# Spin-orbit effects in pentavalent Iridates: Models and materials

Sayantika Bhowal  School of Physical Sciences, Indian Association for the
Cultivation of Science, Jadavpur, Kolkata 700 032, India  current address:
Materials Theory, ETH Zürich, Wolfgang-Pauli-Strasse 27, Zürich 8093,
Switzerland<EMAIL_ADDRESS>Indra Dasgupta  School of Physical
Sciences, Indian Association for the Cultivation of Science, Jadavpur, Kolkata
700 032, India<EMAIL_ADDRESS>
###### Abstract

Spin-orbit effects in heavy 5$d$ transition metal oxides, in particular,
iridates, have received enormous current interest due to the prediction as
well as the realization of a plethora of exotic and unconventional magnetic
properties. While a bulk of these works are based on tetravalent iridates
($d^{5}$), where the counter-intuitive insulating state of the rather extended
5$d$ orbitals are explained by invoking strong spin-orbit coupling, the recent
quest in iridate research has shifted to the other valencies of Ir, of which
pentavalent iridates constitute a notable representative. In contrast to the
tetravalent iridates, spin-orbit entangled electrons in $d^{4}$ systems are
expected to be confined to the $J=0$ singlet state without any resultant
moment or magnetic response. However, it has been recently predicted that,
magnetism in $d^{4}$ systems may occur via magnetic condensation of
excitations across spin-orbit-coupled states. In reality, the magnetism in
Ir5+ systems are often quite debatable both from theoretical as well as
experimental point of view. Here we provide a comprehensive overview of the
spin-orbit coupled $d^{4}$ model systems and its implications in the studied
pentavalent iridates. In particular, we review here the current experimental
and theoretical understanding of the double perovskite ($A_{2}B$YIrO6, $A=$
Sr, Ba, $B=$Y, Sc, Gd), 6H-perovskite (Ba${}_{3}M$Ir2O9, $M=$ Zn, Mg, Sr, Ca),
post-perovskite (NaIrO3), and Hexagonal (Sr3MIrO6) iridates, along with a
number of open questions that require future investigation.

  *

##  1 Introduction

Spin-orbit coupling (SOC) is a relativistic effect, which may be thought of as
an interaction between the intrinsic spin moment of an electron and the
magnetic field, generated in the rest frame of the electron due to its orbital
motion around the positively charged nucleus. This interaction results in a
coupling between the spin ($\vec{S}$) and the orbital ($\vec{L}$) angular
momenta of an electron, lifting the $(2S+1)(2L+1)$ -fold degeneracy of an
atomic energy level, and hence the name “Spin-orbit coupling”. Mathematically,
the coupling emerges naturally from the Dirac Hamiltonian and may be written
as ${\cal H}_{\rm
SOC}=\frac{\hbar^{2}}{2m_{e}^{2}c^{2}}\frac{1}{r}\frac{dV}{dr}(\vec{L}\cdot\vec{S})=\lambda\vec{L}\cdot\vec{S}$,
where the potential $V(r)=-\frac{Ze^{2}}{4\pi\varepsilon_{0}r}$ due to the
nuclear charge $Ze$ determines the coupling constant $\lambda$.

The strong dependence of $\lambda$ on the atomic number $Z$ [1, 2, 3], renders
the inevitable spin-orbit effects in the heavy elements of the periodic table,
leading to a wide variety of fascinating phenomena, which has continued to be
the center of attention in the condensed matter research over a long period of
time. This includes, for example, various topological effects [4, 5, 6, 7, 8],
that stem from the non-trivial geometry of the electronic Bloch bands,
providing not only the platform to realize the well known particles in the
standard model of relativistic high-energy physics in the low energy
excitations of the condensed matter systems, but also leading to the emergence
of the field of quantum computation. In addition, there are Rashba and
Dresselhaus spin-orbit interactions [9, 10, 11] due to a gradient of
electrostatic potential in non-centrosymmetric systems, which constitute the
heart of the present day hot topic of spin-orbitronics research [12, 13, 14,
15, 16, 17, 18, 19, 20]. The present review, however, focuses on a different
aspect of the SOC effects, namely, the interplay between SOC and Coulomb
interaction in a crystalline solid, which often leads to unconventional phases
of matter including the novel spin-orbit Mott insulating state.

While the importance of SOC in correlated systems had already come into notice
[21, 22, 23], the breakthrough came with the pioneering work of Kim et. al
[24] in the year 2008, where the authors have shown that the SOC is absolutely
essential to describe the insulating state of Sr2IrO4, as the rather moderate
Coulomb interaction alone can not explain the opening of an energy gap in the
extended Ir-5$d$ states. As predicted by Jackeli and Khaliullin [25], this
unique combination of SOC and Coulomb interaction can offer the long-awaited
realization of the Kitaev model, that hosts a number of fascinating properties
including anyonic excitations, topological degeneracy, etc., in real condensed
matter systems, replacing the usual Heisenberg spin model. The idea of spin-
orbit assisted Mott insulating state led to extensive studies over the past
decade, in the search for new spin-orbit coupled correlated materials, their
characterizations, as well as reinvestigation of previously synthesized
materials in the light of the new insights on SOC.

Among these various spin-orbit coupled systems, the family of iridates have
received considerable attention, primarily because of the availability of
various crystal geometries as well as different Ir valencies within the same
family. A large section of the studied iridates share the same
Ir${}^{4+}:d^{5}$ electronic configuration as the prototypical spin-orbit Mott
insulator Sr2IrO4, where the different connectivities of the IrO6 structural
unit resulting from diverse crystal geometries, give rise to a plethora of
exotic electronic and magnetic properties of matter such as novel spin-orbit
entangled metallic state, [26, 27, 28, 29, 30], magnetic Kitaev interactions
[31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], various topological
states [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55], e.g., Dirac and Weyl
semimetal, nodal line semi-metals, Axion insulators, quantum spin liquid
states [56, 57, 58], hidden order multipolar state [59, 60, 61, 43, 62],
superconductivity [63, 64], etc. A number of excellent reviews [65, 66, 67,
68, 69, 70, 71] already exist in the literature, addressing many of these
fascinating properties.

The present review, however, focusses on another charge state of iridium, the
5+ charge state, Ir5+ with $d^{4}$ electronic configuration, known as
pentavalent iridates. Similar to the tetravalent iridates, described above,
pentavalent iridates also offer a wide range of structural variety. The
magnetic properties of these iridates are, however, strikingly different owing
to the dramatic effect of spin-orbit interaction on a local $t_{2g}^{4}$
electronic configuration. While in absence of SOC, degenerate $t_{2g}^{4}$
state results in a magnetic $S=1$ state, in presence of strong SOC orbital and
spin angular momenta are coupled to form a non-magnetic $J=0$ state within an
atomic limit. Locally this singlet $J=0$ ground state is separated from the
higher lying triplet $J=1$ state by an energy scale that is determined by the
strength of the SOC, $\lambda$. However, similar to the quantum dimer model
[72], the superexchange interaction between the neighboring multiplets may
cause a dispersion of the triplet state, facilitating the possibility of a
magnetic excitation from $J=0$ to $J=1$ state, depending on the relative
values of the energy scales involved. Such a magnetic excitation maybe viewed
as annihilation of singlet bosons and the simultaneous creation of triplet
bosons, known as triplon excitons. The prediction [73] of such excitonic
magnetism in the nonmagnetic $J=0$ ground state of $d^{4}$ spin-orbit coupled
systems motivated tremendous interest in the pentavalent iridates.

While the first $d^{4}$ Ir-based material was synthesized about sixty years
ago, around 1960’s [74, 75], the interest in the $d^{4}$ spin-orbit coupled
materials is renewed with the recent advancement in the understanding of the
SOC effects. In real solids, the situation is even more complicated in the
presence of additional interactions such as non-cubic crystal field
($\Delta_{\rm NCF}$), bandwidth (W), Hund’s Coupling ($J_{H}$), Coulomb
interaction ($U$), etc., which often compete among each other. Indeed, the
ground state of $d^{4}$ iridates is determined by the delicate balance between
these various competing interactions, leading to the emergence of a number of
unconventional and exotic phases [76, 77, 78, 79, 80]. The magnetic properties
of such subtle ground state in real materials are, however, often quite
debatable both from theoretical as well as experimental point of view.

The present review aims at providing an overview of the spin-orbit physics in
$d^{4}$ iridates, taking explicitly a few notable oxide members of this family
into account. The idea of this review is to focus on some of the key issues in
this field as well as put forward some open questions, inspiring future work
along this direction. We first discuss the theoretical models, developed for
$d^{4}$ spin-orbit coupled systems, and, then, explicitly examine their
implications by reviewing some of the reported $d^{4}$ iridates, synthesized
in the form of double perovskite, post perovskite, 6H perovskite, and
hexagonal structures. Interestingly, the various crystal geometries also
facilitate different connectivities of the IrO6 octahedra, e.g., double
pervoskite and hexagonal iridates contain isolated IrO6 octahedron, while each
IrO6 unit in post pervoskite structure shares its edge as well corner with the
neighboring octahedron. On the other hand, IrO6 octahedra in 6H iridates are
face-shared. Such diverse connectivities, in turn, provide the opportunity to
investigate and compare the interplay between the different competing energy
scales, governed by the specific structural geometries.

##  2 Background of the $J=0$ state and Iridates

In this section, we provide the background of the $J=0$ state. In particular,
we discuss the necessary criteria for the formation of the $J=0$ state,
explain how these requirements, often, can be met in iridates, thereby, making
them good candidate materials to study, and briefly mention the possibility of
excitonic magnetism, which we, further, discuss in detail in section 3, which
has propelled enormous interest in $d^{4}$ iridate research.

###  2.1 Formation of $J=0$ state

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., $\lambda_{5d}>\lambda_{4d}>\lambda_{3d}$, as also
schematically illustrated in Fig. 1 (a).

Interestingly, going from 3$d$ to 5$d$ TM elements, the $d$ orbitals get
spatially more extended [see Fig. 1 (b)]. As a result, the wide 5$d$ orbitals
hybridize strongly with the neighboring ligand atoms, leading to a larger
crystal field splitting in 5$d$ TM compounds compared to the systems with
narrow 3$d$ bands. This has some interesting consequences, which play crucial
role in the formation of $J=0$ state in a $d^{4}$ electronic configuration.
For a simple illustration, we consider TM-O6 octahedral unit. Indeed such
units are quite common in various transition metal 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 3$d$
orbitals in the valence state. Due to weak crystal field splitting ($\Delta$)
in these systems, the Hund’s coupling $J_{\rm H}$ that favors parallel spin
alignment of the electrons in different $d$ orbitals win, and $3d^{4}$
electronic configuration leads to a high spin $S=2$ state
($t^{3}_{2g}e^{1}_{g}$), as seen for example in LaMnO3 [81]. The SOC effects
being rather weak, the crystal field split states are only very weakly
perturbed by SOC. In case of 4$d$ TM elements, however, the 4$d$ orbitals
being wider than the 3$d$ orbitals, $\Delta$ is relatively larger in
magnitude. In this case, $\Delta$ wins over the Hund’s coupling $J_{\rm H}$,
resulting in a low spin state $S=1$, with four electrons in the $t_{2g}$
orbitals ($t^{4}_{2g}e^{0}_{g}$ ) [82, 83]. The formation of the low spin
state is the first step towards the $J=0$ state. However, due to its moderate
value in 4$d$ TMs, often SOC acts only as a perturbation to the $t_{2g}$
states. It is interesting to point out here that the recent report of a $J=0$
ground state in 4$d$ based Ru compound K2RuCl6 within $LS$ coupling scheme
points to the importance of SOC (beyond perturbation) also in 4$d$ based
systems [84]. Nevertheless, the remaining criterion of strong SOC is satisfied
in $5d$ TMs, where strong SOC forms spin-orbital entangled states, destroying
the individual orbital and spin identities of the constituents.

Figure 1:  The formation of $J=0$ state. (a) Schematic illustration of the
different energy scales of 3$d$, 4$d$ and 5$d$ orbitals, where the base of the
triangle denotes the band-width $W$ of the respective orbitals situated at an
increasing height from the base. The increasing and decreasing values of the
different parameters: Coulomb correlation $U$, Hund’s coupling $J_{H}$, SOC
$\lambda$, and crystal field splitting $\Delta$ along the height of the
triangle are indicated by big arrows. (b) Radial distribution
$r^{2}R_{nl}^{2}(r)$ for 3$d$, 4$d$ and 5$d$ orbitals as a function of
distance from the nucleus $r$, indicating that the $5d$ orbitals are more
spatially extended than the rest, leading to larger $W$ as shown in (a). (c)
The evolution of the ground state of a $d^{4}$ atomic system in an ideal
octahedral environment, depending on the relative energy scales of the
parameters. Left: The magnetic high spin $S=2$ state in the $J_{H}>\Delta$
regime, relevant for $3d$ orbitals. Middle: The magnetic low spin $S=1$ state
in the $\Delta>J_{H}$ regime, typical for $4d$ orbitals in presence of weak
SOC. Right: Spin-orbit entangled non-magnetic $J=0$ state in presence of large
SOC, expected for heavy $5d$ orbitals. See text for details.

