# Kitaev magnetism through the prism of lithium iridate

Alexander A. Tsirlin<EMAIL_ADDRESS>Philipp Gegenwart  Experimental
Physics VI, Center for Electronic Correlations and Magnetism, University of
Augsburg, 86159 Augsburg, Germany

###### Abstract

Fifteen years since its inception, Kitaev model still boasts only a narrow
group of material realizations. We review the progress in studying and
understanding one of them, lithium iridate Li2IrO3 available in three
polymorphs that host strong Kitaev interactions on spin lattices of different
dimensionality and topology. We also discuss feasibility, effectiveness, and
repercussions of tuning strategies based on the application of external
pressure and chemical substitutions.

##  I Introduction

Quantum states characterized by long-range entanglement show some of the most
intriguing and enigmatic properties in condensed-matter systems [1].
Understanding the nature of these quantum states and elucidating their
behavior remains a formidable challenge, because relevant microscopic models
are notoriously difficult or even impossible to solve. In this context, Alexei
Kitaev’s discovery of an analytically solvable spin model hosting a quantum
spin liquid ground state received significant attention, reflected for example
in 2300 citations of his original work [2] over the last 15 years and in
Kitaev interactions emerging as the new type of magnetic exchange on par with
the Heisenberg and Dzyaloshinskii-Moriya interactions as textbook examples of
magnetic couplings.

Kitaev spin model was originally formulated for the planar honeycomb lattice
and later extended to different tricoordinated lattices in three dimensions
[3, 4]. The crux of the model is the Ising-like exchange term where spin
direction $\gamma$ varies from one bond to another,

$\mathcal{H}=K\sum_{\langle ij\rangle}S_{i}^{\gamma}S_{j}^{\gamma}$ (1)

($\gamma=X,Y,Z$ depending on the bond). This special setting creates an
exchange frustration, which is quite different from the more common
geometrical frustration, and implies that long-range magnetic order does not
form because no stable spin direction can be found in the magnetically ordered
state. On a more rigorous level, the solution of Eq. (1) is facilitated by
representing each spin operator as a combination of four Majorana fermions
that are separated into flux variables and free “matter” fermions. This
elegant construction has far-reaching implications. Excitations of the Kitaev
spin liquid exhibit an unusual anyonic statistics (they are neither bosons nor
fermions) and create interesting opportunities for topological quantum
computing [2].

Different extensions of the Kitaev model were considered over the last years.
Details of these extended models, along with basic properties of the parent
Kitaev model itself, can be found in several review articles [5, 6, 7].
Another interesting direction has been the search for real-world
manifestations of the Kitaev physics and, specifically, for insulating magnets
dominated by the Kitaev exchange term.

Bringing Kitaev model into the lab requires a spin-orbit-entangled electronic
state of the transition-metal ion. The symmetries of this state and associated
exchange pathways should favor the interaction term of Eq. (1) and prevent
Heisenberg or any other type of exchange. This situation is rather unique and
was thought to be accomplishable only in $4d$ and $5d$ metals with the low-
spin $d^{5}$ state and strong spin-orbit coupling [8]. More recently, the
possibility of a similar microscopic scenario in the high-spin $d^{7}$ states
[9] and selected $4f$ states [10, 11] was put forward, but at the time of this
review most experimental efforts are still focused on a handful of $d^{5}$
compounds of Ir4+ ($5d$) and Ru3+ ($4d$). The purpose of the present article
is to look at some interesting aspects and associated challenges through the
prism of a single material, lithium iridate Li2IrO3, that adopts three
different crystal structures, allows multiple chemical substitution, and
highlights several generic features of Kitaev magnets. For a recent review of
other Kitaev materials see also Refs. [12, 13].

##  II Parent compounds

###  II.1 Crystal structure

Two decades ago solid-state chemists would probably find lithium iridate a
very mundane compound, yet another derivative of the rocksalt structure with
alkaline and transition metals filling octahedral voids of the close-packed
oxygen framework [14, 15]. Similar compounds are known for many of the $4d$,
$5d$, and even $5p$ metals [16] and show little differences in their crystal
chemistry. This fact, along with the high price of iridium and its
undesirability for material science, did not motivate detailed studies.
Indeed, no single attempt of the crystal structure refinement for Li2IrO3 was
performed until 2000’s [17, 18], despite the first reports on the synthesis
having appeared much earlier [19, 20].

Figure 1:  Crystal structures of three Li2IrO3 polymorphs and the respective
magnetic topologies: planar honeycomb ($\alpha$), hyperhoneycomb ($\beta$),
and stripy-honeycomb ($\gamma$) lattices of the Ir4+ ions. Different colors
show $X$-, $Y$-, and $Z$-bonds of the Kitaev model, Eq. (1).

Recent interest triggered by prospects of the Kitaev physics has led to an
extensive reinvestigation of the layered $\alpha$-Li2IrO3 polymorph that was
shown to be magnetically ordered below $T_{N}\simeq 15$ K [21], in contrast to
the earlier report where no magnetic order was found [22]. Further research
culminated in the discovery of two new polymorphs [23, 24, 25] that differ in
the spatial arrangement of Li and Ir within the octahedral voids. This
difference leads to distinct crystallographic symmetries and magnetic
topologies (Fig. 1). Monoclinic $\alpha$-Li2IrO3 has planar honeycomb
structure in the vein of the original Kitaev model. In contrast, orthorhombic
$\beta$\- and $\gamma$-Li2IrO3 feature three-dimensional (3D) hyperhoneycomb
and stripy-honeycomb (or harmonic-honeycomb) lattices, respectively.
Theoretical work on 3D versions of the Kitaev model surged [26, 27, 28, 29,
30, 31, 32] after these new polymorphs were discovered.

