characterised by two counter-rotating spin sublattices. Locally, these states
resemble the ‘period-3 chain’ structure discussed above, which is an
indication for sizeable off-diagonal $\Gamma$ interactions.

The origin of this universality across the three Li2IrO3 polytopes has been
the subject of several works [101, 238, 159, 45, 239], see also Ref. [100].
One key ingredient seems to be the common local geometry of the bonds, which
suggests that the minimal microscopic description is similar in these
compounds. Indeed, the orders observed in $\beta$-Li2IrO3 and $\gamma$-Li2IrO3
have been rationalized [238, 159, 45] within the framework of the minimal NN
$J$-$K$-$\Gamma$ model, with large FM Kitaev interaction, sizeabe $\Gamma$
interactions (with $|\Gamma|\\!<\\!|K|$) and much weaker AFM Heisenberg
exchange $J\\!\ll|\Gamma|$.

Another key ingredient behind this universality seems to be the physics along
individual $xy$ chains and how the corresponding modulated orders on these
chains tile in the whole 3D netowork [101, 45]. The work of Ref. [239] has
revealed a more general mapping that connects most of the ordered states of
the $JK\Gamma$ phase diagram (and not just the ones relevant to
$\beta$-Li2IrO3) to their 2D counterparts on the honeycomb lattice, with the
classical energetics being identical in 2D and 3D.

###  VI.2 The paradigm of $\beta$-Li2IrO3

In the following we focus on the paradigm of $\beta$-Li2IrO3 [31, 34, 94, 115,
99, 240], which stands out among the three polytopes for its quantitative
level of description [45, 46, 48, 50, 240].

Model. Here, the $JK\Gamma$ model takes the general form of Eqs. (11) and
(12), where, as mentioned above, the bond index $\gamma$ runs over five types
of bonds, $x$, $y$, $x^{\prime}$, $y^{\prime}$ and $z$ (see also Fig. 8), with

$K_{\gamma}=K,~{}~{}~{}\Gamma_{\gamma}=\sigma_{\gamma}\Gamma,~{}~{}~{}\Gamma^{\prime}_{\gamma}=0,$
(75)

and $\sigma_{\gamma}\\!=\\!\pm 1$ incorporates the sign structure of the
$\Gamma$ couplings, as discussed in App. .1. For the cubic axes frame of Eq.
(86) taken in Ref. [45], or the equivalent frame of Refs. [238, 159],
$\sigma_{z/x/y^{\prime}}\\!=\\!1$ and $\sigma_{x^{\prime}/y}=-1$, see Fig. 8.

Figure 9: The shaded region shows the portion of the parameter space
$(\theta,\phi)$ [see Eqs. (75) and (76)] that is believed [238, 159, 45] to be
relevant for $\beta$-Li2IrO3. The ‘$K$-region’ and ‘$\Gamma$-region’ host
different types of classical states (with varying wavelengths), and their
boundary (red line) emerges from the hidden, isotropic SO(3) point (red star)
[45]. Quantitative estimates of $J$, $K$ and $\Gamma$ (see Table 1) place
$\beta$-Li2IrO3 at the yellow point shown, very close to the special
frustrated line $\phi=3\pi/2$. For the phases outside the shaded region
(including the white region shown), see detailed analysis in Refs. [238, 159].
Adapted from Ref. [45].

Hidden SU(2) points. In analogy to the 2D honeycomb case, the above model has
hidden SU(2) symmetry points at $K\\!=\\!\Gamma$ and $J\\!=\\!0$, as these
points can be mapped to dual points with
$K^{\prime}\\!=\\!\Gamma^{\prime}\\!=\\!0$ and $J^{\prime}\\!=\\!-K$ [45,
241]. For $K\\!=\\!\Gamma\\!<\\!0$ and $J\\!=\\!0$ the system then maps to an
antifferomagnetic Heisenberg model, whereas for $K\\!=\\!\Gamma\\!>\\!0$ and
$J\\!=\\!0$, the system maps to a FM Heisenberg model. The corresponding
duality transformation is similar to the transformation $\mathcal{T}_{6}$ of
Sec. IV.2.1, and involves six sublattices on isolated $xy$ chains, and 12
sublattices for the hyperhoneycomb lattice (24 for the stripyhoneycomb). The
hidden SU(2) point that is most relevant for $\beta$-Li2IrO3, where $K$ and
$\Gamma$ are both negative, is shown in Fig. 9, where we use the
parametrization

$J=\sin\theta\cos\phi,~{}~{}K=\sin\theta\sin\phi,~{}~{}\Gamma=-\cos\theta.$
(76)

Here, $\phi\\!\in\\![0,2\pi)$ and $\theta\\!\in\\![0,\frac{\pi}{2}]$, and the
minus sign in the last term accounts for the negative sign of $\Gamma$ in
$\beta$-Li2IrO3.