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.,

$\displaystyle{\cal H}_{\rm SOC}$
$\displaystyle=\left(\begin{array}[]{cc}{\cal H}_{+}&0\\\ 0&{\cal
H}_{-}\end{array}\right),$ (3)

where ${\cal H}_{+}$ (${\cal H}_{-}$) in the basis $\\{|yz\downarrow\rangle$,
$|zx\downarrow\rangle$, $|xy\uparrow\rangle\\}$ ($\\{|yz\uparrow\rangle$,
$|zx\uparrow\rangle$, $|xy\downarrow\rangle\\}$) has the following form

$\displaystyle{\cal H}_{\pm}$ $\displaystyle=\left(\begin{array}[]{ccc}0&\mp
i\frac{\lambda}{2}&\pm\frac{\lambda}{2}\\\ \pm
i\frac{\lambda}{2}&0&+i\frac{\lambda}{2}\\\
\pm\frac{\lambda}{2}&-i\frac{\lambda}{2}&0\end{array}\right).$ (7)

We have assumed, here, degenerate $t_{2g}$ states in an ideal octahedral
environment. The diagonalization of ${\cal H}_{\rm SOC}$ yields four-fold
degenerate $\Gamma_{8}$ (-$\frac{\lambda}{2}$) and two-fold degenerate
$\Gamma_{7}$ ($\lambda$) states, where each of the eigenstates of ${\cal
H}_{+}$ has its time-reversal counterpart in the corresponding eigenstate of
${\cal H}_{-}$, forming the Kramer’s doublet. The explicit form of these
eigenstates are given by:

$\displaystyle|1\alpha\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}(|d_{yz\uparrow}\rangle+i|d_{zx\uparrow}\rangle),$
$\displaystyle|1\beta\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|d_{yz\downarrow\rangle}-i|d_{zx\downarrow}\rangle),$
$\displaystyle|2\alpha\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}(2|d_{xy\uparrow}\rangle-|d_{yz\downarrow\rangle}-i|d_{zx\downarrow}\rangle),$
$\displaystyle|2\beta\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}(2|d_{xy\downarrow}\rangle+|d_{yz\uparrow}\rangle-i|d_{zx\uparrow}\rangle),$

and,

$\displaystyle|3\alpha\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(|d_{xy\uparrow}\rangle+|d_{yz\downarrow}\rangle+i|d_{zx\downarrow}\rangle),$
$\displaystyle|3\beta\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(|d_{xy\downarrow}\rangle-|d_{yz\uparrow}\rangle+i|d_{zx\uparrow}\rangle).$

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 $j_{\rm eff}$ = 3/2 and
$j_{\rm eff}$ = 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 5$d$ shell, we can see that the lower lying $j_{\rm
eff}$ = 3/2 state gets completely occupied while the $j_{\rm eff}$ = 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, $j_{\rm eff}$ for
this state may be determined from the effective orbital angular momentum
$l_{\rm eff}=-1$ (see Appendix A for the illustration of this point) and the
spin angular momentum $s=1$, which again give a non-magnetic $j_{\rm
eff}=l_{\rm eff}+s=0$ state. The evolution of the ground state of a $d^{4}$
system in the different regimes of the competing energy scales is
schematically shown in Fig. 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.

Figure 2:  The simplified picture of the magnetic-response of a singlet-
triplet system in presence of a magnetic field $\vec{B}$. In absence of a
magnetic field, $\vec{B}=0$, the singlet $|J=0\rangle$ and the triplet
$|J=1\rangle$ states are separated by an energy gap of $\Delta_{st}$, while in
presence of a magnetic field the triplet state $|J=1\rangle$ undergoes a
Zeeman splitting, leading to splitting of the $|J_{z}=\pm 1\rangle$ states,
with energies $E_{\mp}=\Delta\mp g_{J}\mu_{B}B$. When the strength of the
applied magnetic field $B$ exceeds a critical value
$B_{c}=\Delta/g_{J}\mu_{B}$, the $|J_{z}=+1\rangle$ state lies below the $J=0$
state, leading to a sharp magnetic transition. In reality, the behavior of the
singlet state depends on the matrix elements of the magnetic dipoles between
the non-magnetic ground state and the magnetic excited state.

As an aside, it is interesting to point out that for five electrons in the
5$d$ shell, the $j_{\rm eff}$ = 1/2 state gets partially occupied in addition
to the completely filled $j_{\rm eff}$ = 3/2 state. This half-filled $j_{\rm
eff}$ = 1/2 state is further split into empty upper and filled lower Hubbard
bands in presence of Coulomb interaction, leading to the well known $j_{\rm
eff}$ = 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 [85].

###  2.2 Origin of excitonic magnetism in spin orbit coupled $d^{4}$ systems

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
$|\psi_{0}\rangle$ with the magnetic excited states $|\psi_{n}\rangle$ gives
rise to a magnetic response in presence of a magnetic field ($\vec{B}$). In
the language of perturbation theory, this can be expressed as
$|\psi\rangle=|\psi_{0}\rangle+\sum_{n}\frac{\langle\psi_{n}|(\vec{B}\cdot\vec{J})|\psi_{0}\rangle}{E_{n}-E_{0}}$
[86]. The essence of the behavior of a singlet-triplet system in presence of a
magnetic field is schematically illustrated in Fig. 2.

####  2.2.1 Why Iridates?

The 5$d$ TM oxides, of which iridates are notable members, facilitate the
basic requirements for the formation of a non-magnetic $J=0$ state. In such a
non-magnetic state, however, magnetism may appear by a mechanism very similar
to Van-Vleck paramagentism via excitation to the higher lying triplet $J=1$
state, as suggested by Khaliullin [73]. The possibility of such excitation is
governed by the relative energy scales of the singlet-triplet splitting
($\Delta_{st}$), determined by the SOC constant $\lambda$, and the
superexchange energy, which is $\sim 4t^{2}/U$. When these two energy scales
are comparable, singlet-triplet Van Vleck transitions may occur, resulting in
a magnetism in the apparently non-magnetic $J=0$ state.

####  2.2.2 Singlet-triplet transition: Excitonic magnetism:

Following Khaliullin’s idea [73] 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 Fig. 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
$t(a_{i}^{\dagger}a_{j}+b_{i}^{\dagger}b_{j}+H.c.)$, 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 Fig. 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., $A=\\{a^{2}bc\\},B=\\{ab^{2}c\\},$ and
$C=\\{abc^{2}\\}$. 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., $L_{x}=-i(B^{\dagger}C-C^{\dagger}B)$, etc.

Figure 3:  The schematic illustration of the 180∘ $d-p-d$ bond. The figure
shows the two active orbitals, $d_{xz}$ and $d_{xy}$, along a bond in the
$x$-direction for corner shared octahedral geometry.

In the strong coupling limit ($U>>t,\lambda,J_{H}$), this results in a spin-
orbital Hamiltonian, having the form $H=(t^{2}/U)\sum_{\langle
ij\rangle}[(\vec{S}_{i}\cdot\vec{S}_{j}+1)O^{(\gamma)}_{ij}+(L_{i}^{\gamma})^{2}+(L_{j}^{\gamma})^{2}]$,
where $O^{(\gamma)}$ 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
$O^{(c)}=(L^{x}_{i}L^{x}_{j})^{2}+(L^{y}_{i}L^{y}_{j})^{2}+L_{i}^{x}L_{i}^{y}L_{j}^{y}L_{j}^{x}+L_{i}^{y}L_{i}^{x}L_{j}^{x}L_{j}^{y}$.
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,
$|0\rangle=\frac{1}{\sqrt{3}}(|1,-1\rangle-|0,0\rangle+|-1,1\rangle)$ and the
first excited state $J=1$ triplet at an energy $\Delta_{st}$ determined by
$\lambda$,
$|T_{0}\rangle=\frac{1}{\sqrt{2}}(|1,-1\rangle-|-1,1\rangle),|T_{\pm
1}\rangle=\pm\frac{1}{\sqrt{2}}(|\pm 1,0\rangle-|0,\pm 1\rangle)$. 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” $\vec{T}$ with the
Cartesian components
$T_{x}=\frac{1}{i\sqrt{2}}(T_{1}-T_{-1}),T_{y}=\frac{1}{\sqrt{2}}(T_{1}+T_{-1}),$
and $T_{z}=iT_{0}$, and the “spin”
$\vec{J}=-i(\vec{T}^{\dagger}\times\vec{T})$. For example,
$\vec{S}=-i\sqrt{\frac{2}{3}}(\vec{T}-\vec{T}^{\dagger})+\frac{1}{2}\vec{J}$,
and
$\vec{L}=i\sqrt{\frac{2}{3}}(\vec{T}-\vec{T}^{\dagger})+\frac{1}{2}\vec{J}$.
Using these relations we can immediately write down the magnetic moment for a
$t_{2g}^{4}$ configuration in terms of $\vec{T}$ and $\vec{J}$, viz.,
$\vec{M}=2\vec{S}-\vec{L}=-i\sqrt{6}(\vec{T}-\vec{T}^{\dagger})+g_{J}\vec{J}$,
where $g_{J}=\frac{1}{2}$. Clearly, the magnetism in a $J=0$ state is driven
solely by the $\vec{T}$ exciton as the second term in $\vec{M}$ does not
contribute for a singlet state, $J=0$, and hence, the magnetism is referred to
as “excitonic magnetism”.

The transformation $H(L,S)\rightarrow H(\vec{T},\vec{J})$, as described above,
gives us the desired effective singlet-triplet model [73],

$H_{\rm
eff}=\Delta_{st}\sum_{i}T_{i}^{\dagger}T_{i}+\kappa\frac{2}{9}\sum_{ij}[\vec{T}_{i}^{\dagger}\cdot\vec{T}_{j}-\frac{7}{16}(\vec{T}_{i}\cdot\vec{T}_{j}+H.c.)].$
(8)

Here $\kappa$ denotes the exchange interaction $4t^{2}/U$ and terms upto
quadratic in $\vec{T}$ 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 $\vec{T}$ 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
$\kappa<\kappa_{c}$, where $\kappa_{c}$ is some critical value of $\kappa$,
the magnetic excitation has a finite gap
$\lambda\sqrt{1-(\kappa/\kappa_{c})}$, indicating a paramagnetic phase. At
$\kappa=\kappa_{c}$ this gap closes and a magnetic phase transition takes
place due to condensation of the $\vec{T}$ bosons.

Interestingly, soon after this proposal of excitonic magnetism, Cao et. al
[76] experimentally showed the existence of magnetism in the $d^{4}$ 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
LABEL:tab1), 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
$\lambda/2S$ [86, 87], where $\lambda$ is the atomic SOC, and $S$ is the total
spin, one might anticipate the SOC strength would be weaker in $d^{4}$ systems
with $S=1$ as compared to tetra-valent iridates with $S=1/2$. Such weak SOC
tends to favor excitonic magnetism in $d^{4}$ 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
[88] based on density matrix renormalization also suggests spin-orbit induced
transition to an excitonic anti-ferromagnetic condensate even at an
intermediate Coulomb interaction regime.

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 [77,
79, 89, 90, 91, 92, 93, 94, 95]. 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 [78]. The easy availability of the $d^{4}$
electronic configuration in iridates, compared to other 5$d$ TM oxides, have
further boosted the search for the realization of $J=0$ state and/or the
excitonic magnetism in Iridates.