Polymorphism related to the different distribution of cations within the
closed-packed oxygen framework is not uncommon for ternary oxides. Some of
them even adopt same symmetries as $\alpha$\- and $\beta$-Li2IrO3 [15],
although detailed knowledge on thermodynamic stability of different polymorphs
in relation to their chemistry and composition is presently missing. In the
Li2IrO3 case, $\alpha$\- and $\gamma$-polymorphs transform into the
$\beta$-polymorph upon heating above 1000 ∘C [23, 33, 34], suggesting that
$\beta$-Li2IrO3 should be the thermodynamically stable form of lithium
iridate.

Figure 2:  Crystal growth of $\alpha$\- and $\beta$-Li2IrO3 from separated
educts. Metallic Li and Ir are placed in different parts of the crucible with
intermediate spikes serving as crystallization centers. The typical crystals
are shown on the right. Part of the figure is adapted from Ref. [33] (CC-BY).

In recent years, Li2IrO3 gained its appeal also among solid-state chemists.
New experiments on Li deintercalation [35, 36, 37] highlighted the unusual
anion-redox mechanism, the partial oxidation of oxygen upon extraction of Li,
that can be used to increase reversible capacity of battery materials [38].
These two lines of researchs, batteries and magnetism, developed entirely
independently from each other and have yet to explore possible interesting
connections, such as tuning or charge doping of Li2IrO3 via controllable
electrochemical intercalation.

###  II.2 Crystal growth

Iridate compounds are notorious for their difficult crystal growth, but
Li2IrO3 is exceptional even among those. The proclivity of Ir4+ toward
reduction and oxygen release prevents the usage of floating-zone method,
because oxygen pressures required to stabilize the Ir4+ state are very high
and essentially unfeasible. In the case of lithium iridate, decomposition
takes place around 1200 ∘C well before melting point is reached. On the other
hand, sizable vapor pressures of both alkaline-metal oxides and iridium oxide
below decomposition temperature facilitate crystal growth from gas phase via
some kind of vapor transport. This process is most prominent in Na2IrO3 where
mm-size single crystals are grown by a simple annealing of pre-reacted powder
sample [39].

The growth of Li2IrO3 is more tricky in the sense that only tiny crystals with
the lateral dimensions of 0.1-0.2 mm can be obtained by simple annealing [40,
34]. Moreover, crystals of the $\alpha$-polymorph can not be grown in this
way, because vapor transport remains weak in the temperature range where this
phase is stable. The method of separated educts [33] leads to improved results
and increases the typical crystal size to about 0.5 mm. Here, lithium and
iridium metals are used as reactants and separated in space inside the
crucible. Spikes are deliberately placed between the reactants and serve as
crystallization centers during the vapor growth from oxidized Li and Ir (Fig.
2). Changing reaction temperature allows growth of both $\alpha$\- and
$\beta$-polymorphs [33, 41, 42].

Whereas $\beta$\- and $\gamma$-Li2IrO3 are believed to be free from crystal
defects, $\alpha$-Li2IrO3 crystals suffer from both twinning and stacking
faults [33, 42]. While the latter is a notorious problem of all layered
materials, the former results from an accidental match between the $a$ and $c$
lattice parameters. This twinning issue is unique to $\alpha$-Li2IrO3 and
renders it a less convenient Kitaev material, especially in the context of
various tuning attempts discussed in Sec. III below. Crystal quality has
drastic influence on the magnetic behavior. Stacking faults broaden magnetic
ordering transition and eventually eliminate it [43], resulting in a state
that ostensibly resembles a spin liquid but hardly constitutes one.

###  II.3 Magnetic anisotropy

Magnetic response of the Li2IrO3 polymorphs is highly anisotropic (cf. Fig. 3
for the $\beta$-polymorph). At high temperatures, paramagnetic effective
moments are in the range of $1.5-2.0\,\mu_{B}$ for all field directions [21,
44, 41, 25], as expected for Ir4+ in the regular octahedral environment
($j_{\rm eff}=\frac{1}{2}$ state independently confirmed by resonant inelastic
x-ray spectroscopy [45, 46] and optical measurements [47, 48, 49, 50]). This
renders on-site magnetic anisotropy, such as $g$-tensor anisotropy, weak. A
much stronger anisotropy arises from the exchange couplings. Curie-Weiss
temperatures for the same polymorph not only differ in absolute values, but
also show different signs depending on the field direction (Fig. 3a) [41, 25].
While this anisotropy potentially carries useful information on the size of
Kitaev and other exchange terms [51], their direct calculation from the
magnetic susceptibility proved to be difficult. The problem may lie in single-
ion effects (electronic excitations at the Ir4+ site) that cause deviations
from the simple Curie-Weiss behavior [52].

Figure 3:  Anisotropic magnetic response of $\beta$-Li2IrO3 [41]. (a) Inverse
magnetic susceptibility for three field directions; the Curie-Weiss fits
(dotted lines) have a similar slope but very different intercepts (Curie-Weiss
temperatures). (b) Field-dependent magnetization reveals the transition at
$B_{c}\simeq 2.7$ T for $B\,\|\,b$; the color bar illustrates a gradual
replacement of the incommensurate $Q\neq 0$ order by the commensurate $Q=0$
mode of the quantum paramagnet that does not break any symmetries and,
therefore, does not cause a magnetic transition as a function of temperature
above $B_{c}$. (c) Temperature-field phase diagram for $B\,\|\,b$; the phase
boundary is probed by thermodynamic measurements. The $Q\neq 0$ (counter-
rotating spirals) and $Q=0$ modes are shown schematically and do not reflect
all details of the real magnetic structure [23].