Relevant region of the phase diagram. Remarkably, all experimental data
reported so far [31, 34, 94, 115, 99, 240] are consistent with parameters
close to $J\\!=\\!0.4~{}\text{meV}$, $K\\!=\\!-18~{}\text{meV}$ and
$\Gamma\\!=\\!-10~{}\text{meV}$ [45, 46, 48, 50, 240]. These parameters place
$\beta$-Li2IrO3 somewhere inside the so-called $K$-region of Fig. 9 [45]. This
material is therefore extremely close to the special $K$-$\Gamma$ line, which
hosts a two-parameter family of degenerate classical ground states, similar to
‘period-3 chain’ family of states with counter-rotating spin sublattices,
discussed in Sec. V.3.1. The weak and positive Heisenberg coupling $J$ lifts
this degeneracy and selects configurations with the 120∘ structure, in analogy
with the selection of the 120∘ state of Fig. 6 by a negative $J$ in the 2D
honeycomb case. With increasing $J$, this configuration gets distorted,
leading to an IC state with propagation wavevector ${\bf Q}\\!=\\!(0.574,0,0)$
in orthorhombic units [31, 34] at zero-field and below $T_{I}\\!=\\!38$ K.

The IC order of $\beta$-Li2IrO3 can be treated as a long-distance twisting of
the closest commensurate period-3 state, with ${\bf Q}=\frac{2}{3}\hat{\bf
a}$, in orthorhombic units of $\frac{2\pi}{a}$ [45]. This ansatz is amenable
to a semi-analytical treatment, and delivers results that are consistent with
almost all experimental findings so far [31, 34, 94, 115, 99, 240], both in
zero field and at finite fields. The quantitative success of this ansatz is
another evidence of the close proximity of the IC order to the line
$\phi\\!=\\!3\pi/2$.

Physics under a magnetic field. The application of a magnetic field along
${\bf b}$ axis (see Fig.8) suppresses the zero-field order with a critical
field of $H_{\bf b}=2.8$ T [94]. This rapid decline of the IC order is a
signature of the smallness of $J$, since in the ${\bf b}$-direction the
critical field only depends on $J$ [46]. For fields along ${\bf a}$ and ${\bf
c}$, the system shows a much weaker response, with the IC order remaining
robust and the magnetization being linear up to the maximum fields measured
[94, 96]. Such a strongly anisotropic response signifies a large separation of
energy scales between $J$ and $\Gamma$. This large separation also manifests
in the structure of the magnetic excitations measured by Resonant Inelastic
X-ray Scattering (RIXS) [115], inelastic neutron scattering (INS) and time-
domain THz spectroscopy [99], and in the magnetic Raman response [240].

##  VII Quasi-1D $K$-$\Gamma$ models

The challenges associated with the 2D $K$-$\Gamma$ Honeycomb model (see Sec.
V.3) have motivated a number of studies which approach the 2D lattice as a
collection of 1D chains. This idea has worked quite well for the Heisenberg-
Kitaev model, see e.g., Ref. [242]. This has lead to several quasi-1D versions
of the model which are numerically and analytically more accessible than the
2D model and, as it turns out, have very rich physics on their own.

###  VII.1 Kitaev-$\Gamma$ chain

We begin with the geometry of an isolated $xy$ chain (the case of $yz$ and
$zx$ chains can be defined analogously), depicted as

(77)

where, for later purposes, we have also labeled the sites according to the
six-sublattice decomposition of (27) along $xy$ chains.

####  VII.1.1 Symmetries and dualities

Before we discuss the phase diagram of the 1D $K$-$\Gamma$ chain model, it is
worth emphasizing that the symmetries and dualities of the 2D honeycomb model
that we examined in Sec. IV (and in particular, the ones involving the
$K$-$\Gamma$ model) carry over to the 1D chain as well. To see this, one
simply needs to isolate an $xy$ chain from the honeycomb lattice, and, for
each given transformation of Sec. IV (symmetry or duality), remove any parts
of the operation that pertain to sites not belonging to the given chain. For
example, the local symmetries $W_{h}$ of Eq. (18) reduce to three-body
operations involving three consecutive sites of the chain. More explicitly,
for the first three consecutive sites of (77), labeled by 2, 1 and 6, the
local symmetry is

$\mathsf{C}_{2x}(2)\mathsf{C}_{2z}(1)\mathsf{C}_{2y}(6)\,.$ (78)

Likewise, the four-sublattice transformation $\mathcal{T}_{4}^{\prime}$
discussed in Sec. IV.4.1 for the 2D model, which maps the FM to the AFM Kitaev
point, is still present in the chain model, and its form is that of Eq. (44)
if we keep the 4-sublattice labelling of (31) for the chain. Finally, the
points $K\\!=\\!\Gamma$ are hidden SU(2) points with $\widetilde{J}\\!=\\!-K$,
as in the 2D model. The associated transformation $\mathcal{T}_{6}$ has again
the form of Eq. (28) when using the labelling of the six-site decomposition of
(77).