[longtable]LTcapwidth=table

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 | Compound | Space | Transport | Magnetic |  $\mu_{\rm eff}/$Ir | Ref.
---|---|---|---|---|---|---
structure |  | group | property | ordering | ($\mu_{B}$) |
| Sr2YIrO6 | $P2_{1}/n$ | Insulator | AFM | 0.91 | [76, 77]
|  | $Fm\bar{3}m$ | Insulator | None | 0.21 | [96]
| Ba2YIrO6 | $Fm\bar{3}m$ | Insulator | None | 0.3 | [77, 79]
|  |  | Insulator | None | 0.44 |  [89]-[94]
|  | $P2_{1}/n$ | - | - | - | [97]
Double |  |  | Insulator | AFM | 1.44 | [95]
perovskite | Sr2GdIrO6 |  $Pn$-$3$ | Insulator | FM | - | [76, 77]
(Isolated) | Sr2ScIrO6 | $P2_{1}/n$ | Insulator | None | 0.16 | [98, 99, 100]
| Sr2LuIrO6 | $P2_{1}/n$ | - | None | - | [97]
| Ba2ScIrO6 | $Fm\bar{3}m$ | Insulator | None | 0.39 | [98, 99]
| Ba2LuIrO6 | $P2_{1}/n$ | - | None | - | [97]
| Ba2LaIrO6 | $P2_{1}/n$ | - | None | - | [97]
|  | $R\bar{3}$ | - | - | - | [101]
| Ba2NdIrO6 | Cubic | - | - | - | [102]
| Ba2SmIrO6 | Cubic | - | - | - | [102]
| Ba2DyIrO6 | Cubic | - | - | - | [102]
| Sr2InIrO6 | $P2_{1}/n$ | Insulator | None | 1.608 | [103]
| Sr2CoIrO6 | $I2/m$ | Insulator | AFM | 0.47 | [104, 105, 106]
| Sr2FeIrO6 | $I2/m$ | Insulator | AFM | - | [103]
|  | $I\bar{1}$ | Insulator | AFM | - | [107]
| Ca2FeIrO6 | $I\bar{1}$ | Insulator | AFM | - | [107]
| La2LiIrO6 | $Pmm2$ | - | None | 1.42 | [108, 109]
| BaLaMgIrO6 | Pseudocubic | - | None | - | [108]
| Bi2NaIrO6 | $P2_{1}/n$ | Insulator | None | 0.19 | [110]
| SrLaMgIrO6 | $P2_{1}/n$ | Insulator | None | 0.61 | [111]
| SrLaZnIrO6 | $P2_{1}/n$ | Insulator | None | 0.46 | [111]
| SrLaNiIrO6 | $P2_{1}/n$ | Insulator | AFM | 0.19 | [111]
Post perovskite | NaIrO3 | $Cmcm$ | Insulator | NM | 0 | [77, 112, 113]
(Corner & |  |  |  |  |  |
edge shared) |  |  |  |  |  |
| Ba3ZnIr2O9 | $P6_{3}/mmc$ | Insulator | SOL | 0.2 | [78]
6H-perovskite | Ba3MgIr2O9 | $P6_{3}/mmc$ | Insulator | None | 0.5-0.6 | [114, 115]
(Face shared) | Ba3CaIr2O9 | $C2/c$ | Insulator | None | - | [114, 115]
| Ba3SrIr2O9 | $C2/c$ | Insulator | None | - | [114, 115]
| Ba4LiIr3O12 | $P6_{3}/mmc$ | Insulator | Magnetic | - | [116]
| Ba4NaIr3O12 | $P6_{3}/mmc$ | Insulator | - | - | [116]
| Ba3CdIr2O9 | $C2/c$ | Insulator | None | 0.3 | [117]
Hexagonal | Sr3NaIrO6 | $R\bar{3}c$ | Insulator | None | - | [118, 119]
(Isolated) | Sr3LiIrO6 | $R\bar{3}c$ | Insulator | None | - | [118, 119, 120]
Pyrochlore | Cd2Ir2O7 | FCC | - | - | - | [121]
(Corner shared) | Ca2Ir2O7 | FCC | - | - | - | [121]
Others | KIrO3 | $Pn3$ | - | - | 1.04 | [122]
| Ba0.5IrO3 | $I23$ | Metal | - | - | [121]
| Sr0.5IrO3 | BCC | - | - | - | [121]
| Na3Cd2IrO6 | $C2/m$ | - | - | - | [123]

##  3 Microscopic model and breakdown of J=0 state

The various comparative energy scales, e.g., $\Delta$, $J_{\rm H}$ and
$\lambda$ as discussed above, in 5$d$ TM oxides enrich the physics of $d^{4}$
iridates. The presence of magnetism in different iridates out of apparently
non-magnetic individual Ir atoms piqued enormous scientific curiosity in the
$d^{4}$ systems. While the possibility of excitonic magnetism has already been
proposed, the understanding of the role of different competing parameters can
provide the answer to the two important questions, that are of primary
interest, namely, which parameter (or parameters) might drive the
uncoventional magnetism in $d^{4}$ spin-orbit coupled systems at the first
place. Secondly what would be the nature of this emergent magnetism. Generic
theoretical models, both atomic as well as beyond atomic limit, developed in
this context, are proven to be useful to gain valuable insight into the
physics of the $d^{4}$ spin-orbit coupled systems. Here we give a brief
overview of these results that essentially provide the ground to analyze the
magnetic response of the real materials.

###  3.1 Atomic model and non-cubic crystal field effect

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 $\Delta$, only the $t_{2g}$ orbitals are considered at each atomic
site. The corresponding atomic many-body Hamiltonian, spanned in the Hilbert
space of dimension ${}^{6}C_{4}=15$, includes the electron-electron
interactions as well as the SOC interaction, viz.,

${\mathcal{H}}_{\rm atom}~{}=~{}{\mathcal{H}}_{int}+{\mathcal{H}}_{SO}.$ (9)

Here the Kanamori Hamiltonian ${\mathcal{H}}_{int}$ contains the intra-orbital
and the inter-orbital Coulomb interactions $U$ and $U^{\prime}$, which are
related to each other via the Hund’s coupling $J_{H}$,viz.,
$U~{}=~{}U^{\prime}+2J_{H}$. The schematic illustration of the various
interaction terms are given in Fig. 4. The explicit form of the Hamiltonian at
an atomic site $i$ is given by [100, 125, 126, 127]

$\displaystyle{\mathcal{H}}^{i}_{int}$ $\displaystyle=$ $\displaystyle
U\sum_{l}n_{i,l\uparrow}n_{i,l\downarrow}+\frac{U^{\prime}-J_{H}}{2}\sum_{l\neq
m,\sigma}n_{i,l\sigma}n_{i,m\sigma}+\frac{U^{\prime}}{2}\sum_{\sigma\neq\sigma^{\prime}}\sum_{l\neq
m}n_{i,l\sigma}n_{i,m\sigma^{\prime}}$ $\displaystyle-$
$\displaystyle\frac{J_{H}}{2}\sum_{l\neq
m}(d_{i,m\uparrow}^{\dagger}d_{i,m\downarrow}d_{i,l\downarrow}^{\dagger}d_{i,l\uparrow}+d_{i,m\uparrow}^{\dagger}d_{i,m\downarrow}^{\dagger}d_{i,l\uparrow}d_{i,l\downarrow}+h.c.)$

and the spin-orbit interaction is

$\displaystyle{\mathcal{H}}^{i}_{SO}$ $\displaystyle=$
$\displaystyle\frac{{\dot{\iota}}\lambda}{2}\sum_{lmn}\epsilon_{lmn}\sum_{\sigma\sigma^{\prime}}\sigma_{\sigma\sigma^{\prime}}^{n}d_{i,l\sigma}^{\dagger}d_{i,m\sigma^{\prime}}.$
(10)

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

Figure 4:  Schematic illustration of the Kanamori interaction terms, intra-
orbital Coulomb interaction $U$ (left), Hund’s Coupling $J_{H}$ (middle), and
inter-orbital Coulomb interaction $U^{\prime}$ (right). For simplicity two
orbitals and two electrons are considered, the different arrangements of which
leads to different energies $E$ depending on the various interactions.  Figure
5: Emergence of magnetism in a spin-orbit coupled atomic $d^{4}$ model. (a)
The energy evolution of the many-body states of the atomic model, Eq. 9, as a
function of SOC for the $d^{4}$ electronic configuration. The ground state of
the atomic model is a non-magnetic $J=0$ state. (b) The emergence of magnetism
in the phase diagram of an atomic model in presence of non-cubic crystal field
$\Delta_{\rm NCF}$, showing the variation of the ground state expectation
value of $J^{2}$ operator in the parameter space of SOC $\lambda$ and
$\Delta_{\rm NCF}$ for a fixed value of $J_{H}/U=$ 0.1. $\Delta_{\rm NCF}$ in
(b) represents the energy separation between the subsequent $t_{2g}$ levels,
the degeneracy of which is broken completely and are energetically equispaced.
The figure is reprinted from Ref. [100], ©2020, with permission from Elsevier.

The exact diagonalization of the atomic model, Eq. 9, 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 Fig.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 $\lambda$, 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 $\lambda$ (In the limit
$J_{H}>>\lambda$, $E_{J=0}=E_{0}-4U+7J_{H}-\lambda$ and
$E_{J=2}=E_{0}-4U+7J_{H}+\lambda/2$; for $J_{H}<<\lambda$,
$E_{J=0}=E_{0}-4U+7J_{H}-2\lambda$ and $E_{J=2}=E_{0}-4U+7J_{H}-\lambda/2$)
but their energy difference is always $3\lambda/2$. The energy eigen value of
the $J=1$ state, on the other hand, always have the same analytical form
$E_{J=1}=E_{0}-4U+7J_{H}-\lambda/2$ [128].

It is noteworthy that in the simple atomic model in Eq. 9, 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,
${\mathcal{H}}^{i}_{\rm
NCF}=\sum_{lm\sigma}\varepsilon_{lm}d_{i,l\sigma}^{\dagger}d_{i,m\sigma}$ in
the Hamiltonian, Eq. 9, where the onsite energies of the different orbitals
are depicted by $\varepsilon_{lm}$ and the difference in the onsite energies
is the measure of non-cubic crystal field $\Delta_{\rm NCF}$ . The phase
diagram, Fig. 5 (b), for the ground state shows the emergence of magnetism
($J\neq 0$) in the $d^{4}$ system due to non-cubic crystal field. The
competition between $\Delta_{\rm NCF}$ and SOC $\lambda$ is evident from this
phase diagram. While in the strong SOC regimes ($\lambda\gg\Delta_{\rm NCF}$)
the non-magnetic $J=0$ state remains intact, magnetism appears in the energy
regime $\Delta_{\rm NCF}\gg\lambda$ [see Fig. 5 (b)].

###  3.2 Two-site model: Band structure effect

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 Eq. 9, hopping between two atomic sites are
also allowed. The two-site many body model Hamiltonian in a Hilbert space of
${}^{12}C_{8}=495$ basis states reads as

${\mathcal{H}}~{}=~{}{\mathcal{H}}_{t}+\sum_{i}({\mathcal{H}}^{i}_{int}+{\mathcal{H}}^{i}_{SO}),$
(11)

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

$\displaystyle{\mathcal{H}}_{t}$ $\displaystyle=$
$\displaystyle\sum_{ij}\sum_{l,m}\sum_{\sigma,\sigma^{\prime}}(t_{ij}^{l\sigma,m\sigma^{\prime}}d_{il\sigma}^{\dagger}d_{jm\sigma^{\prime}}+{\rm
H.C.})$ (12)

and the second and the third terms in Eq. 11 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 [129]

$\displaystyle{\mathcal{H}}^{i}_{int}$ $\displaystyle=$
$\displaystyle\frac{U-3J_{H}}{2}\hat{N}_{i}(\hat{N}_{i}-1)+\frac{5}{2}J_{H}\hat{N}_{i}-2J_{H}\hat{S}_{i}^{2}-\frac{1}{2}J_{H}\hat{L}_{i}^{2}.$
(13)

A diagonal hopping in the orbital and spin space i.e.,
$t_{ij}^{l\sigma,m\sigma^{\prime}}\rightarrow
t\delta_{lm}\delta_{\sigma\sigma^{\prime}}$ is assumed, where all three
$t_{2g}$ orbitals actively participate in the hopping. The exact
diagonalization results of the above Hamiltonian, as discussed in Ref. [129],
are more reliable in the Mott limit ($U/t\gg 1$). 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 $U$, $\lambda$ and $t$ for
a chosen value of $J_{H}$= $0.2U$ is shown in Fig. 6 (a). Interestingly, the
phase transition $J=0\rightarrow J=1$ at a critical value of $\lambda/t$ 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 Fig.
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 3$d$
Mott insulators, here the local moments are not robust and deviate from their
atomic values.

Figure 6: Schematic illustrations of the emergence of magnetism in two-site
$d^{4}$ spin-orbit coupled model system. (a) The schematic phase diagram in
the parameter space of $t/\lambda$ and $t/U$, showing the transition from the
non-magnetic singlet $J=0$ state to magnetic $J=1$ and $J=2$ states with
increasing hopping. (b) The variation of the expectation values of the local
$L_{i}^{2}$, $S_{i}^{2}$ and $J_{i}^{2}$ operators and the total $J^{2}$
operator as functions of $\lambda/t$, indicating the discontinuous jump of the
total angular momenta $J_{i}^{2}$ at each site and $J^{2}$ for the two-site
system across the phase transition, while the local spin and orbital angular
momenta $L_{i}^{2}$ and $S_{i}^{2}$ do not change. (c) The FM to AFM
transition with increasing SOC for three sets of $(t,J_{H})$ as indicated in
the figure. The different superexchange paths for the (d) FM and the (e) AFM
interactions. Figures (a), (b), (d), and (e) are reprinted with permission
from Ref. [129] ©2015 by the American Physical Society and (c) is reprinted
from Ref. [100], ©2020, with permission from Elsevier.