$\beta$\- and $\gamma$-polymorphs additionally show a very anisotropic
response in field-dependent magnetization at low temperatures [25, 40]. The
field applied along one special direction, which is crystallographic $b$ for
the crystal structures shown in Fig. 1, leads to a partial polarization with a
field-induced transition around $B_{c}=2.7$ T (Fig. 3b) accompanied by a
seeming loss of magnetic order: no phase transition as a function of
temperature can be seen for this field direction above $B_{c}$ [40, 53, 41].
This behavior is often taken as a hallmark of Kitaev physics, because several
candidate materials, including $\alpha$-RuCl3 [54, 55, 56] and, more recently,
Co2+ oxides [57, 58], all revealed a similar evolution. Naively, one may
expect Kitaev spin liquid to appear in $\beta$-Li2IrO3 above $B_{c}$ for
$B\,\|\,b$, but this field-induced state is in fact a rather mundane quantum
paramagnet, as we will see shortly.

Another important remark at this point is that magnetic anisotropy, including
anisotropy of the magnetization, is not a consequence of strong Kitaev
interactions per se. Despite the underlying exchange anisotropy, Kitaev model
itself [Eq. (1)] is in fact isotropic and would show equal Curie-Weiss
temperatures for all directions of the applied field [51]. The strongly
anisotropic magnetic response reviewed in this section indicates salient
deviations of all Li2IrO3 polymorphs from the simple Kitaev model.

###  II.4 Magnetic order and interactions

The departure of all Li2IrO3 polymorphs from the pure Kitaev model is cemented
by the presence of long-range magnetic order below $T_{N}=15$ K [59]
($\alpha$), $37-38$ K [23] ($\beta$), and 40 K [60] ($\gamma$). Magnetic
structures are incommensurate, non-coplanar, and bear surprising resemblances
to each other. All of them feature zigzag chains with counter-rotating spin
spirals (Fig. 3c), quite unusual magnetic entities that signal strong Kitaev
interactions and should not appear in conventional antiferromagnets dominated
by Heisenberg exchange [32].

Further experiments showed that the field-induced state of $\beta$-Li2IrO3 for
$B\,\|\,b$ and $B>B_{c}$ lacks long-range magnetic order but features strong
commensurate ($Q=0$) correlations, with ferromagnetic alignment of the
$b$-components of the spin and antiferromagnetic, zigzag-like alignment of the
$ac$-components [40]. When magnetic field is applied along the $b$ direction,
such a state does not break any symmetries and does not lead to a phase
transition upon cooling (Fig. 3c). It is in fact similar to a fully polarized
state of an isotropic ferromagnet. This explains the apparent absence of
magnetic order above $B_{c}$ in the sense that no phase transition takes place
as a function of temperature; phase transition is replaced by a crossover. The
resulting low-temperature state is by no means a spin liquid and should be
classified as quantum paramagnet [41].

The type of magnetic order can be indicative of underlying magnetic
interactions. In the case of $j_{\rm eff}=\frac{1}{2}$ moments of Ir4+, at
least three main terms have to be considered [61],

$\mathcal{H}_{12}=J\mathbf{S}_{1}\mathbf{S}_{2}+KS_{1}^{\gamma}S_{2}^{\gamma}+\Gamma(S_{1}^{\alpha}S_{2}^{\beta}+S_{1}^{\beta}S_{2}^{\alpha}),$
(2)

where $J$ is Heisenberg exchange, $K$ is Kitaev exchange, and $\Gamma$ is off-
diagonal anisotropy. The index $\gamma$ is one of the Kitaev axes ($X,Y,Z$)
and changes from bond to bond, while $\alpha$ and $\beta$ denote spin
components perpendicular to $S^{\gamma}$.

Magnetic order observed in $\beta$\- and $\gamma$-Li2IrO3 can be rationalized
within the framework of this minimal $JK\Gamma$ model for nearest-neighbor
interactions and with ferromagnetic Kitaev term $K$ [62, 63, 64]. This model
also explains similarities of magnetic order across the different polymorphs
[65]. Moreover, field-induced phase transition and the $B_{c}$ value of
$\beta$-Li2IrO3 gauge $J\simeq 0.3$ meV [66], suggesting that at least
$\beta$-Li2IrO3 should be dominated by $K$ and $\Gamma$, while $J$ is
negligible. The smallness of an excitation gap below $B_{c}$ [67] puts an
additional $|K|\simeq|\Gamma|$ constraint and allows a quite accurate
positioning of $\beta$-Li2IrO3 on the phase diagram, in reasonable agreement
with the ab initio results [68, 69]. The absolute values of both $|K|$ and
$|\Gamma|$ are in the range of $10-15$ meV [69, 41].

Less is known about the microscopic parameters of $\alpha$-Li2IrO3. Magnetic
structure of this compound was so far rationalized only within an extended
model that breaks equivalence of interactions on the $XY$\- and $Z$-bonds of
the honeycomb lattice [59]. Such disparity is indeed allowed by symmetry in
all Li2IrO3 polymorphs (and also in other Kitaev candidates), but its role is
generally believed to be minor [51]. The situation may be different in
$\alpha$-Li2IrO3, though. Moreover, the terms beyond Eq. (2), including
Dzyaloshinskii-Moriya interactions and long-range Heisenberg interactions, may
not be insignificant [51]. It is also worth noting that a large deal of
confusion in the microscopic analysis [51, 70] was caused by the inaccurate
crystallographic data used for $\alpha$-Li2IrO3 prior to 2016. The results
obtained for crystal structures other than the experimental one from Ref. [33]
(or, alternatively, the fully relaxed one [51]) may be strongly influenced by
this ambiguity.

