FM interaction in absence of SOC between two Ir ions with d⁴ electronic configuration, the spin–orbit entangled Γ₇ and Γ₈ states at the neighboring Ir sites, originating from the Ir-t_2g orbitals in presence of SOC, interact with each other anti-ferromagnetically (see figures 12(e) and (f)). The key to the AFM interaction in presence of SOC is the unusual hopping between pseudo-spins. Unlike the hopping between real spin that are always spin-conserved, the spin–orbital entanglement also allows for certain hopping between two states with different pseudo-spins, e.g., Γ₈ electrons can virtually hop from |1α⟩ (|1β⟩) to |3β⟩ (|3α⟩) state. These electrons, which are promoted to the Γ₇ state further can gain energy by hopping to the other Γ₇ state as illustrated in figure 12(f). This later hopping is, however, forbidden if the interaction is ferromagnetic, thereby stabilizing an AFM interaction between the closely spaced Ir atoms forming dimers. Such AFM interaction manifests in the negative Curie–Weiss temperature θ_CW ∼ −30 K with an effective moment ∼0.2μ_B /Ir in the susceptibility measurements. The magnetic moment at the Ir site represents the hopping assisted excitation from J = 0 state to J = 1 state. We come to the detail discussion of this point in the latter part of this section.

The AFM interaction within the Ir–Ir dimer leads to the formation of SOS state. Each of these SOSs, further, anti-ferromagnetically interact with six other neighboring SOSs located on a frustrated triangular network within the a–b plane and three out of the plane neighbors, corresponding to the Ir–Ir distances ${d}_{\text{Ir}}^{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r},ab}$ and ${d}_{\text{Ir}}^{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r},c}$ in table 3 respectively. The energy scales of these intra-dimer ${J}_{d}^{\text{intra}}$ (−14.6 meV, with neighboring single Ir ion) and inter-dimer interactions ${J}_{d}^{\text{inter}}$ (−1.5 meV for six neighbors within the ab plane and −1.6 meV with three neighbors perpendicular to the ab plane) being comparable, a strong magnetic frustration is developed in the system, giving rise to a spin–orbital liquid state, as evident from the muon-spin rotation, magnetization and heat capacity measurements [76].

The magnetic properties of the Mg compound is very similar to the Zn compound, showing noticeable AFM interaction from magnetic susceptibility measurement, although the heat capacity and μSR measurements suggest a weaker frustration in this case. The weaker frustration in the Mg compound results from the stronger ${J}_{d}^{\text{intra}}$ (−21.3 meV) compared to ${J}_{d}^{\text{inter}}$ (−0.9 meV within the a–b plane and −1.4 meV out-of-plane interaction) in the system. On the other hand, the strong distortion of the IrO₆ octahedra in the Sr and the Ca compounds break the local inversion symmetry, giving rise to uncompensated nearest neighbor Dzyaloshinskii–Moriya (DM) interactions. These DM interactions are expected to cause the canting of the spins, resulting in non-collinear magnetic structures. Indeed, explicit DFT calculations [112] with the symmetry allowed magnetic structures corresponding to an assumed propagation vector $\overrightarrow {k}=(0,0,0)$ shows that the lowest energy magnetic configuration corresponds to the magnetic space group C2/c', in which the Ir ions forming the dimer have anti-parallel spin components along the x, and z directions while due to canting have a parallel y component. This net y component of spin moments from a pair of neighboring Ir ions, however, is anti-parallel to the net y moment of another pair of Ir ions, situated along the c direction, leading to a net zero moment in the system. The weak dimeric feature observed in the magnetic susceptibility of Ba₃CaIr₂O₉ may be an outcome of the antiparallel inter-dimer arrangement of moments in the C2/c' magnetic structure as shown in figure 12(g). On the contrary, similar analysis [112] showed that another symmetry allowed magnetic configuration C2'/c' is energetically favorable for the Sr compound. Interestingly, this magnetic configuration allows for a net FM moment in the system, possibly leading to a ferro-magnetic like feature in magnetic susceptibility under low magnetic fields. Thus, the family Ba₃ MIr₂O₉ with M = Zn, Mg, Ca, Sr constitutes an intriguing platform to investigate the effect of local structural distortion in the magnetic properties of spin–orbit entangled d⁴ state.