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 Fig. 3), so that
$t_{ij}^{l\sigma,m\sigma^{\prime}}\rightarrow
t\delta_{lm}(1-\delta_{km})\delta_{\sigma\sigma^{\prime}}$. 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., $t_{ij}^{l\sigma,m\sigma^{\prime}}\rightarrow
t\delta_{lm}\delta_{km}\delta_{\sigma\sigma^{\prime}}$. 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 $d^{4}$
spin-orbit coupled systems [130]. A next question that naturally occurs is the
nature of the magnetism. Meetei et. al [129] 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 Ref. [129], based on the simple perturbative analysis. The
atomic ground state in the $d^{4}$-$d^{4}$ configuration with $L_{i}=1$ and
$S_{i}=1$ has a total energy, $E_{0}=12U-26J_{H}$, computed using Eq. 13. The
hopping in Eq. 12 acts as a pertubation and the ground state gains energy via
virtual hopping. For FM superexchange, the intermediate $d^{3}$-$d^{5}$ state
[see Fig. 6 (d)] with $S_{1}=3/2,L_{1}=0,S_{2}=1/2,$ and $L_{2}=1$ has the
total energy $E_{\rm FM}=13U-29J_{H}$. This results in an energy gain by the
FM state $\Delta E_{\rm FM}=-\frac{2t^{2}}{U-3J_{H}}$. Similarly, the
intermediate paths for the AFM superexchange, as schematically illustrated in
Fig. 6 (e), gains an energy $\Delta E_{\rm
AFM}=-\frac{2}{3}\frac{t^{2}}{U-3J_{H}}-\frac{14}{6}\frac{t^{2}}{U}-\frac{t^{2}}{U+2J_{H}}$.
Comparing the two energy scales $\Delta E_{\rm FM}$ and $\Delta E_{\rm AFM}$,
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 [100] 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 $d^{4}$ systems. While
ferromagnetism is stabilized in absence of SOC ($\lambda=0$), with increase in
$\lambda$ ferromagnetic interaction become weaker, and eventually at a
critical value $\lambda_{c}$, an AFM interaction is favored. The critical
value of $\lambda_{c}$ depends on the strength of the other parameters of the
Hamiltonian as shown in Fig. 6 (c). In particular, a larger Hund’s coupling
tends to increase the $\lambda_{c}$, thereby favoring the ferromagnetism.
Dynamical mean-field theory study combined with the continuous-time quantum
Monte Carlo [131] also supports these findings.

Inspired by the results of the two site exact diagonalization, the lattice
problem of magnetism in $d^{4}$ systems has also been analysed by constructing
an effective spin-orbital lattice model [129, 130, 132].

The effective model was employed by Svoboda et. al [130] 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 Eq. 11, can be found in Ref. [130] 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 Fig. 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 $-t^{2}/U$, failing to favor
ferromagnetism.

Figure 7: Graphical illustration of the orbital frustration for $N_{orb}=2$
model. The orbital frustration may be described by fixing the type of
interaction along bond 1 to either (a) AFM or (b) FM. Once the interaction
along bond 1 is fixed, it can be seen from (a) and (b) that the interaction
along bond 3 corresponds to a higher energy configuration, leading to orbital
frustration. The double occupied orbitals at each site of the square lattice
are indicated in the figure. The schematic illustrations in (a) and (b) are
reprinted with permission from Ref. [130] ©2017 by the American Physical
Society.

In addition, Ref. [130] 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 Fig. 7,
considering the case of $N_{orb}=2$. In a $d^{4}$ 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 Fig. 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 Fig. 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 Fig. 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
Fig. 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 $\lambda\ll zJ_{SE}$ ($z$
being the coordination number) with strong orbital effects in absence of
strong octahedral distortions.

Figure 8: Schematic illustration of the phase diagram, adapted with permission
from Ref. [133] ©2013 Springer Nature, showing the metal to insulator
transition in the parameter space of $U/D$ and $\lambda/D$, where $D$ is the
half-bandwidth.

###  3.3 Metal-insulator transition

We now turn to the transport properties of $d^{4}$ spin-orbit coupled systems.
The transport properties in $d^{4}$ systems, as studied in Ref. [133] using
the rotational invariant Gutzwiller approximation (RIGA) method as well as the
dynamical mean-field theory (DMFT). The calculations reveal metal-insulator
transition as a function of SOC constant $\lambda$ 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 $\lambda$ and $U$ (see Fig. 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 $j_{\rm eff}=3/2$ and $j_{\rm eff}=1/2$
bands, separated from each other with an energy gap of $\approx 3\lambda/2$.
With increase in $\lambda$, the higher lying $j_{\rm eff}=1/2$ bands tend to
get empty as the electrons move to the $j_{\rm eff}=3/2$ bands. Consequently,
at a critical value of $\lambda/D$, $D$ being the half-bandwidth, the $j_{\rm
eff}=3/2$ bands are completely occupied, resulting in a transition to the band
insulating state from a metallic state. Moving to the other limit where
$\lambda=0$, 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.

Coming back to the discussion of the phase diagram in Fig. 8, we can see that
the phase diagram may be subdivided into three different regions: (I) The
regime $0<\lambda/D<0.24$, 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 $d^{5}$ filling as well [23]. In this Mott phase, the local
moment at each site vanishes, forming a spin-orbital singlet state and the
ground state is simply the product state of the local singlets at each site.
(II) The next regime is $0.24<\lambda/D<1.33$. 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 $\lambda$ results in a band insulating state. Increasing $U$
further pushes the system into Mott regime. (III) Finally, in the regime
$\lambda/D>1.33$, strong SOC leads to completely empty $j_{\rm eff}=1/2$ 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.

##  4 Materials

The various theoretical predictions [73, 129] described in the previous
section, for the $d^{4}$ spin-orbit coupled systems generated enormous
interest in pentavalent iridates for the realization of excitonic magnetism.
The search is further, triggered by the observation of novel magnetism in
$d^{4}$ double perovsite system Sr2YIrO6 [76], followed by a series of other
$d^{4}$ iridates. The observed magnetism in iridates is, however, quite
enigmatic. The ground state properties of these materials are full of
controversial and conflicting results. While a large number of materials have
been studied and characterized in past years, here we focus on a few
particular materials that have received considerable attention in the $d^{4}$
iridate research.

The pentavalent iridates, that we discuss in this review, can be broadly
categorized into four classes according to their crystal geometry : (a) double
perovskite (b) post perovskite (c) 6H perovskite and (d) hexagonal iridates.
The crystal structure plays a key role in deciding magnetism in iridates, so
the ground state properties of the iridates, belonging to the same geometry
class, are expected to be quite similar. Interestingly, the ground state of
$d^{4}$ iridates is so delicate that even small changes in the fine details of
the structure can lead to dramatic changes in the ground state properties, as
we now proceed to discuss below.

###  4.1 Double perovskite iridates

The general formula for the double perovskite iridates, that we discuss here,
is A2BIrO6, 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 Fig. 9).

Owing to the compositional and structural flexibility, double perovskite
iridates have particularly received enormous attention. The interest in the
$d^{4}$ 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 $d^{4}$ electronic configuration was reported
even earlier, e.g., in another double perovskite system Sr2CoIrO6 [104] (this
is even before the theoretical prediction by Khaliullin [73]). In this work
Narayanan et. al. [104] studied the structural, electronic and magnetic
properties of La2-xSrxCoIrO6 for $0\leq x\leq 2$, 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., $P2_{1}/n+I2/m\rightarrow I2/m\rightarrow I4/m\rightarrow
Fm\bar{3}m$, and exhibits AFM ordering as revealed from magnetization
measurements. Interestingly, the estimated effective magnetic moment for the
system $\mu_{\rm eff}=5.1(1)\mu_{B}/$ f.u. suggests for magnetic contribution
from the Ir5+ ion, as the contribution ($\mu_{\rm eff}=4.9\mu_{B}/$ Co) for
the Co3+ ion alone can not describe the estimated moment value. This small
moment, however, remains undetected in the neutron powder diffraction (NPD)
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 [106]. 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.

Figure 9: Schematic illustration of the two different scenario, proposed for
double perovskite iridate Ba2YIrO6. (a) Crystal structure of a double
perovskite iridate (top) and the geometric frustration of the face-centered
lattice formed by the Ir ions (bottom). (b) The summary of the two conflicting
scenario, discussed in theory and experiment, regarding the magnetism in
Ba2YIrO6. While the scenario (I) supports the breakdown of the $J=0$ state,
scenario (II) suggests the presence of $J=0$ state. According to (I), the
experimentally observed moment is intrinsic and it appears due to band
structure effect, illustrated in (c), where the double perovskite iridate lies
in the intermediate regime of $\lambda\leq W$. The scenario (II), on the
contrary, suggests absence of any moment at the Ir site and the observed
moment in the experiment is ascribed to the extrinsic effects, such as anti-
site disorder, impurity, magnetic defect etc. The figure (d) in this scenario
shows the weak dispersion of the triplet excitation spectrum, failing to
overcome singlet-triplet energy gap. The figures for the first scenario is
taken from Ref. [77] and the second scenario is reprinted with permission from
Ref. [92] ©2017 by the American Physical Society.  Figure 10: The densities of
states (DOS) for Ba2YIrO6, computed within (a) GGA, (b) GGA+SOC, and (c)
GGA+SOC+U, considering ferromagnetic spin configuration. The dominant orbital
characters are indicated in the figure. The GGA spin polarized calculation
gives a half-metallic solution in (a). Inclusion of SOC leads to changes in
the DOS, as indicated in (b). The inset in (b) shows the mixing between
$j_{\rm eff}=1/2$ (shaded region) and $j_{\rm eff}=3/2$ (solid line) states,
obtained from a non-magnetic Ir-$t_{2g}$ only tight-binding model in presence
of SOC, indicating the origin of finite moment at the Ir-site. Application of
Coulomb interaction $U$ fails to open up a gap within the ferromagnetic spin
configuration, while the energy gap is present in the computed energetically
lowest type-I anti-ferromagnetic configuration as shown in the inset of (c).

A series of such double perovskite iridates A2R3+Ir5+O6 (A= Sr, Ba and R=La,
Y, Sc, Lu) were synthesized long ago in 1999 by Wakeshima et. al. [97] 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. [76]
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 $\mu_{\rm eff}$ at the Ir site is 0.91 $\mu_{B}$
and the Curie-Weiss temperature $\theta_{\rm CW}$ is as large as $-229$ K,
indicating the presence of AFM interaction as well as strong frustration in
the system. Sr2YIrO6 crystallizes in monoclinic $P2_{1}/n$ 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 $d^{4}$ double perovskite iridate
Sr2GdIrO6 in the same work [76] showed that unlike the Y-compound, the Gd-
compound orders ferro-magnetically at $T_{c}=2.3$ K with a comparable Curie
Weiss temperature $\theta_{\rm CW}=1.2$ K. The saturation moment was found to
be $M_{s}\sim 7\mu_{B}$/fu consistent with the Gd3+ (4$f^{7}$) 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 [77]. 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 [134].
The conclusions of Ref. [77] 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
$Fm\bar{3}m$ 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 Fig. 9 (b)]. Interestingly, despite the
different non-cubic crystal field, the bandwidths in this series of double
perovskites are comparable and large in magnitude ($\sim 1.2$ eV) that leads
to significant mixing between $j_{\rm eff}=1/2$ and $j_{\rm eff}=3/2$ states
[see inset of Fig. 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 Fig. 10.

In complete contrast to Refs. [76, 77], DMFT calculations [90] 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 $j_{\rm eff}=1/2$ and $j_{\rm eff}=3/2$ 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. [89] revisited and studied in detail the
structural, magnetic, and thermodynamic properties of Ba2YIrO6, which was
originally synthesized more than 45 years ago [102]. This study confirmed the
ideal cubic structure of Ba2YIrO6, in contrast to the monoclinic symmetry
reported earlier [97], and showed that the system does not order down to 0.4
K. However, the Curie-Weiss fit yields a Curie Weiss temperature
$\theta_{CW}\sim-8.9$ K and an effective moment of $0.44\mu_{B}$/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 $j_{\rm eff}=1/2$ state is part of this full
$j_{5/2}$ manifold [60]). 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 $J_{\rm H}=0.5$ eV in contrast to Ref. [90]. The origin
of the moment at the Ir site remains elusive from this work.