###  II.5 Magnetic excitations

Incommensurate long-range order causes complex magnon spectra that were indeed
observed in the energy range up to 12 meV in $\alpha$-Li2IrO3 [71] and up to
16 meV in $\beta$-Li2IrO3 [72]. Despite the strong exchange anisotropy, magnon
gaps in zero field are negligible [71, 67]. In the case of $\beta$-polymorph,
field-induced phase transition opens a gap above $B_{c}$, similar to another
Kitaev material $\alpha$-RuCl3 where this gap opening has been studied
extensively [73, 74, 75].

The presence of sizable or even dominant Kitaev interactions led to an idea
that excitations other than magnons may appear at higher energies above the
one-magnon part of the spectrum, or above $T_{N}$ where long-range magnetic
order disappears. Indeed, an excitation continuum has been seen above 20 meV
in different Li2IrO3 polymorphs using resonant inelastic x-ray scattering [76,
72] and Raman spectroscopy [77, 78, 79]. This phenomenology resembles not only
other Kitaev materials [80, 81], but also frustrated magnets with predominant
Heisenberg interactions [82]. Consequently, it remains debated whether the
excitation continuum manifests Majorana excitations of the underlying Kitaev
model [83] or arises from magnon breakdown triggered by the strong exchange
anisotropy [84].

Interestingly, the continuum persists well above $T_{N}$ [76, 72, 81] where
magnon description should become irrelevant. Calculations for the pure Kitaev
model reveal interesting similarities to the experimental observations in this
temperature range [76, 81], although it remains unclear whether additional
interactions manifested by the spectral structure below $T_{N}$ [81] can be
simply neglected when magnetic order disappears. Calculating finite-
temperature response for extended Kitaev models [85] remains a significant
challenge that essentially constraints possible interpretations of the
continuum to fermionic excitations of the pure Kitaev model, whereas other
scenarios await a thorough theoretical study. The excitation continuum present
far above $T_{N}$ appears to be a generic feature of the Kitaev iridates and
may be used to classify them as “proximate Kitaev spin liquids”, although
details of this purported connection remain to be explored.

###  II.6 Open questions

Finally, we summarize main open questions regarding three Li2IrO3 polymorphs.
First, the exact microscopic scenario of $\alpha$-Li2IrO3 is far from being
settled. This Kitaev material was among the first to appear [21] and initially
appealed researchers with its apparent simplicity, but nonetheless it eludes
complete understanding even after a decade of extensive research. It also
shows the peculiar spatial anisotropy of the underlying honeycomb lattice.
This effect was not seen in any other Kitaev material and deserves a more
thorough investigation, especially in the light of newly prepared substituted
compounds (Sec. III.2) where pre-conditions for deviating from isotropic
Kitaev interactions become even stronger.

Second, $\beta$-Li2IrO3 shows one interesting feature that goes beyond the
already established microscopic scenario. A very weak remnant magnetization
develops below 100 K, well above $T_{N}=37-38$ K. Muon spin relaxation
confirms bulk nature of this putative order, although no oscillations typical
for a long-range-ordered state appear at 100 K. An unusual type of order may
thus coexist or probably compete with the conventional incommensurate long-
range order that develops below $T_{N}$ [34]. Further experiments, such as
thermal expansion measurements, may be interesting to probe the effect of this
100 K anomaly on the lattice. It is also interesting whether aspects like Li
deficiency or Li dynamics can be relevant in this case.

Third, the incommensurate non-coplanar order of Li2IrO3 is quite unusual in
its own right. The common wisdom is that non-collinear magnetic orders are
very sensitive to the applied field and readily turn into collinear spin-
density waves [86] or exotic spin textures like skyrmions [87]. At first
glance, the field-induced transition of $\beta$-Li2IrO3 in $B\,\|\,b$ seems
fairly mundane because it leads to a state with commensurate spin-spin
correlations, but the way this new state sets in is also quite unusual.
Commensurate correlations appear already below $B_{c}$ and compete with the
incommensurate order [40]. This is very different from conventional
antiferromagnets where collinear state abruptly transforms into a spin-flop
state when field is applied along the magnetic easy axis.

Theoretical studies based on the $JK\Gamma$ model highlight an interesting
evolution of incommensurate Li2IrO3 orders in magnetic field [88].
$\beta$-Li2IrO3 is predicted to undergo a field-induced transition also for
$B\,\|\,c$ around 13 T [88], the field feasible for different experimental
probes of magnetic order and phase transitions. In the case of
$\gamma$-Li2IrO3, torque measurements in high fields revealed an unusual
sawtooth shape discussed in the context of chiral spin order [89], although
calculations for realistic spin models with strong exchange anisotropy offer a
more conventional explanation of this finding [90].

##  III Tuning

Figure 4:  Interaction parameters of Eq. (2) vs. Ir–O–Ir bridging angle [51].
Kitaev limit is approached around 100 deg. Colored bars schematically show the
bridging angles in some of the candidate materials listed in Table 1. The bar
length reflects the distribution of angles between the $X$-, $Y$-, and
$Z$-bonds and between compounds of the same series: $\alpha$-Li2IrO3 [33] and
$\beta$-Li2IrO3 [24], Na2IrO3 [91], $\alpha$-MgIrO3 and $\alpha$-ZnIrO3 [92],
Ag3LiIr2O6 [93], and Cu3NaIr2O6 [94]. Pressure evolution is shown for
$\beta$-Li2IrO3 based on the data from Ref. [69].