Figure 10: Zero-temperature quantum phase diagram of the $K$-$\Gamma$ model on
(a) spin-$1/2$ chain [211, 243, 212], (b) spin-1 chain [244], and (c) two-leg
spin-1/2 ladder [213], with the parametrization $K=\cos\phi$ and
$\Gamma=\sin\phi$. Blue stars show the positions of the hidden SU(2) symmetry
points with $K\\!=\\!\Gamma$, whereas red stars in (a-b) show the positions of
the hidden SU(2) points with $K\\!=\\!-\Gamma$, see text.

Altogether then, the dualities and symmetries of the planar 2D model carry
over to the 1D chain model in a straightforward way. What is not immediately
evident, however, is that the chain model has, in fact, some additional
symmetries and dualities compared to the 2D model. We will discuss the three
most notable examples:

i) At the pure Kitaev point, the chain has local symmetries defined on bonds
[245]. These correspond to $\pi$-rotations (of the two spins of the bond)
around the ${\bf x}$-axis (${\bf y}$-axis) for a $y$-type (respectively,
$x$-type) of bond. For the bonds (1,2) and (1, 6) of (77), for example, the
corresponding local symmetries take the form

$\mathsf{C}_{2x}(2)\mathsf{C}_{2x}(1)~{}~{}\text{and}~{}~{}\mathsf{C}_{2y}(1)\mathsf{C}_{2y}(6)\,,$
(79)

respectively. We can now recognize that the local 3-site symmetries of Eq.
(78) are the product of the bond-symmetries of (79). As shown by Sen et al.
[245], the presence of these bond symmetries gives rise to a qualitative
difference between integer and half-odd-integer values of $S$, which stems
from the fact that for integer $S$, all the bond symmetries commute with each
other, but for half-odd-integer $S$ this is not true any longer. One of the
immediate consequences is that all the eigenstates of the chain with half-odd-
integer $S$ are $2^{N/2}$-fold degenerate.

ii) Compared to the 2D model, the chain model has an additional pair of hidden
SU(2) points, at $K\\!=\\!-\Gamma$ (with $\widetilde{J}\\!=\\!-K$) [45, 211].
For the site-labelling of (77), the associated transformation
$\mathcal{T}_{6}^{\prime}$ reads:

$\begin{array}[]{c}\text{6-sublattice}\\\ \text{transf.}~{}\mathcal{T}_{6}^{\prime}\end{array}:~{}~{}\begin{array}[]{c | rrr}\text{label}~{}j&S_{j}^{x}&S_{j}^{y}&S_{j}^{z}\\\ \hline\cr 1&\widetilde{S}_{1}^{x}&\widetilde{S}_{1}^{y}&\widetilde{S}_{1}^{z}\\\ 2&\widetilde{S}_{2}^{z}&-\widetilde{S}_{2}^{y}&\widetilde{S}_{2}^{x}\\\ 3&\widetilde{S}_{3}^{y}&-\widetilde{S}_{3}^{z}&-\widetilde{S}_{3}^{x}\\\ 4&-\widetilde{S}_{4}^{y}&-\widetilde{S}_{4}^{x}&-\widetilde{S}_{4}^{z}\\\ 5&-\widetilde{S}_{5}^{z}&\widetilde{S}_{5}^{x}&-\widetilde{S}_{5}^{y}\\\ 6&-\widetilde{S}_{6}^{x}&\widetilde{S}_{6}^{z}&\widetilde{S}_{6}^{y}\\\ \hline\cr\end{array}$ (80)

Indeed, taking, e.g., the $y$-bond $(1,2)$ of (77) and using (80) with
$J\\!=\\!\Gamma^{\prime}\\!=\\!0$ and $K\\!=\\!-\Gamma$ gives