4.3.2. Hopping induced magnetism

As already discussed, a spontaneous magnetic moment, due to the large Ir–Ir hopping, is developed at the Ir site in Ba₃ MIr₂O₉ (M = Zn, Mg, Ca, Sr). In contrast to the double perovskite iridates, where the Ir atoms are separated apart, in the 6H perovskite structure the Ir atoms form dimer, which constitute the basic building block of the 6H structure. This strong Ir–Ir dimer interaction manifests itself in the RIXS spectrum, that probes the low energy excitations of these iridates. In contrast to the double perovskite iridates, the RIXS spectrum in Ba₃ MIr₂O₉ has distinct features, as shown in figure 13(b), which cannot be captured within an atomic model. A meaningful description of such RIXS spectrum requires at least a two-site model with finite inter-site hopping, emphasizing the importance of the band structure effect on the ground state properties of these series of 6H iridates. Here, we provide an overview of the band structure effect and its implications to the RIXS spectra observed for Ba₃ MIr₂O₉ systems.

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We begin with a brief description of the RIXS process. RIXS is a second order 'photon-in photon-out' spectroscopic process, as illustrated schematically in figure 13(a). In case of d⁴ iridates, the incoming photon with frequency ω₁ excites the Ir-2p core electron to the Ir-d, t_2g unoccupied states (corresponding to the L₃ absorption edge) forming an intermediate state $\vert n\rangle =\vert 2{p}_{\frac{3}{2}}^{\enspace \text{hole}}{t}_{2g}^{5}\rangle$ with core-hole. To fill up the hole in the core level, a different t_2g electron corresponding to the initial ${t}_{2g}^{4}$ configuration may come back to the 2p state resulting in an emission of a photon of frequency ω₂ and a final state $\vert f\rangle =\vert {t}_{2g}^{4}\rangle$. Note that the final state |f⟩ can be any kind of arrangement of four electrons within the three t_2g orbitals and the energy difference ω₁ − ω₂ corresponds to these d–d excitation energies. The estimation of these energies are, therefore, beyond the scope of single-particle description and it is important to consider a many-body Hamiltonian with proper interactions.

The typical RIXS spectrum for Ba₃ MIr₂O₉ is shown in figure 13(b). Such a spectrum, in particular the additional peaks as shown in the figure, cannot be described by the triplet (J = 0 → J = 1) and quintet (J = 0 → J = 2) spin–orbit excitons of an atomic model for spin–orbit coupled d⁴ systems. Indeed, the two-peak feature is the characteristic of a two-site model with a finite inter-site hopping (see figure 13(c)), indicating the necessity to go beyond the atomic limit to incorporate the non-local effects, such as non-cubic crystal field and the hopping, present within the solid.

The spin–orbit exciton energies in the RIXS spectra may be mapped into the corresponding differences in the energy eigenvalues computed using exact diagonalization of the two-site model (similar to the Hamiltonian discussed in section 3.2, equation (6) with an additional non-cubic crystal field term). In the two-site model, the atomic states interact with each other in presence of significant hopping giving rise to a new set of spin–orbit coupled states, e.g., atomic states J = 1 (J₁) at each sites interact with each other to give rise to a new set of nine states, which we denote here by J₁–J₁ for simplicity. The resulting new sets of spin–orbit coupled states are: J₀–J₀ (1), J₀–J₁ (6), J₁–J₁ (9), J₀–J₂ (10), J₁–J₂ (30), J₂–J₂ (25), etc, where the number within the parenthesis indicates the number of states arising due to interaction between the two-atomic states. The excitation from the J₀–J₀ state to the states J₁–J₁ and J₀–J₂ states describe the broad peak around ∼0.4 eV in the RIXS spectrum, observed in the experiments.