The presence of magnetic long-range order, as well as other apparent
deviations of all Li2IrO3 polymorphs from the pure Kitaev model, raised the
question whether a suitable tuning could enhance Kitaev interaction term $K$
at the cost of reducing the $J$ and $\Gamma$ terms of Eq. (2). On the
microscopic level, one expects $J$, $K$, and $\Gamma$ to be strongly dependent
on the local geometry and especially on the Ir–O–Ir bridging angles in the
crystal structure (Fig. 4) [51, 70].

The $\alpha$\- and $\beta$-polymorphs of Li2IrO3 feature the bridging angles
of $94-95^{\circ}$ that should result in $K<0$ and $\Gamma\simeq|K|$ with a
relatively smaller $|J|\ll|K|$ [51]. This interaction regime was established
for $\beta$-Li2IrO3 indeed and can be expected for the other polymorphs, too
(Sec. II.4). Reducing the bridging angle toward $90^{\circ}$ has the unwanted
effect of lowering $K$ and enhancing $\Gamma$. On the other hand, increasing
the angle toward $100^{\circ}$ seems to be the right choice. Here, different
hopping parameters may conspire to suppress $J$ and $\Gamma$, while keeping
$K$ large and ferromagnetic [61, 51], manifesting proximity to the Kitaev
limit.

In the following, we review different tuning strategies from this microscopic
perspective. Pressure experiments define how the bridging angles and,
consequently, interaction terms of Eq. (2) are affected by changes in the unit
cell volume. Volume changes are then used to assess different compounds
prepared by chemical substitution.

###  III.1 Pressure and Strain

Pressure has drastic effect on Li2IrO3. Structural phase transition detected
at room temperature around $P_{c}=3.8$ GPa in $\alpha$-Li2IrO3
($C2/m\rightarrow P\bar{1}$) [95] and $3.8-4.4$ GPa in $\beta$-Li2IrO3
($Fddd\rightarrow C2/c$) [46, 96, 97] leads to magnetism collapse via an
abrupt shortening of $\frac{1}{3}$ of the Ir–Ir bonds on the (hyper)honeycomb
lattice. The high-pressure phase is dimerized and entirely non-magnetic,
because unpaired electrons of Ir are now involved into the formation of metal-
metal bonds manifested by the short Ir–Ir distances in the crystal structure.

This instability seems to be generic for all Kitaev magnets. It has been
observed in $\alpha$-RuCl3 already below 1 GPa [98, 99] and can be expected in
Na2IrO3 too, albeit at much higher pressures [100] where experimental
situation remains controversial [50, 101, 102]. Prominent changes in the
electronic spectrum measured by resonant inelastic x-ray scattering,
especially the suppression of the characteristic peak of the $j_{\rm
eff}=\frac{3}{2}\rightarrow\frac{1}{2}$ excitation around 0.7 eV, confirm the
breakdown of the $j_{\rm eff}=\frac{1}{2}$ state of iridium in the dimerized
phase [103, 46].

Figure 5:  Pressure evolution of $\beta$-Li2IrO3: (a) breakdown of magnetic
order around 1.4 GPa according to magnetic susceptibility and muon spin
relaxation data [69]; (b) structural transformations from low-temperature
x-ray diffraction [96]. Note that $P_{c}$ shifts toward lower pressures upon
cooling. Additionally, an intermediate $P2_{1}/n$ phase and the phase mixture
appear at low temperatures.

Pressure evolution before the structural phase transition is peculiar too.
Branching ratio measured by x-ray absorption reveals strong changes with
pressure even below $P_{c}$ [103, 104] and suggests that the electronic state
of Ir4+, as well as magnetic interactions, may be altered significantly. In
the case of $\alpha$-Li2IrO3, abrupt changes already below 0.2 GPa were
speculated based on powder data [103], although concurrent and subsequent
x-ray diffraction [95], optical [105], and Raman [79] studies on single
crystals did not reveal any additional structural phase transition below
$P_{c}$. It is likely that $\alpha$-Li2IrO3 evolves non-monotonically under
pressure, but retains its crystal structure till the onset of magnetism
collapse and structural dimerization at $P_{c}$. Whether the incommensurately
ordered magnetic ground state of $\alpha$-Li2IrO3 persists up to $P_{c}$
remains to be studied.

Table 1:  Comparison of parent ($\alpha$-Li2IrO3, Na2IrO3) and substituted
compounds: ionic radius $r$ for the six-fold coordinated ion [106], lattice
parameters ($a$, $b$, $c$, $\beta$), and unit cell volume $V$ per formula
unit. Lattice parameters of Cu2IrO3 ($C2/c$) and $\alpha$-AIrO3 with A = Mg,
Zn, Cd ($R\bar{3}$) have been re-calculated for the $C2/m$ unit cell of
$\alpha$-Li2IrO3 to facilitate the comparison.