In the meantime, independent experiments [135, 136] probed the effect of
structural distortions on the magnetism of the series of double perovskites
Ba2-xSrxYIrO6 ($0\leq x\leq 1$). Both studies show no evidence for magnetism
down to 2 K. The effective moment, reported in Ref. [135], is rather small,
0.16 $\mu_{B}$/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. [95], 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 1.44 $\mu_{B}$/Ir in Ba2YIrO6 and a large
Curie-Weiss temperature of -149 K. The authors questioned the large singlet-
triplet gap predicted earlier [90], 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. [96] even
questioned the previously reported monoclinic symmetry of Sr2YIrO6 [76, 135]
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
$\mu_{\rm eff}=0.21\mu_{B}$/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
[91], anti-site disorder [92], Ir4+/Ir6+ magnetic defects [93], excluding the
possibility of any excitonic magnetism [see scenario II in Fig. 9 (b)].

On the contrary, Nag et. al. [79] pointed out that the temperature dependence
of the magnetic response below 10K for Ba2YIrO6 as revealed from their $\mu$SR
studies is rather unusual. In particular, the temperature dependence of the
stretched exponent parameter $\beta$ of the fitted muon polarization P(t) =
$\exp-\left(\alpha t\right)^{\beta}$ cannot be corroborated with a $J=0$
state, for which no such temperature dependence of $\beta$ 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 spin-orbital singlets, 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 4$d$ analogue Ba2YMoO6 [137], supporting the
scenario I in Fig. 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 [79,
138] that facilitate a quantitative estimation of the parameters $\lambda$ 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
$\lambda\approx 0.39-0.42$ and $J_{H}\approx 0.24-0.26$ eV [79, 138]. While
theoretical studies [92, 139, 140] suggest excitonic magnetism is unlikely in
Ba2YIrO6 due to large estimated singlet-triplet gap in RIXS measurement, Ref.
[95] 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 [141] show the presence of type-I AFM
state in Ba2YIrO6, similar to the initial DFT results [77], confirming the
possibility of interesting physics beyond the atomic limit. The insulating
state in Ref. [141], however, occurs at a higher $U$ than in Ref. [77] ($U=4$
vs $U=2$ eV). The magnetism in Ref. [141] 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 [98, 99,
100]. Similar to the Y compounds, Ba-Sc compound crystallizes in the cubic
$Fm\bar{3}m$ structure while Sr-Sc compound undergoes monoclinic distortion of
the $P2_{1}/n$ space group. Experimentally, the systems do not exhibit long
range magnetic order while an effective moment of 0.16 $\mu_{B}$/Ir for the Sr
compound and even larger moment of 0.39 $\mu_{B}$/Ir for the Ba compound was
estimated. The generated magnetic moment was attributed to the hopping induced
de-localization of holes in these systems [99, 100].

The compositional flexibility within the general formula of A2BIrO6 leads to a
large number of pentavalent double perovskite iridates, that includes but not
limited to the listed materials in Table LABEL:tab1. 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 3$d$-5$d$ interaction, combining the strong
correlation of 3$d$ TM ion to the strong SOC of 5$d$ TM ion. While this offers
a broad area of research spanned by some excellent works [103, 107, 105, 106],
we do not discuss them in the present review.

###  4.2 Post-perovskite iridate

The first post-perovskite oxide with pentavalent atom NaIrO3, containing Ir in
the $d^{4}$ configuration, was synthesized using high-pressure solid state
method [112]. The post perovskite NaIrO3 crystallizes in the layered
orthorhombic $Cmcm$ space group. In contrast to the corner shared IrO6
octahedra in the double perovksite 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 Fig. 11 (a) and (b), these sheets are,
further, stacked along the $b$ direction, separated by the Na atoms.

Interestingly, the magnetic measurement as well as DFT calculations [112]
confirm the nonmagnetic state of Ir in NaIrO3. This was, thus, considered an
ideal example for the realization of the $J=0$ nonmagnetic state [112], 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 Refs. [77] and [113] addressed
these issues, providing insights to the answer to these questions, which we
now proceed to discuss.

Figure 11: The $S=0$, insulating state of NaIrO3. (a) The crystal structure of
NaIrO3 in the orthorhombic $Cmcm$ space group. (b) The distorted IrO6
octahedra and their connection in the $ac$ plane. Different bond lengths and
bond angles are also indicated in the figure. (c) The resulting non-degenerate
Ir-$t_{2g}$ states, $t_{1},t_{2}$, and $t_{3}$, and the non-cubic crystal
field splitting between the states. The large $\Delta_{\rm NCF}$ leads to
$S=0$ state, rather than $J=0$ state. (d) The schematic illustration of the
phase diagram, adapted with permission from Ref. [113], in the parameter space
of $U$ and $\lambda$ relevant for NaIrO3. The realistic parameters for NaIrO3
($U=2$, $\lambda=0.33$ eV, and $J_{H}=U/4$) correspond to a band insulating
state, indicated by the black triangle in the figure. (f) The computed density
of states for NaIrO3, adopted from Ref. [77], considering the experimentally
reported crystal structure [112] (bottom) and the optimized structure as in
Table. 2 (top), indicating correlation effects may not be important for the
experimentally observed insulating behavior.

####  4.2.1 $S=0$ instead of $J=0$ state in ${\rm NaIrO_{3}}$:

According to the theory that supports the magnetism at the Ir site in the
pentavalent double perovskite iridates, as discussed in section 4.1,
emphasizes the crucial role of band structure effects in preventing the
realization of the desired $J=0$ state. The band structure effects become
particularly important due to the three-dimensional connectivity of the Ir
atoms. Therefore, the reduced dimensionality of the connecting Ir atoms,
resulting from the two-dimensional Ir sheets in the post perovskite structure
(see Fig. 11) raises the hope to find the $J=0$ state in NaIrO3.

However, in addition to the desired layered structure, the IrO6 octahedra in
NaIrO3 also suffer from strong distortions [see Fig. 11 (b)]. The
$\angle$O-Ir-O bond angle deviates significantly from the ideal 90∘ angle to
about $\approx$ 73.2∘, while the edge and corner sharing $\angle$Ir-O-Ir
angles are respectively 106.8∘ and 140.4∘. The Ir-O bond lengths also differ
by about $0.02$ Å. Such a high distortion leads to strong non-cubic crystal
field, that removes the degeneracy of the $t_{2g}$ orbitals completely,
splitting it into three non-degenerate energy levels. The energy separation
between these levels were estimated to be $\Delta_{\rm NCF}\approx$ 0.57-0.59
eV [77], which is comparable to the typical energy scale of the Hund’s
coupling $J_{H}\approx 0.4-0.6$ eV in 5$d$ TM oxides. As a result,
$\Delta_{\rm NCF}$ dominates over $J_{H}$, stabilizing an electronic
configuration $(\uparrow\downarrow,\uparrow\downarrow,0)$ [see Fig. 11(c)]
that leads to a non-magnetic state.

Clearly, the non-magnetic state in NaIrO3 is distortion driven and has noting
to do with the SOC. Thus, it is better described by an $S=0$ state, rather
than a $J=0$ state. The stabilization of the non-magnetic state even without
SOC in the DFT calculations [112, 77] further supports this finding.

####  4.2.2 Insulating state:

In contrast to the observed insulating behavior in the experiments, DFT
calculations show a metallic state in the non-magnetic ground state [77, 112].
Application of SOC and Coulomb correlation $U$ do not affect this scenario.
This leads to the puzzling insulating behavior in NaIrO3.

Failing to describe the insulating behavior within DFT+SOC+U approach, Ref.
[113] made an attempt to investigate the insulating state with the
LDA+Gutzwiller method. Starting from a low energy tight-binding model, derived
from the LDA calculation, with atomic SOC in it, they succeeded in obtaining
an insulating solution for NaIrO3 for reasonable values of SOC
($\lambda\approx 0.33$ eV) and $U=$2 eV, relevant for NaIrO3. Furthermore,
they identified the insulating state of NaIrO3 as a “band insulator” rather
than a Mott insulator [see Fig. 11 (d)].

Ref. [77] further emphasized the influence of the structural geometry in
driving the insulating state in NaIrO3. Structural optimization, keeping the
experimental symmetry and the lattice parameters intact, showed that the
position of the light oxygen atoms were changed significantly (see Table 2),
leading to an almost insulating state, as depicted in Fig. 11 (e). This
further supports earlier finding of band insulating state in NaIrO3,
indicating correlation effects may not be significant in NaIrO3. With these
new theoretical insights, it might be worthwhile to revisit the transport
properties of NaIrO3, possibly on single crystal samples, motivating future
experiments in this direction.

Table 2:  Comparison of the Wyckoff positions for the experimental and the
optimized crystal structure of the primitive unit cell of NaIrO3, as reported
in Ref. [77].

  * Atoms | Experimental structure | Optimized structure
---|---|---
| x | y | z | x | y | z
Na | 0.0 | 0.2507 | 0.25 | 0.0 | 0.2513 | 0.25
Ir | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
O1 | 0.5 | 0.43766 | 0.25 | 0.5 | 0.42621 | 0.25
O2 | 0.5 | 0.10159 | 0.05681 | 0.5 | 0.11652 | 0.05852

###  4.3 6H perovskite iridates

We now discuss the $d^{4}$ 6H-perovskite iridates with the general formula
Ba${}_{3}M$Ir2O9. 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 [115] back in 2006 and, then, further revisited with an idea to
understand the SOC effects [78, 114, 80, 117]. 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 [115, 102, 142, 143, 144, 145]. 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 ($d^{5}$). 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
$Ba_{3}MIr_{2}O_{9}$:

The basic crystal structure of Ba3MIr2O9 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 Fig. 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 [115, 78, 114] show that while
Ba3ZnIr2O9 and Ba3MgIr2O9 with the smaller cations Zn and Mg crystallize in
the hexagonal $P6_{3}/mmc$ 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 $\Delta_{\rm NCF}$, the monoclinic symmetry of the Ca and Sr
compounds leads to complete removal of the degeneracy of the $t_{2g}$
orbitals, denoted by $t_{1},t_{2},$ and $t_{3}$ in Fig. 12 (d), with the
corresponding energy differences $\Delta_{1}$ and $\Delta_{2}$ respectively
(see Fig. 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 ($\Delta_{\rm NCF}$ or $\Delta_{1}$,
$\Delta_{2}$) are listed in Table 3.

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 [146], later a combined experimental and theoretical analysis
[78] 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 $d^{4}$ electronic
configuration, the spin-orbit entangled $\Gamma_{7}$ and $\Gamma_{8}$ states
at the neighboring Ir sites, originating from the Ir-$t_{2g}$ orbitals in
presence of SOC, interact with each other anti-ferromagnetically [see Fig.
12(e) and (f)]. The key to the AFM interaction in presence of SOC is the
unusual hopping between pesudo-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.,
$\Gamma_{8}$ electrons can virtually hop from $|1\alpha\rangle$
($|1\beta\rangle$) to $|3\beta\rangle$ ($|3\alpha\rangle$) state. These
electrons, which are promoted to the $\Gamma_{7}$ state further can gain
energy by hopping to the other $\Gamma_{7}$ state as illustrated in Fig 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 $\theta_{\rm CW}\sim$ -30 K with an effective moment $\sim$0.2
$\mu_{B}$/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.

Figure 12: Structural distortion and magnetic properties of Ba3MIr2O9. (a)
Crystal structure of Ba3MIr2O9. The structure consists of face-shared Ir2O9
units, which are arranged on a triangular network, shown in (b). The intra-
($d^{intra}$) and inter-dimer ($d^{inter,ab}$ and $d^{inter,c}$) distances are
indicated in (a) and (b). (c) The Ir2O9 unit, that undergoes different types
of distortion for $M=$ Zn, Mg, Ca, and Sr (see Table 3). This results in a
trigonal crystal field splitting $\Delta_{\rm NCF}$ in Zn and Mg compounds
(left), while the $t_{2g}$ degeneracy is completely removed for Sr and Ca
compounds (right), shown schematically in (d). The non-degenerate $t_{2g}$
states for Sr and Ca are denoted by $t_{1},t_{2}$, and $t_{3}$ and the energy
separations between the consecutive levels are denoted by $\Delta_{1}$ and
$\Delta_{2}$ respectively. The cartoons, adopted from ref. [78], showing (e)
the FM and (f) the AFM interactions between Ir5+ ions in absence and presence
of SOC (see text for details). (g) and (h) show the symmetry allowed magnetic
structures corresponding to the magnetic space groups $C2^{\prime}/c^{\prime}$
and $C2/c^{\prime}$ which are found to be energetically lowest magnetic
configurations for Sr and Ca compounds respectively. Figures (g) and (h) are
adopted from Ref. [114]. Table 3:  The comparison of the structural parameters
and the resulting non-cubic crystal field splitting in Ba${}_{3}M$Ir2O9,
$M=$Zn, Mg, Ca and Sr.