An estimation of the various parameters, starting from a realistic set of values extracted from the ab initio calculations, may be obtained when the theoretically computed excitation energies coincide with the extracted energy loss values within the uncertainty of the measurement (full width at half maximum of the Lorentzian function). Note that unlike the atomic model, the energies of the spin–orbit excitons obtained from the two-site model depend on the Coulomb repulsion U possibly due to the superexchange energy scale ∼t²/U involved in the latter case. The estimated Hund's coupling J_H is about ∼0.45 eV, which is comparable to the other 5d⁵ materials but higher than the d⁴ double perovskite iridates [136, 144]. More interestingly, the SOC λ is highly suppressed to a value of ∼0.26 eV, highlighting the decisive role of hopping in these 6H iridates. The large intra-dimer hopping and the strongly suppressed SOC leads to deviation of the two-site ground state from the ideal J = 0 state by about 10%, resulting in a non-zero moment as also evident from the ground state expectation value of ⟨J²⟩ (see figures 13(d) and (e)). In a nut shell, Ba₃ MIr₂O₉ is an exemplary d⁴ system, where the hopping dramatically affects the spin–orbit coupled ground state, leading to formation of moment at the 'atomically non-magnetic' Ir site.

The pentavalent iridate Sr₃ MIrO₆ crystallizes in the hexagonal structure with the $R\overline{3}c$ space group symmetry, which is isostructural to the parent compound Sr₄IrO₆. The doping with alkaline metals M, e.g., Na, Li, K at the Sr site results in a reduced charge state of Ir, viz, the pentavalent iridate, the subject of the present review. To the best of our knowledge these pentavalent iridates have been synthesized as early as 1996 [117, 118, 122]. The crystal structure of Sr₃ MIrO₆ is quite interesting in the sense that the Ir atoms are far apart from each other, constituting isolated IrO₆ octahedral unit. These octahedra share their face with the neighboring MO₆ trigonal prism, forming thereby an alternating chain of IrO₆–SrO₆ along the crystallographic c-axis. The crystal structure of Sr₃ MIrO₆ consists of such well separated alternating chains, as shown in figure 14. The large Ir–Ir distance and the lower connectivity of Ir atoms results in a narrow Ir-t_2g bands, which, in turn, gives rise to the hope of realizing the J = 0 state. Interestingly, existing experiments [117, 118] indicate a temperature-independent paramagnetic behavior for Sr₃ MIrO₆, with M = Na, Li, which further points towards a non-magnetic J = 0 state.

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With this viewpoint, recently DFT calculations [116] have been performed on both Na and Li compounds. These calculations find a non-magnetic state for these two systems. With the projection on to the j_eff states, the authors show that the j_eff = 3/2 and j_eff = 1/2 states are very well separated in energy without any mixing between them. Interestingly, the J = 0 states, proposed for this system, are insulating in nature even without Coulomb correlation. This is in sharp contrast to the other d⁴ iridates, discussed previously in this review, where finite Coulomb correlation is always required to open up an energy gap. The insulating property of these systems are, therefore, better described as SOC driven band insulating state, rather than the Mott insulating state. A simple explanation to this behavior lies within the reduced dimensionality of the structure, resulted from the isolated IrO₆ octahedra, as discussed before. As result of this structural low dimensionality, the bandwidth also reduces which, in turn, favors an insulating state. Notably, the bandwidth of these materials are almost half of the Ir-t_2g bandwidths found in the double perovskite and 6H perovskite structures which may be conducive for the realization of the J = 0 state.

Interestingly, recent experiments [145] however point towards magnetism at the Ir site for both Na and Li compounds, with even larger effective magnetic moment. Notably, the IrO₆ octahedra in these systems are not ideal and undergo distortions. It might be, therefore, interesting to revisit these materials both experimentally and theoretically with the new insights at hand to confirm or refute the presence of J = 0 non-magnetic state in Sr₃ MIrO₆. It is important to point out here that the isostructural d⁵ sister compound Sr₃NiIrO₆ exhibits a strong single-ion magnetic anisotropy [146]. In presence of any magnetism in the d⁴ counterparts, it would be furthermore interesting to see whether such single-ion magnetic anisotropy is also present in pentavalent iridates.