$r$ (Å) $a$ (Å) $b$ (Å) $c$ (Å) $\beta$ (deg) $V$ (Å3) Ref. $\alpha$-Li2IrO3
0.68 5.175 8.936 5.119 109.83 55.68 [33] Na2IrO3 0.97 5.427 9.395 5.614 109.04
67.65 [91] $\alpha$-MgIrO3 0.66 5.158 8.935 4.418 90 53.46 [92]
$\alpha$-ZnIrO3 0.74 5.199 9.005 4.445 90 54.20 [92] $\alpha$-CdIrO3 0.97
5.368 9.297 4.781 90 61.60 [107] Cu3LiIr2O6 0.96 5.291 9.146 6.011 107.20
69.44 [94] Cu3NaIr2O6 0.96 5.390 9.300 5.971 108.64 71.28 [94] Cu2IrO3 0.96
5.393 9.311 5.961 107.51 71.37 [108] Ag3LiIr2O6 1.26 5.283 9.132 6.482 105.68
75.26 [93] H3LiIr2O6 – 5.349 9.243 4.873 111.44 56.07 [109]

A different behavior was reported for the $\beta$-polymorph. Here,
thermodynamic measurements and muon spin relaxation suggest the breakdown of
magnetic order already at 1.4 GPa, well below $P_{c}$, followed by the
formation of a mixed state that blends static and dynamic spins in the form of
coexisting spin glass and spin liquid (Fig. 5a) [69]. Interestingly, the
dimerization pressure $P_{c}$ shows a strong temperature dependence [96], but
it does not seem that $P_{c}$ simply drops to 1.4 GPa at low temperatures,
making the breakdown of magnetic order a mere consequence of structural
dimerization. More likely, a mixture of structurally dimerized and non-
dimerized phases forms above 1.4 GPa [96] and potentially explains the mixed
magnetic state detected by muons [69]. Alternatively, a previously unforeseen
intermediate phase with a more complex crystal structure may set in at 1.4 GPa
and precede the full dimerization (Fig. 5b) [96].

The behavior of $\beta$-Li2IrO3 right above 1.4 GPa is thus of significant
interest. It is so far the only Kitaev magnet where pressure was instrumental
in stabilizing a magnetically disordered state without a complete loss of
local magnetic moments. A similar breakdown of magnetic order may occur in
$\gamma$-Li2IrO3 at a very similar pressure [110], but its interplay with
structural changes remains to be studied.

Another important aspect of pressure tuning is that the reduction in the unit
cell volume lowers the Ir–O–Ir angles and brings them closer to $90^{\circ}$
(Fig. 4) [69]. Indeed, ab initio calculations based on the experimentally
determined structural parameters of $\beta$-Li2IrO3 under pressure revealed
the enhancement of $\Gamma$ and the reduction in $|K|$ [69], the direction
opposite to the one initially desired. Along with the problem of competing
structural instabilities and imminent magnetism collapse, this renders
hydrostatic pressure a less than optimal tuning strategy for reaching Kitaev
limit in iridates. In fact, negative pressure or tensile strain [111] may be
more promising in this case.

Strain tuning has been discussed as a viable route toward the Kitaev limit
also for $\alpha$-RuCl3 [112], but experimental work in this direction remains
scarce. Given that large strain would be required to reach the Kitaev limit,
epitaxial growth on suitably chosen substrates should be the most promising
strategy. Thin films of $\alpha$-Li2IrO3 have been grown on Y-doped ZrO2 by
pulsed laser deposition and proved to be polycrystalline in nature [113]. In
contrast, metal-organic aerosol deposition allows epitaxial growth on
different substrates, although such films show mostly paramagnetic response
down to low temperatures [114]. Growing structurally ordered thin films of
Kitaev iridates remains an interesting problem and will be a natural avenue
for future research.

###  III.2 Chemical substitutions

Pressure and strain effect can be also emulated by suitable chemical
substitutions. Ion-exchange reactions have been used to prepare a gamut of new
compounds where Li in $\alpha$-Li2IrO3 (or sometimes Na in Na2IrO3) is fully
or partially replaced by other monovalent or divalent ions, see Table 1 and
Fig. 6. The larger ions like Ag+ may lead to only a partial (75 %)
substitution because Li atoms centering Ir hexagons can not be exchanged.
Other ions allow the complete substitution. All these compounds, except
H3LiIr2O6, are prepared by topotactic ion exchange at $300-400$ ∘C, well below
the synthesis temperature of Li2IrO3. They are available as polycrystalline
samples only. Similar to other compounds prepared by the ion exchange [115,
116], chemically substituted iridates are most likely metastable and should
decompose if annealed at high temperatures.

Unit cell volumes of these compounds follow ionic radii of the substituted
elements (Table 1). Smaller divalent metals (Mg, Zn) do not change the volume
significantly. In contrast, monovalent metals lead to a significant expansion
of the Li2IrO3 lattice. This further correlates with the presence of long-
range magnetic order in $\alpha$-MgIrO3 and $\alpha$-ZnIrO3 below 32 and 47 K,
respectively [92]. On the other hand, most of the Cu- and Ag-substituted
compounds show disordered magnetism of some sort [108, 117, 118, 93], albeit
with one exception [119].

At first glance, the suppression of long-range magnetic order in the Cu- and
Ag-substituted iridates is consistent with the trend of approaching Kitaev
limit upon lattice expansion. However, Na2IrO3 serves as another example of a
similarly expanded $\alpha$-Li2IrO3 structure (Table 1), and it does show
long-range magnetic order [39, 120, 121]. In that case, nearest-neighbor $J$
and $\Gamma$ are indeed small111Note however that sizable $J$ and $\Gamma$ are
used in Ref. [81] to model the excitation spectra. while $K$ is large and
ferromagnetic in agreement with the preceding discussion, but third-neighbor
interaction $J_{3}$ intervenes to stabilize antiferromagnetic zigzag order
[51]. Whether or not this $J_{3}$ appears also in the Cu- and Ag-substituted
iridates remains an interesting problem for further investigation. In the case
of the (Na1-xLix)2IrO3 solid solutions that allow a gradual compression of
Na2IrO3 up to $x\simeq 0.24$, long-range magnetic order persists [122]
suggesting the continuous presence of $J_{3}$. However, the type of magnetic
order in Li-doped Na2IrO3 is not known with certainty [123, 124]. The
miscibility gap above $x\simeq 0.24$ is caused by the fact that Li can be
introduced in the honeycomb planes only (Fig. 6) [122].