  * | Ba3ZnIr2O9 | Ba3MgIr2O9 | Ba3CaIr2O9 | Ba3SrIr2O9
---|---|---|---|---
a (Å) | 5.77034 | 5.7705 | 5.9091 | 5.9979
b (Å) | 5.77034 | 5.7705 | 10.2370 | 10.3278
c (Å) | 14.3441 | 14.3216 | 14.7624 | 15.1621
$\beta$ (∘) |  |  | 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
$d^{intra}_{\rm Ir}$ (Å) | 2.75 | 2.76 | 2.735 | 2.69
$d^{inter,c}_{\rm Ir}$ (Å) | 5.54 | 5.52 | 5.52, 5.78, 5.86 | 5.86, 5.99, 6.11
$d^{inter,ab}_{\rm Ir}$ (Å)I | 5.78 | 5.77 | 5.909, 5.91, 5.91 | 5.97, 5.97, 5.99
$\angle$Ir-O1-Ir (∘) | 85.92 | 85.96 | 85.75 | 82.55
$\angle$Ir-O2-Ir (∘) |  |  | 80.06 | 79.50
$\Delta_{\rm NCF}$ (eV) | 0.04 | 0.06 | 0.07, 0.24 | 0.09, 0.16

The AFM interaction within the Ir-Ir dimer leads to the formation of spin-
orbital singlet (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 $d^{inter,ab}_{\rm Ir}$
and $d^{inter,c}_{\rm Ir}$ in Table 3 respectively. The energy scales of these
intra-dimer $J^{intra}_{d}$ (-14.6 meV, with neighboring single Ir ion) and
inter-dimer interactions $J^{inter}_{d}$ (-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 [78].

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 $\mu$SR measurements suggest a weaker
frustration in this case. The weaker frustration in the Mg compound results
from the stronger $J^{intra}_{d}$ (-21.3 meV) compared to $J^{inter}_{d}$
(-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 [114]
with the symmetry allowed magnetic structures corresponding to an assumed
propagation vector $\vec{k}=(0,0,0)$ shows that the lowest energy magnetic
configuration corresponds to the magnetic space group $C2/c^{\prime}$, 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 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 $C2/c^{\prime}$ magnetic structure as shown in Fig. 12(g). On
the contrary, similar analysis [114] showed that another symmetry allowed
magnetic configuration $C2^{\prime}/c^{\prime}$ 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 Ba3MIr2O9
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 $d^{4}$ 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 Ba3MIr2O9 ($M=$ 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 Ba3MIr2O9 has distinct features, as shown in
Fig. 13 (b), which can not 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 Ba3MIr2O9 systems.

Figure 13: The role of hopping in the RIXS spectrum of the 6H iridates. (a)
Schematic illustration of the RIXS process in general. (b) The low-energy RIXS
spectra from the SOC states in Ba3ZnIr2O9 (BZIO). In marked contrast to the
corresponding spectra in Ba2YIrO6 (BYIO), the spectra in 6H perovskite depicts
the presence of additional inelastic peaks and spectral-weight transfer to
lower energies. (c) The energy states obtained from an atomic model and a two-
site model. The higher energy states with low intensity are not shown. The
feature of the observed spectra for 6H iridates in (b), although can not be
explained within an atomic model but is a characteristic of a two-site model
indicating the crucial role of hopping in the present iridates. (d) The
deviation of the ground state $\psi_{g}$ of a two-site model, relevant for 6H
iridates, from the $J=0$ state. (e) The same can be concluded from the phase
diagram, showing the variation of the expectation value of $J^{2}$ operator in
the parameter space of $t/U$ and $\lambda/U$. The black triangle in the phase
diagram corresponds to the parameters $t$ = 0.1 eV, $\Delta_{\rm NCF}$ = 0.02
eV, $J_{\rm H}$ = 0.5 eV, $\lambda$ = 0.25 eV and $U$ = 2.0 eV, relevant for
the iridates.

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
Fig. 13 (a). In case of $d^{4}$ iridates, the incoming photon with frequency
$\omega_{1}$ excites the Ir-$2p$ core electron to the Ir-$d$, $t_{2g}$
unoccupied states (corresponding to the L3 absorption edge) forming an
intermediate state $|n\rangle=|2p_{\frac{3}{2}}^{\rm hole}t_{2g}^{5}\rangle$
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
$\omega_{2}$ and a final state $|f\rangle=|t_{2g}^{4}\rangle$. Note that the
final state $|f\rangle$ can be any kind of arrangement of four electrons
within the three $t_{2g}$ orbitals and the energy difference
$\omega_{1}-\omega_{2}$ 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 Ba3MIr2O9 is shown in Fig. 13 (b). Such a
spectrum, in particular the additional peaks as shown in the figure, can not
be described by the triplet ($J=0\rightarrow J=1$) and quintet
($J=0\rightarrow J=2$) spin-orbit excitons of an atomic model for spin-orbit
coupled $d^{4}$ systems. Indeed, the two-peak feature is the characteristic of
a two-site model with a finite inter-site hopping [see Fig. 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, Eq. 11 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 $J=1$ ($J_{1}$) at each sites interact with each other to
give rise to a new set of nine states, which we denote here by $J_{1}-J_{1}$
for simplicity. The resulting new sets of spin-orbit coupled states are:
$J_{0}-J_{0}$ (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, where the number within the
parenthesis indicates the number of states arising due to interaction between
the two-atomic states. The excitation from the $J_{0}-J_{0}$ state to the
states $J_{1}-J_{1}$ and $J_{0}-J_{2}$ states describe the broad peak around
$\sim 0.4$ 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 $\sim
t^{2}/U$ involved in the latter case. The estimated Hund’s coupling $J_{H}$ is
about $\sim 0.45$ eV, which is comparable to the other $5d^{5}$ materials but
higher than the $d^{4}$ double perovskite iridates [138, 147]. More
interestingly, the SOC $\lambda$ is highly suppressed to a value of $\sim
0.26$ 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 $\langle J^{2}\rangle$ [see Fig. 13 (d) and (e)]. In a
nut shell, Ba3MIr2O9 is an exemplary $d^{4}$ system, where the hopping
dramatically affects the spin-orbit coupled ground state, leading to formation
of moment at the “atomically non-magnetic” Ir site.

###  4.4 Hexagonal Iridates Sr3MIrO6

The pentavalent iridate Sr3MIrO6 crystallizes in the hexagonal structure with
the $R\bar{3}c$ 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 [119, 120, 124].
The crystal structure of Sr3MIrO6 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 Sr3MIrO6 consists of such
well separated alternating chains, as shown in Fig. 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 [119, 120] indicate a
temperature-independent paramagnetic behavior for Sr3MIrO6, with $M=$Na, Li,
which further points towards a non-magnetic $J=0$ state.

Figure 14:  Crystal structure of Sr3MIrO6, showing the alternating chain of
IrO6-SrO6 running along the crystallographic $c$-axis. The face-shared IrO6
and MO6 unit is shown separately. The Ir atoms, both within the chain
($d_{intra}$), and between the chains ($d_{inter}$), are quite well separated
in the structure, which leads to narrow Ir-$t_{2g}$ bands.

With this viewpoint, recently DFT calculations [118] 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 $j_{\rm eff}$ states, the
authors show that the $j_{\rm eff}=3/2$ and $j_{\rm eff}=1/2$ 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 $d^{4}$ 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 [148] 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 Sr3MIrO6. It is
important to point out here that the isostructural $d^{5}$ sister compound
Sr3NiIrO6 exhibits a strong single-ion magnetic anisotropy [149]. In presence
of any magnetism in the $d^{4}$ counterparts, it would be furthermore
interesting to see whether such single-ion magnetic anisotropy is also present
in pentavalent iridates.

##  5 Outlook

Understanding electronic and magnetic properties of iridates is a rapidly
growing field of research, a part of which are pentavalent iridates. While
most of these pentavalent iridates have been synthesized long ago, e.g., the
first $d^{4}$ Ir-based material was synthesized as early as 1960’s [74, 75],
understanding and revealing many of the interesting properties of these
materials remain elusive until recently. The surge in the interest of
discovering intriguing properties of $d^{4}$ iridates are primarily driven by
the theoretical proposal of excitonic magnetism in these materials. While
theoretical and experimental understanding of the ground state of real
pentavalent iridate systems are not always conclusive, the enormous curiosity
to understand the electronic and magnetic properties of these materials over
the past decade builds up the solid ground to further tune and manipulate the
states to get desired properties of matter. While the review of each of the
members of this giant family of materials is beyond the scope of the present
article, the present review provides an overall view of this family,
highlighting the important issues that have been of interest to the scientific
community and focussing on a few specific materials that have received
considerable attention over the time.

It is hard to draw any conclusion to the iridates research as the field is
still at its peak with more than 300 publications over the past five years
(according to the “iridates” keyword search in the web of science) and many
more avenues to be explored in future. We, therefore, end this review by
drawing the attention of the readers to a few interesting open questions or
directions that still need to be answered or ventured.

The ground state properties of $d^{4}$ double perovskite iridates have been at
the centre of debate and controversies. The Ir atoms do not order magnetically
in most of the $d^{4}$ iridates, including the double perovskite systems. A
more clear and better understanding of the ground states of these systems may
be obtained by studying the magnetic properties of these systems under
pressure or strain. Intuitively speaking, the tensile strain would separate
the Ir ions far apart, leading to more atomic like scenario, where the spin-
orbit effects prevail. On the other hand, the compressive strain would place
the Ir ions close to each other, and if the strain is strong enough a
competition between SOC and covalency may result in. This, in turn, may lead
to the onset of long range order. It might be interesting to see the effect of
strain on the spin-orbit excitons as well, which may be probed in RIXS
experiments. The spin-orbit excitons have received a great deal of attention
in recent times, particularly driven by the fact that its condensates near
phase transitions exhibit dispersive Higgs mode, a long-sought-after physical
phenomenon [150]. External perturbations such as pressure or strain may
trigger this critical transition points, that results from the competition
between various non-local effects and spin-orbit effect, which may be
difficult to attain intrinsically in the bulk iridates.

The strain effects might also shed light into the insulating nature of the
iridates. While most of the iridates are found to be insulating, their origin
of the insulating behavior, as discussed in the literature, is actually quite
different. For example, while NaIrO3 and Sr3MIrO6 are close to band insulating
state [113, 118], double perovskite iridate Ba2YIrO6 is described as a Mott
insulator [89]. It is noteworthy that Ref. [141] also indicated possible
connection between the AFM configuration and the insulating state. Structural
modifications with pressure or strain might be useful to understand and
analyze the transport properties of the different $d^{4}$ iridates, revealing
also the connection between magnetism and insulating state, if any. It is
interesting to point out here that the tetravalent perovskite iridate exhibits
dramatical changes in the electrical conducting properties in the two-
dimensional limit [151, 152]. The corresponding behavior of the pentavalent
counterparts may be worth investigating.

While the magnetic interactions in the $d^{4}$ iridates are either very weak
or completely absent, it would also be interesting to look for higher order
“hidden” parameter, magnetic and/or non-magnetic, which may act as primary
order parameters, governing the properties of the materials. Since SOC, in
general, is often found to promote such hidden orders, $d^{4}$ iridates are
excellent materials to investigate them. Interestingly, such hidden orders
have been predicted and also experimentally verified for $5d^{2}$ double
perovskites [153, 154]. Since the $d^{4}$ electronic configuration is
particle-hole analogue of $d^{2}$ configuration (two electrons in $t_{2g}$
orbitals are equivalent to four holes in the $t_{2g}$ orbitals), it would be
interesting to explore the studied pentavalent iridates from this point of
view.

Undoubtedly there are still plenty of avenues to be explored both
theoretically and experimentally, motivating many more future researches on
the pentavalent iridates in the coming years.

## Acknowledgements

We take this opportunity to deeply acknowledge Avinash V. Mahajan, Sugata Ray,
Tanusri Saha-Dasgupta, and Santu Baidya, Atasi Chakraborty, Shreemoyee
Ganguly, Abhishek Nag, and Swarup K. Panda, who provided us with new insights
and inspiring ideas and fruitful collaborations. We thank all our
collaborators in the iridate research over the last few years: Fabrice Bert,
Pabitra K. Biswas, I. Carlomagno, Anna Delin, Anna Efimenko, Olle Eriksson,
Paul G. Freeman, Adrian D. Hillier, Mitsuru Itoh, Som Datta Kaushik, Ying Li,
Martin Mansson, Roland Mathieu, Philippe Mendels, Carlo Meneghini, Srimanta
Middey, Larse Nordström, Jean-Christophe Orain, Henrik M. Ronnow, Marco M.
Sala, Tapati Sarkar, Dipankar Das Sarma, Denis Sheptyakov, Vasudeva Siruguri,
Mark T. F. Telling, Roser Valentí. We also thank Bernhard Keimer, Giniyat
Khaliullin, Daniel I. Khomskii, Arun Paramekanti, Sashi Satpathy, and Nandini
Trivedi for many useful discussions. I. D. thanks Science and Engineering
Research Board (SERB), India (Project No.EMR/2016/005925) and Department of
Science and Technology-Technical Research Centre (DST-TRC) for financial
support.