$KS_{1}^{y}S_{2}^{y}+\Gamma(S_{1}^{x}S_{2}^{z}+S_{1}^{z}S_{2}^{x})\to-K\widetilde{{\bf
S}}_{1}\cdot\widetilde{{\bf S}}_{2}\,,$ (81)

which is again a Heisenberg coupling with $\widetilde{J}\\!=\\!-K$. Altogether
then, it follows that when $K$ is positive, the points $K\\!=\\!\Gamma$ and
$K\\!=\\!-\Gamma$ both map to a FM Heisenberg model, and must therefore show
FM ordering at low $T$. By contrast, when $K$ is negative, the points
$K\\!=\\!\Gamma$ and $K\\!=\\!-\Gamma$ both map to the AFM Heisenberg model,
and we therefore get a gapless critical phase [246].

iii) The chain model has an additional duality that maps
$(K,\Gamma)\to(K,-\Gamma)$. The associated operation is the global
$\pi$-rotation around the $z$ axis in spin space alone. This operation maps
$(S_{i}^{x},S_{i}^{y},S_{i}^{z})\\!\mapsto\\!(-S_{i}^{x},-S_{i}^{y},S_{i}^{z})$
for all sites $i$, which effectively reverses the sign of $\Gamma$. Thus, the
model with $K$ and $\Gamma$ of opposite signs can be mapped to the model with
the same signs. This duality, which can also result from the three-sublattice
transformation $\mathcal{T}_{3}$ of Ref. [211], is not present in the 2D
model.

####  VII.1.2 Phase diagram for $S=1/2$

We begin with the physics of the pure Kitaev points, which goes back to the
study of Ref. [151]. As for the 2D and 3D cases, these points can be mapped to
free fermion models via a Jordan-Wigner transformation [151], and it has been
shown that the ground states are critical (gapless) phases. In the anisotropic
version of the model with different couplings on the two types of bonds (i.e.,
with $K_{x}\\!\neq\\!K_{y}$), a nonzero gap opens up as soon as we depart from
the special point $K_{x}\\!=\\!K_{y}$. The transition between the two gapped
phases on the two sides of this critical point does not involve any change of
symmetry, but is an example of a change from one topological order to another,
each associated with a separate hidden nonlocal order parameter [151].

When the $\Gamma$ coupling is introduced, the Jordan-Wigner transformation is
no longer useful, but the model has been studied using non-Abelian
bosonization and DMRG techniques [211, 243, 212]. The resulting phase diagram
is shown in Fig. 10 (a), with the parametrization $K\\!=\\!\cos\phi$ and
$\Gamma\\!=\\!\sin\phi$. Note that due to the duality which maps
$(K,\Gamma)\mapsto(K,-\Gamma)$, the points $\phi$ and $2\pi-\phi$ are dual to
each other. In total, there are seven extended phases:

i) There are two (dual) FM orders, denoted by FM and FM’ in Fig. 10 (a),
surrounding the two hidden SU(2) points $K\\!=\\!\pm\Gamma$ with positive $K$.
These orders are gapped everywhere except at the hidden points themselves,
where the ordering breaks a continuous symmetry, leading to Goldstone modes.
The ordered phase is characterized by the O${}_{h}\rightarrow$ D4 symmetry
breaking, where Oh is the full octahedral group and Dn represents the dihedral
group of order 2$n$. This phase has a six-fold degenerate ground state, with
moments along the $\pm\hat{x}$, $\pm\hat{y}$, and $\pm\hat{z}$ [211].

ii) The phases surrounding the two hidden AFM SU(2) points
($K\\!=\\!\pm\Gamma$, with negative $K$), remain gapless in extended regions
of parameter space [denoted by AFM and AFM’ in Fig. 10 (a)] and feature an
emergent SU(2)1 symmetry. The pure $\Gamma$ points are special members of
these extended (dual) phases, and can be described as Luttinger liquids with
central charge 1.

iii) While the FM and AFM mapping is valid in the Kitaev-$\Gamma$ chain, their
stability under a finite $\Gamma$ term is very different. As shown in the
phase diagram, the FM Kitaev point is immediately unstable and becomes a
critical point as soon as $\Gamma$ is introduced. On the other hand, the AFM
Kitaev phase (C) remains stable for weak enough $\Gamma$.

iv) Finally, there are two (dual) magnetic orders, denoted by SN in Fig. 10
(a), which are sandwiched between the AFM Kitaev phase (C) and the FM (FM’)
phases. These orders are characterized by two types of modulations, with
2-site and 6-site periodicities [243]. A finite spin nematicity (SN) on a NN
bond is another signature that differentiates these phases from their
surrounding AFM Kitaev and FM orders.