Figure 6:  Chemical substitutions in $\alpha$-Li2IrO3. Parent structure
contains Li atoms between the honeycomb layers (black), as well as in the
centers of Ir hexagons (dark red). Only the former Li atoms are exchanged in
Ag3LiIr2O6 and H3LiIr2O6, whereas a complete substitution is achieved in
$\alpha$-MgIrO3 and Cu2IrO3.

Additionally, all substituted iridates are prone to stacking disorder [125,
117]. The $\alpha$-Li2IrO3 precursor is likely to contain stacking faults
itself [43, 33], and their concentration can only increase when one removes
ions that hold together the Ir honeycomb layers. Since the ion exchange takes
place at a relatively low temperature, the system should be far from
thermodynamic equilibrium and retain all the defects, both contained in the
parent material and accumulated during the ion exchange. The problem of
stacking disorder appears most prominently in Ag3LiIr2O6 that has been
prepared with different amounts of stacking faults. Samples with a low
concentration of stacking faults show long-range magnetic order [119, 126,
127], whereas samples with a high concentration of stacking faults develop a
disordered magnetic state resembling a spin liquid [93]. This observation is
in line with the previous work on pure $\alpha$-Li2IrO3 where magnetic
ordering transition is also suppressed when the concentration of stacking
faults increases [43].

Yet another problem, now specific to the Cu substitution, is partial charge
transfer. Experiments on Cu2IrO3 revealed that about 10% of Cu+ is oxidized
into Cu2+ with the respective amount of Ir4+ reduced toward Ir3+ [117]. This
brings one additional dimension, chemical disorder, which becomes even more
prominent in hydrogenated lithium iridate, H3LiIr2O6.

###  III.3 Chemical disorder

Cu2IrO3 features a mixture of four different species: Ir4+, Cu2+, Ir3+, and
Cu+. The first two are magnetic, while the second two are not. It was argued
that Cu2+ resides in the center of Ir hexagons [117] and turns one of the
adjacent Ir atoms into Ir3+, thus diluting the honeycomb spin lattice [118].
Muons detect 60% static and 40% dynamic spins at low temperatures [117]. An
enthusiastic interpretation of this behavior is that Cu2IrO3 without charge
transfer would manifest the Kitaev spin liquid, but Cu2+ ions create impurity
centers where static spins precipitate [117]. On the other hand, in Ag3LiIr2O6
a similar phenomenology is caused by stacking faults alone [93, 119],
suggesting that structural disorder should play a crucial role in Cu2IrO3,
too, and may be the only cause of its purported spin-liquid behavior. This
abundant disorder will certainly complicate an unambiguous interpretation of
continuum-like spectral features reported for Cu2IrO3 recently [128, 129].

Unlike other chemically substituted iridates, H3LiIr2O6 is prepared
hydrothermally by treating $\alpha$-Li2IrO3 in sulfuric acid [130], and single
crystals that even exceed $\alpha$-Li2IrO3 in size can be obtained [131]. The
compound lacks magnetic order, reveals persistent spin dynamics down to at
least 50 mK, and overall resembles a spin liquid [130]. On the other hand, not
only are stacking faults abundant [109], but also hydrogen atoms can take
different positions between the Ir layers [132].

In contrast to monovalent metals, H+ does not stay at the midpoint between the
two layers and forms a short covalent bond to oxygen instead (Fig. 6). This
creates local O–H dipoles probed by dielectric measurements [133, 134]. At
high temperatures, the H+ ions tunnel between two equivalent sites in a
paraelectric state. Below 100 K, the slowing down of this proton dynamics
[134] leads to [LiIr2O6] honeycomb layers randomly decorated by hydrogen atoms
in an effectively static configuration. Ab initio calculations pinpoint strong
changes in magnetic interactions depending on the positions of hydrogen and
their bonding to oxygen [132, 135], an effect also well-known for $3d$
transition metals [136].

Random positions of hydrogen should thus lead to an extreme randomness of
magnetic interactions in H3LiIr2O6. This effect alone is sufficient to break
down long-range magnetic order and induce a magnetically disordered state not
necessarily related to the Kitaev spin liquid. Abundant stacking faults [109]
will further facilitate this type of behavior. A prominent feature of
H3LiIr2O6 is a pileup of low-energy states probed by specific heat
measurements and showing a strong dependence on the magnetic field. These
states can be traced back to Majorana fermions in a bond-disordered Kitaev
spin liquid [137], although such phenomenology is by no means specific to this
scenario [138]. It can be also accounted for by a random-singlet state arising
from quenched disorder even in the absence of Kitaev interactions [139].
Thermodynamic properties of the recently prepared Sr0.5IrO3 are well captured
by this model [140].

It is also instructive to review several studies that introduced disorder
deliberately. Both magnetic Ru4+ [141] and non-magnetic Ti4+ [142] doping lead
to a rapid suppression of the long-range order and formation of a glassy
state. These observations further illustrate that even minor structural
disorder may result in the absence of a magnetic transition. For honeycomb
Na2(Ir1-xTix)O3, the spin-glass freezing temperature as well as the Curie-
Weiss temperature extrapolate to zero at the site-percolation threshold of 30%
[142], which is indeed expected in a honeycomb magnet with predominant
nearest-neighbor couplings [143] and justifies the application of Eq. (2) in
the microscopic analysis. On the other hand, for $\alpha$-Li2(Ir1-xTix)O3 the
Curie-Weiss temperature as well as the freezing temperature clearly extend
beyond this 30% threshold up to at least $x\simeq 0.5$, thus giving evidence
for interactions beyond nearest neighbors, which are indeed discussed
theoretically for this compound [51].