## Appendix A: $l_{\rm eff}=-1$ state

In presence of strong octahedral crystal field, atomic Ir-$d$ levels are split
into three-fold degenerate $t_{2g}$ and an $e_{g}$ doublet. Now by writing
$L_{x}~{}=~{}\frac{1}{2}(L_{+}+L_{-})$ and
$L_{y}~{}=~{}\frac{1}{2i}(L_{+}-L_{-})$ in terms of ladder operators and
following the orbital momentum algebra,

$\displaystyle L_{z}|l,m_{l}\rangle$ $\displaystyle=$ $\displaystyle
m_{l}|l,m_{l}\rangle$ (14) $\displaystyle L_{\pm}|l,m_{l}\rangle$
$\displaystyle=$ $\displaystyle\sqrt{l(l+1)-m_{l}(m_{l}\pm 1)}|l,m_{l}\pm
1\rangle$

we can calculate the matrix elements of $\langle t_{2g}|{\bf
L}|t_{2g}\rangle$,

$\displaystyle L_{x}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&i\\\
0&-i&0\end{array}\right)L_{y}=\left(\begin{array}[]{ccc}0&0&-i\\\ 0&0&0\\\
i&0&0\end{array}\right)L_{z}=\left(\begin{array}[]{ccc}0&i&0\\\ -i&0&0\\\
0&0&0\end{array}\right),$ (24)

where $t_{2g}\equiv\\{d_{yz},d_{xz},d_{xy}\\}$. Note that here we have dropped
out the $\langle t_{2g}|{\bf L}|e_{g}\rangle$ terms, assuming that the
$t_{2g}$ states are completely decoupled from the $e_{g}$ states, due to large
separation in the energy scale, determined by $\Delta$. In reality, however,
the $t_{2g}-e_{g}$ mixing may not be ignored. As discussed in Ref. [155], the
mixing may be crucial in interpreting the branching ratio, measured in
experiments as well as in theory to construct effective models for the purpose
of studying the magnetic phases and magnetic excitations. Nevertheless, if
such mixing be ignored, the projection of the L operator for the $d$-electrons
on the $t_{2g}$ manifold, as shown in Eq. 24 may be identified as L$(t_{2g})$
= -L$(p)$, where L$(p)$ is the matrix representation of the L operator in the
basis of $p\equiv\\{p_{x},p_{y},p_{z}\\}$, indicating that the $t_{2g}$
orbitals effectively behave like $p$-orbitals having angular momentum $l_{\rm
eff}$ =-1, but with an extra negative sign.