##  IV Conclusions and outlook

As the first decade of Li2IrO3 research comes to an end, it is time to
summarize several important insights and lessons gained from this material, as
well as outline possible directions for future studies:

First, structural diversity of Li2IrO3 polymorphs has been the main trigger
for exploring Kitaev model beyond planar honeycomb geometry. 3D versions of
this model received broad attention only after they materialized in $\beta$
and $\gamma$-polymorphs of Li2IrO3. One exciting prediction of subsequent
theoretical studies is the thermodynamic phase transition separating Kitaev
spin liquid on the hyperhoneycomb lattice from the paramagnetic phase [26,
27]. Experimental verification of this prediction pends a 3D hyperhoneycomb
material, which is suitably tuned to the Kitaev limit.

Second, Li2IrO3 polymorphs uncovered highly non-trivial magnetically ordered
states with counter-rotating spin spirals. This incommensurate order is driven
by anisotropic exchange terms and especially by the off-diagonal anisotropy
$\Gamma$ that was previously not on radar. First experimental studies reveal
magnons as excitations of these unusual incommensurate states at low energies,
but also detect continuum-like features at higher energies where
fractionalizion or magnon breakdown probably become relevant (Sec. II.5).
Detailed study of spectral properties for different Li2IrO3 polymorphs is
clearly warranted, also from the perspective of topological magnons that are
already predicted [144] as excitations of commensurate states obtained by
quenching counter-rotating spirals in the applied field.

Third, both 3D polymorphs of Li2IrO3 revealed a peculiar response to the
applied field. They deviate from the intuitive behavior of Heisenberg
antiferromagnets, which remain long-range-ordered in a spin-flop phase until
fully saturated. In Kitaev magnets, exchange anisotropy and especially the
off-diagonal $\Gamma$ term can lead to a quick suppression of magnetic long-
range order in external field well before saturation is reached. This may
create an impression that the material turns into a spin-liquid state, but
experiments on $\beta$-Li2IrO3 clearly challenge such an interpretation. The
field-induced state of this compound is a quantum paramagnet that does not
break any symmetries and does not undergo any phase transitions as a function
of temperature, but the underlying spin-spin correlations resemble
conventional long-range order. This type of behavior can be easily confused
with a field-induced spin-liquid state and requires caution in analyzing basic
experimental signatures, such as presence or absence of thermodynamic phase
transitions, in anisotropic magnets.

Fourth, layered crystal structure of $\alpha$-Li2O3 has turned into a toolbox
of new materials prepared by chemical substitution. It is probably not
coincidental that these substitutions are easiest to perform in the material
most prone to stacking faults and other types of disorder. Defects of the
$\alpha$-Li2IrO3 precursor seem to be preserved and even enhanced upon
chemical substitution. Several new Kitaev materials with the disordered
magnetic ground state have been discovered, but randomness caused by
structural disorder plays a crucial role in all of them. This structural
disorder makes connections to the Kitaev spin liquid elusive even if these
connections exist.

Ionic substitutions in 3D Li2IrO3 polymorphs could be a viable strategy,
because those polymorphs are structurally well ordered and naturally immune to
the stacking faults. Nevertheless, other types of defects should be carefully
watched out. Many recent claims of the spin-liquid behavior are related to
materials with structural imperfections of some sort, and instances of spin-
liquid mimicry are not uncommon [145]. Spin dynamics driven by randomness can
be interesting in its own right [82], but manifold nature of defects in
substituted iridates (stacking faults, Cu2+ impurities, random positions of
hydrogen) requires a suitable gauge to quantify the amount of disorder and its
influence on exchange couplings. Strain tuning in epitaxial thin films and
uniaxially compressed bulk crystals could be an alternative that eliminates
the disorder or at least makes it easier to control. This interesting
direction awaits further experimental development.

Li2IrO3 has been a challenging material for magnetism studies. Its central
aspects, including magnetic order and field-induced transitions, could never
be resolved using polycrystalline samples. Despite long and arduous efforts,
typical crystal size remains well below 1 mm for any of the polymorphs, but
even these tiny crystals offered a comprehensive insight and firmly
established the role of Kitaev interactions in this material. Several state-
of-the-art techniques – cantilever magnetometry, ac-calorimetry, resonant
x-ray scattering (elastic and inelastic) – have been instrumental in studying
sub-mm size Li2IrO3 crystals. Further development of these techniques, along
with continuous efforts in synthesis and crystal growth, will open new
opportunities in understanding Li2IrO3 and its peculiar physics.

Acknowledgments

Fruitful research on Li2IrO3 and its derivatives would not be possible without
continuous contributions and support from many of our colleagues, in
particular Anton Jesche, Yogesh Singh, Friedrich Freund, Soham Manni,
Alexander Zubtsovskii, Ina Pietsch, Mayukh Majumder, Rudra Manna, Radu Coldea,
Stephen Winter, Roser Valentí, and Ioannis Rousochatzakis. We acknowledge
financial support by the German Research Foundation (DFG) via the Project No.
107745057 (TRR80). AAT was also supported by the Federal Ministry for
Education and Research through the Sofja Kovalevskaya Award of Alexander von
Humboldt Foundation.

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