## References

## References

  * [1] For hydrogenic systems, the SOC constant $\lambda$ strongly increases with the atomic number Z, $\lambda\propto Z^{4}$. However, alternative treatment (Landau and Lifshitz, 1965) shows that for the outer electrons, that are responsible for the properties in the solid, $\lambda\propto Z^{2}$[2, 3]. Both these dependences though are only rough guides and the value of SOC in a real material can not be determined by any of these treatment, they however emphasize the strong dependence of $\lambda$ on the atomic number Z.
  * [2] L. Landau and E. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Course of Theoretical Physics (Elsevier Science, Amsterdam, 1981).
  * [3] Shanavas K V, Popovíc Z S and Satpathy S 2014 Theoretical model for Rashba spin-orbit interaction indelectrons Phys. Rev. B 90 165108
  * [4] Kane C L and Mele E J 2005 Z2 Topological Order and the Quantum Spin Hall Effect Phys. Rev. Lett. 95 146802
  * [5] Fu L and Kane C L 2008 Superconducting Proximity Effect and Majorana Fermionsat the Surface of a Topological Insulator Phys. Rev. Lett. 100 096407
  * [6] Hasan M Z and Kane C L 2010 Colloquium: Topological insulators Rev. Mod. Phys. 82 3045-67
  * [7] Qi X-L and Zhang S-C 2011 Topological insulators and superconductors Rev. Mod. Phys. 83 1057-110
  * [8] Ando Y 2013 Topological insulator materials J. Phys. Soc. Japan 82 102001
  * [9] Rashba E I, Sheka V I and Tela F T 1959 Collected Papers (Leningrad), v.II, 162-176 (in Russian), English translation: Supplemental Material to the paper by Bihlmayer G, Rader O and Winkler R 2015 Focus on the Rashba effect, New J. Phys. 17 050202
  * [10] Bychkov Y A and Rashba E I 1984 Properties of a 2D electron gas with a lifted spectrum degeneracy Sov. Phys. - JETP Lett. 39 78-81
  * [11] For a recent review see: Manchon A, Koo H C, Nitta J, Frolov S M and Duine R 2015 A New perspectives for Rashba spin–orbit coupling Nat. Mater. 14 871-882
  * [12] Culcer D, Keser A, Li Y and Tkachov G 2020 Transport in two-dimensional topological materials: recent developments in experiment and theory 2D Mater. 7 022007
  * [13] Andrzej P, Rodríguez K and Kapcia K J 2018 Superconducting monolayer deposited on substrate: Effects of the spin-orbit coupling induced by proximity effects Phys. Rev. Materials 2 024801
  * [14] Hikino S 2018 Magnetization Reversal by Tuning Rashba Spin–Orbit Interaction and Josephson Phase in a Ferromagnetic Josephson Junction J. Phys. Soc. Jpn. 87 074707
  * [15] Tkachov G 2019 Probing the magnetoelectric effect in noncentrosymmetric superconductors by equal-spin Andreev tunneling J. Phys.: Condens. Matter 31 055301
  * [16] Greening R W, Smith D A, Lim Y, Jiang Z, Barber J, Dail S, Heremans J J and Emori S 2020 Current-induced spin-orbit field in permalloy interfaced with ultrathin Ti and Cu Appl. Phys. Lett. 116 052402
  * [17] Zarezad A N and Abouie J 2020 Transport in two-dimensional Rashba electron systems doped with interacting magnetic impurities Phys. Rev. B 101 115412
  * [18] Alidoust M 2020 Critical supercurrent and $\phi_{0}$ state for probing a persistent spin helix Phys. Rev. B 101 155123
  * [19] Alidoust M and Halterman K 2015 Long-range spin-triplet correlations and edge spin currents in diffusive spin-orbit coupled SNS hybrids with a single spin-active interface J. Phys.: Condens. Matter 27 235301
  * [20] Alidoust M, Shen C and Zutíc I 2021 Cubic spin-orbit coupling and anomalous Josephson effect in planar junctions Phys. Rev. B 103 L060503
  * [21] Perfetti L, Georges A, Florens S, Biermann S, Mitrovic S, Berger H, Tomm Y, Höchst H and Grioni M 2003 Spectroscopic signatures of a bandwidth-controlled Mott transition at the surface of 1T-TaSe2 Phys. Rev. Lett. 90 166401
  * [22] Perfetti L, Loukakos P A, Lisowski M, Bovensiepen U, Berger H, Biermann S, Cornaglia P S, Georges A and Wolf M 2006 Time evolution of the electronic structure of 1T-TaS2 through the insulator-metal transition Phys. Rev. Lett. 97 067402
  * [23] Liu G-Q, Antonov V N, Jepsen O and Andersen O K 2008 Coulomb-enhanced spin–orbit splitting: the missing piece in the Sr2RhO4 puzzle Phys. Rev. Lett. 101 026408
  * [24] Kim B J et al 2008 Novel Jeff = 1/2 Mott state induced by relativistic spin–orbit coupling in Sr2IrO4 Phys. Rev. Lett. 101 076402
  * [25] Jackeli G and Khaliullin G 2009 Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models Phys. Rev. Lett. 102 017205
  * [26] Hirata Y et al 2013 Complex orbital state stabilized by strong spin-orbit coupling in a metallic iridium oxide IrO2 Phys. Rev. B 87 161111(R)
  * [27] Panda S K, Bhowal S, Delin A, Eriksson O and Dasgupta I 2014 Effect of spin orbit coupling and Hubbard $U$ on the electronic structure of IrO2 Phys. Rev. B 89 155102
  * [28] Sun Y, Zhang Y, Liu C -X, Felser C and Yan B 2017 Dirac nodal lines and induced spin Hall effect in metallic rutile oxides Phys. Rev. B 95 235104
  * [29] Nelson J N et al 2019 Dirac nodal lines protected against spin-orbit interaction in IrO2 Phys. Rev. Materials 3 064205
  * [30] Sen K et al 2020 Strange semimetal dynamics in SrIrO3 Nat. Commun. 11 4270
  * [31] Liu X, Berlijn T, Yin W-G, Ku W, Tsvelik A, Kim Y-J, Gretarsson H, Singh Y, Gegenwart P and Hill J P 2011 Long-range magnetic ordering in Na2IrO3 Phys. Rev. B 83 220403(R)
  * [32] Ye F, Chi S, Cao H, Chakoumakos B C, Fernandez-Baca J A, Custelcean R, Qi T F, Korneta O B and Cao G 2012 Direct evidence of a zigzag spin-chain structure in the honeycomb lattice: A neutron and x-ray diffraction investigation of single-crystal Na2IrO3 Phys. Rev. B 85 180403(R)
  * [33] Singh Y, Manni S, Reuther J, Berlijn T, Thomale R, Ku W, Trebst S, and Gegenwart P 2012 Relevance of the Heisenberg-Kitaev Model for the Honeycomb Lattice Iridates A2IrO3 Phys. Rev. Lett. 108 127203
  * [34] Sizyuk Y, Price C, Wölfle P, and Perkins N B 2014 Importance of anisotropic exchange interactions in honeycomb iridates: Minimal model for zigzag antiferromagnetic order in Na2IrO3 Phys. Rev. B 90 155126
  * [35] Rau J G, Lee E K-H, and Kee H-Y 2014 Generic Spin Model for the Honeycomb Iridates beyond the Kitaev Limit Phys. Rev. Lett. 112 077204
  * [36] Biffin A et al 2014 Noncoplanar and Counterrotating Incommensurate Magnetic Order Stabilized by Kitaev Interactions in $\gamma$-Li2IrO3 Phys. Rev. Lett. 113 197201
  * [37] Biffin A, Johnson R D, Choi S, Freund F, Manni S, Bombardi A, Manuel P, Gegenwart P and Coldea R 2014 Unconventional magnetic order on the hyperhoneycomb Kitaev lattice in $\beta$-Li2IrO3: Full solution via magnetic resonant x-ray diffraction Phys. Rev. B 90 205116
  * [38] Kimchi I and Vishwanath A 2014 Kitaev-Heisenberg models for iridates on the triangular, hyperkagome, kagome, fcc,and pyrochlore lattices Phys. Rev. B 89 014414
  * [39] Becker M, Hermanns M, Bauer B, Garst M and Trebst S 2015 Spin-orbit physics of $j=1/2$ Mott insulators on the triangular lattice Phys. Rev. B 91 155135
  * [40] Chun S H et al 2015 Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na2IrO3 Nat. Phys. 11 462
  * [41] Takayama T et al 2015 Hyperhoneycomb Iridate $\beta$-Li2IrO3 as a Platform for Kitaev Magnetism Phys. Rev. Lett. 114 077202
  * [42] Catuneanu A, Rau J G, Kim H-S and Kee H-Y 2015 Magnetic orders proximal to the Kitaev limit in frustrated triangular systems: Application to Ba3IrTi2O9 Phys. Rev. B 92 165108
  * [43] Bhowal S, Ganguly S and Dasgupta I 2019 Spin-orbit coupling driven novel magnetism in $d^{5}$ 6H-perovskite iridates Ba3IrTi2O9 and $Ba_{3}$TiIr2O9 J. Phys.: Condens. Matter. 31 85802
  * [44] Aczel AA et. al. 2019 Revisiting the Kitaev material candidacy of Ir4+ double perovskite iridates, Phys. Rev. B 99 134417
  * [45] Pesin D and Balents L 2010 Mott physics and band topology in materials withstrong spin-orbit interaction Nat. Phys. 6 373-381
  * [46] Wan X, Turner A M, Vishwanath A and Savrasov Y S 2011 Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates Phys. Rev. B 83 205101
  * [47] Carter J-M, Shankar V V, Zeb M A and Kee H-Y 2012 Semimetal and Topological Insulator in Perovskite Iridates Phys. Rev. B 85 115105
  * [48] Ueda K et. al. 2012 Variation of Charge Dynamics in the Course of Metal-Insulator Transition for Pyrochlore-Type Nd2Ir2O7 Phys. Rev. Lett. 109 136402
  * [49] Zhang H, Haule K and Vanderbilt D 2017 Metal-Insulator Transition and Topological Properties of Pyrochlore Iridates Phys. Rev. Lett. 118 026404
  * [50] Sun Y, Zhang Y, Liu C -X, Felser C. and Yan B. 2017 Dirac nodal lines and induced spin Hall effect in metallic rutile oxides Phys. Rev. B 95 235104
  * [51] Ueda K, Kaneko R, Ishizuka H, Fujioka, J, Nagaosa N and Tokura Y 2018 Spontaneous Hall effect in the Weyl semimetal candidate of all-in all-out pyrochlore iridate Nat. Commun. 9 3032
  * [52] Sushkov A B, Hofmann J B, Jenkins G S, Ishikawa J, Nakatsuji S, Sarma S D and Drew H D 2015 Optical evidence for a Weyl semimetal state in pyrochlore Eu2Ir2O7 Phys. Rev. B 92 241108(R)
  * [53] Xu X et al 2019 Strong spin-orbit coupling and Dirac nodal lines in the three-dimensional electronic structure of metallic rutile IrO2 Phys. Rev. B 99 195106
  * [54] Nelson J N et al 2019 Dirac nodal lines protected against spin-orbit interaction in IrO2 Phys. Rev. Mater. 3 064205
  * [55] Bhowal S and Satpathy S. 2019 Dirac nodal lines and large spin Hall effect in the 6H-perovskite iridate Ba3TiIr2O9 Phys. Rev. B 100 115101
  * [56] Okamoto Y, Minoru N, Hiroko A-K and Takagi H 2007 Spin-Liquid State in the $S=1/2$ Hyperkagome Antiferromagnet Na4Ir3O8 Phys. Rev. Lett. 99 137207
  * [57] Dey T, Mahajan A V, Khuntia P, Baenitz M, Koteswararao B and Chou F C 2012 Spin-liquid behavior in $J_{\rm eff}=1/2$ triangular lattice compound Ba3IrTi2O9 Phys. Rev. B 86 140405(R)
  * [58] Lee, W-J et al 2017 Putative spin liquid in the triangle-based iridate Ba3IrTi2O9 Phys. Rev. B 96 014432
  * [59] Lovesey S W and Khalyavin D D 2014 Strange magnetic multipoles and neutron diffraction by an iridate perovskite (Sr2IrO4) J. Phys.: Condens. Matter. 26 322201
  * [60] Ganguly S, Grånäs O, and Nordström L 2015 Nontrivial order parameter in Sr2IrO4 Phys. Rev. B 91 020404(R)
  * [61] Zhao L et al 2016 Evidence of an odd-parity hidden order in a spin–orbit coupled correlated iridate Nat. Phys. Lett. 12 32
  * [62] Ganguly S and Bhowal S 2020 Study of nontrivial magnetism in 3d-5d transition metal based double perovskites Phys. Rev. B 101 085104
  * [63] Kim K, Sung N H, Denlinger J D and Kim B J 2016 Observation of a d-wave gap in electron-doped Sr2IrO4 Nat. Phys. Lett. 12 37
  * [64] Petrzhik A M et al 2019 Superconducting current and low-energy states in a mesa-heterostructure interlayeredwith a strontium iridate film with strong spin-orbit interaction Phys. Rev. B 100 024501
  * [65] Witczak-Krempa W, Chen G, Kim Y B and Balents L 2014 Correlated Quantum Phenomena in the Strong Spin-Orbit Regime Annu. Rev. Condens. Matter Phys. 5 57
  * [66] Bertinshaw J, Kim Y K, Khaliullin G and Kim B J 2019 Square Lattice Iridates Annu. Rev. Condens. Matter Phys. 10 315
  * [67] Schaffer R, Lee E K-H, Yang B-J and Kim Y B2016 Recent progress on correlated electron systems with strong spin–orbit coupling Rep. Prog. Phys. 79 094504
  * [68] Rau J G, Lee E K-H and Kee H-Y 2016 Spin-Orbit Physics Giving Rise to Novel Phases in Correlated Systems: Iridates and Related Materials Annu. Rev. Condens. Matter Phys. 7 195
  * [69] Martins C, Aichhorn M and Biermann S 2017 Coulomb correlations in 4d and 5d oxides from first principle - or how spin-orbit materials choose their effective orbital degeneracies J. Phys.: Condens. Matter 29 263001
  * [70] Cao G and Schlottmann P 2018 The challenge of spin-orbit-tuned ground states in iridates: a key issues review Rep. Prog. Phys. 81 042502
  * [71] Winter S M et al 2017 Models and materials for generalized Kitaev magnetism J. Phys.: Condens. Matter 29 493002
  * [72] Sachdev S and Keimer B 2011 Quantum criticality Phys. Today 64 2 29
  * [73] Khaliullin G 2013 Excitonic Magnetism in Van Vleck–type $d^{4}$ Mott Insulators Phys. Rev. Lett. 111 197201
  * [74] Figgis B N and Lewis J 1960 Modern Coordination Chemistry Interscience Publishers 451
  * [75] Earnshaw A, Figgis B, Lewis J and Peacock R 1961 The Magnetic Properties of Some $d^{4}$-Complexes J. Chem. Soc. 601 3132
  * [76] Cao G, Qi T F, Li L, Terzic J, Yuan S J, DeLong L E, Murthy G and Kaul R K 2014 Novel Magnetism of Ir5+ (5$d^{4}$) Ions in the Double Perovskite Sr2YIrO6 Phys. Rev. Lett. 112 056402
  * [77] Bhowal S, Baidya S, Dasgupta I and Saha-Dasgupta T 2015 Breakdown of $J=0$ nonmagnetic state in $d^{4}$ iridate double perovskites: A first-principles study Phys. Rev. B 92 121113(R)
  * [78] Nag A et al 2016 Origin of the Spin-Orbital Liquid State in a Nearly $J=0$ Iridate Ba3ZnIr2O9 Phys. Rev. Lett. 116 097205
  * [79] Nag A et al 2018 Origin of magnetic moments and presence of spin-orbit singlets in Ba2YIrO6 Phys. Rev. B 98 014431
  * [80] Nag A, Bhowal S, Sala M M, Efimenko A, Dasgupta I and Ray S 2019 Hopping-Induced Ground-State Magnetism in 6H Perovskite Iridates Phys. Rev. Lett. 123 017201
  * [81] Nanda B R K and Satpathy S 2008 Effects of strain on orbital ordering and magnetism at perovskite oxide interfaces: LaMnO3/SrMnO3 Phys. Rev. B 78 054427
  * [82] Kumar R et al 2019 Unconventional magnetism in the 4$d^{4}$-based $S=1$ honeycomb system Ag3LiRu2O6 Phys. Rev. B 99 054417
  * [83] Chen Q et al 2019 Large Positive Zero-Field Splitting in the Cluster Magnet Ba3CeRu2O9 J. Am. Chem. Soc. 141 9928-9936
  * [84] Takahashi H et al 2021 Nonmagnetic $J=0$ State and Spin-Orbit Excitations in K2RuCl6 arXiv:2107.10046
  * [85] G. Khaliullin and S. Okamoto 2002 Quantum Behavior of Orbitals in Ferromagnetic Titanates: Novel Orderings and Excitations Phys. Rev. Lett. 89 167201
  * [86] Khomskii D I 2014 Transition Metal Compounds, University Printing House, Cambridge CB2 8BS, United Kingdom.
  * [87] Dai D, Xiang H and Whangbo M-H 2008 Effects of Spin-Orbit Coupling on Magnetic Properties of Discrete and Extended Magnetic Systems J Comput Chem 29 2187-2209
  * [88] Kaushal N, Soni R, Nocera A, Alvarez G and Dagotto E 2020 BCS-BEC crossover in a ($t_{2g}$)4 excitonic magnet Phys. Rev. B 101 245147
  * [89] Dey T et al 2016 Ba2YIrO6: A cubic double perovskite material with Ir5+ ions Phys. Rev. B 93 014434
  * [90] Pajskr K, Nov́ak P, Pokorný V, Kolorenc J, Arita R and Kunes J 2016 On the possibility of excitonic magnetism in Ir double perovskites Phys. Rev. B 93 035129
  * [91] Hammerath F et al 2017 Diluted paramagnetic impurities in nonmagnetic Ba2YIrO6 Phys. Rev. B 96 165108
  * [92] Chen Q et al 2017 Magnetism out of antisite disorder in the $J=0$ compound Ba2YIrO6 Phys. Rev. B 96 144423
  * [93] Fuchs S et al 2018 Unraveling the Nature of Magnetism of the 5$d^{4}$ Double Perovskite Ba2YIrO6 Phys. Rev. Lett. 120 237204
  * [94] Gong H, Kim K, Kim B H, Kim B, Kim J, Min B I 2018 Is the ground state of 5$d^{4}$ double-perovskite Iridate Ba2YIrO6 magnetic or nonmagnetic? J. Magn. Magn. Mater. 454 66-70
  * [95] Terizc J, Zheng H, Ye F, Zhao H D, Schlottmann P, De Long L E, Yuan S J and Cao G 2017 Evidence for a low-temperature magnetic ground state in double-perovskite iridates with Ir5+(5$d^{4}$) ions Phys. Rev. B 96 064436
  * [96] Corredor L T et al 2017 Iridium double perovskite Sr2YIrO6: A combined structural and specific heat study Phys. Rev. B 95 064418
  * [97] Wakeshima M, Harada D and Hinatsu Y 1999 Crystal structures and magnetic properties of ordered perovskites A2R3+Ir5+O6 (A = Sr, Ba; R = Sc, Y, La, Lu) J. Alloy Compd. 287 130-136
  * [98] Kayser P, Kennedy B J , Ranjbar B, Kimpton J A and Avdeev M 2017 Spin-Orbit Coupling Controlled Ground State in the Ir(V) Perovskites A2ScIrO6 (A = Ba or Sr) Inorg. Chem. 56 2204-2209
  * [99] Chakraborty A and Dasgupta I 2019 Study of electronic structure and magnetism in $d^{4}$ double perovskite iridates A2ScIrO6 (A=Ba, Sr) J. Magn. Magn. Mater. 492 165708
  * [100] Bhowal S and Dasgupta I 2020 Interplay of covalency, non-cubic crystal field, and spin-orbit coupling : A comparative study of $d^{5}$, $d^{4}$, and $d^{3}$ double perovskite iridates Sr2MIrO6 (M=Ce, Sc, Ca) J. Magn. Magn. Mater. 507 166827
  * [101] Fu W T and Ijdo D J W 2005 Re-examination of the structure of Ba2MIrO6 (M=La, Y) : space group revised J. Alloys Compd.  394 L5-L8
  * [102] Thumm I, Treiber U and Kemmler-Sack S 1980 Über Sauerstoffperowskite mit fünf- und vierwertigem Iridium Verbindungen vom Typ Ba2B3+lr5+06 und Ba3B3+lr${}^{4.5+}_{2}$09 J. Solid State Chem. 35 156-166
  * [103] Laguna-Marco M A et al 2015 Electronic structure, local magnetism, and spin-orbit effects of Ir(IV)-, Ir(V)-, and Ir(VI)-based compounds Phys. Rev. B 91 214433
  * [104] Narayanan N et al 2010 Temperature and composition dependence of crystal structures and magnetic and electronic properties of the double perovskites La2-xSrxCoIrO6 ($0\leq x\leq 2$) Phys. Rev. B 82 024403
  * [105] Kolchinskaya A et al 2012 Magnetism and spin-orbit coupling in Ir-based double perovskites La2-xSrxCoIrO6 Phys. Rev. B 85 224422
  * [106] Agrestini S et al 2019 Nature of the magnetism of iridium in the double perovskite Sr2CoIrO6 Phys. Rev. B 100 014443
  * [107] Kharkwal K C et al 2020 Structure, magnetism, and electronic properties in 3$d$-5$d$ based double perovskite (Sr1-xCax)2FeIrO6 ($0\leq x\leq 1$) Phys. Rev. B 102 174401
  * [108] Powell A V, Gore J G and Battle P D 1993 The magnetic properties of iridium in mixed-metal oxides J. Alloys Comp. 201 73
  * [109] Hayashi K, Demazeau G, Pouchard M and Hagenmuller P 1980 Preparation and Magnetic study of a New Iridium (V) perovskite LaLi0.5Ir${}_{0}.5$O3 Mater.Res. Bull. 15 461
  * [110] Prasad B, Doert T, Felser C and Jansen M 2018 On $J_{\rm eff}=0$ Ground State Iridates(V): Tracking Residual Paramagnetism in New Bi2NaIrO6 Chem. Eur. J. , 24 16762 - 16765
  * [111] Wolff K K, Agrestini S, Tanaka A, Jansen M and Tjeng L H 2017 Comparative Study of Potentially $J_{\rm eff}=0$ Ground State Iridium(V) in SrLaNiIrO6, SrLaMgIrO6, and SrLaZnIrO6 Z Anorg Allg. Chem.  643 2095
  * [112] Bremholm M, Dutton S E, Stephens P W and Cava R J 2011 NaIrO3-a pentavalent post-perovskite J. Solid State Chem. 184 601-7