####  VII.1.3 Integer spins: Spin-1 $K$-$\Gamma$ model

Quasi-1D models of higher spin have been also studied [245, 244]. While the
spin-1 Kitaev chain is not exactly solvable, numerical works show that the
ground state is translationally invariant and lives in the ‘flux-free’ sector,
where all mutually-commuting bond symmetries of Eq. (79) equal to $+1$ [245,
244]. Compared to the ground state, the first excited states feature one bond
with flux $-1$, leading to an $N$-fold degeneracy, where $N$ is the number of
bonds [244]. Due to the unusual excitations, the Kitaev phase displays a
nontrivial double-peak structure in the specific heat. In contrast to the
spin-$1/2$ Kitaev honeycomb model, whose low-temperature peak comes from the
freezing of vison fluxes, the low-temperature peak in the spin-1 Kitaev chain
is related to the freezing of $\mathbb{Z}_{2}$ degrees of freedom and is
relevant to the highly degenerate low-lying excited states. Moreover, the
thermal entropy is released gradually without generating a plateau in the
crossover temperature region, which is again different from the Kitaev
honeycomb model.

The introduction of the $\Gamma$ coupling leads to a rich phase diagram which
contains six phases, see Fig. 10 (b). These phases include: i) a gapped FM
Kitaev (FK) phase and a gapped AFM Kitaev phase (AK), ii) two FM phases (FM
and FM’) surrounding the two hidden FM SU(2) points, and iii) two Haldane
phases surrounding the two hidden AFM SU(2) points, which are gapped, unlike
the spin-1/2 case.

###  VII.2 Spin-1/2 Kitaev-$\Gamma$ ladder

While the Kitaev-$\Gamma$ chain highlights the interplay of these two
interactions leading to both ordered and disordered phases, the link to the
physics of the 2D Kitaev-$\Gamma$ model is weak. The difference between the
Kitaev-$\Gamma$ chain behavior and numerical results obtained for the 2D
Kitaev-$\Gamma$ model is obviously from the missing $z$-bond, and one may
wonder if the minimal geometry such as that of the 2-leg Kitaev-$\Gamma$
ladder, depicted as

(82)

may capture the phases found in the 2D model. This geometry has been the
subject of several studies with Kitaev-like models [151, 152, 242, 43, 247,
248, 213, 249, 250].

####  VII.2.1 Symmetries and dualities

At the pure Kitaev point, the local plaquette symmetries of the 2D model [see
Eq. (18) for spin-1/2 or its extension in Eq. (19) for higher spins] are still
present in the 2-leg ladder. Furthermore, the hidden SU(2) points at
$K\\!=\\!\Gamma$ remain present in the parameter space of the ladder, see blue
stars in the phase diagram presented in Fig. 10 (c). Contrary to the chain,
however, the $T_{6}^{\prime}$ mapping of Eq. (80) is lost in the ladder
geometry due to the presence of the $z$-bonds that link the two chains of the
ladder. As a result, the points $K\\!=\\!-\Gamma$ are not hidden SU(2) points.
The same situation is encountered in the $K$-$\Gamma$ honeycomb model.
Similarly, the duality that maps $(K,\Gamma)$ to $(K,-\Gamma)$ in the chain
model is not present in the 2-leg ladder. Therefore the points $\phi$ and
$2\pi-\phi$ are not dual.

####  VII.2.2 Phase diagram

The phase diagram of the spin-1/2 $K$-$\Gamma$ ladder is shown in Fig. 10 (c),
as obtained by numerical studies [43, 213]. In total, there are seven extended
phases, with the following main features:

i) The pure Kitaev ladder at $\phi\\!=\\!0$ and $\pi$ is exactly solvable, and
at both points it is in a disordered gapped phase [151], featuring a long-
range nonlocal string order parameter [242]. The Kitaev phases survive for
small enough $\Gamma$, leading to the extended phases denoted by AK (AFM
Kitaev) near $K=1$ and $\Gamma=0$ and FK (FM Kiteav) near $K=-1$ and
$\Gamma=0$ in Fig. 10 (c).

ii) The hidden SU(2) point at $\phi\\!=\\!\pi/4$ maps to the spin-1/2
Heisenberg ladder with $\widetilde{J}\\!=\\!-K\\!<\\!0$, which explains the FM
order surrounding this point. Likewise, the second hidden SU(2) point at
$\phi=5\pi/4$ maps to the AFM spin-1/2 Heisenberg ladder, which gives rise to
a rung singlet (RS) phase surrounding this point. This phase is gapped with
short-range correlations [251, 252]. Note that, in the 2D limit, the FM and RS
phases will eventually give way to the FM and Néel phases, respectively, whose
stability regions are outside the parameter space of Fig. 5.

iii) Near $\Gamma\\!=\\!1$ and $K\\!=\\!0$, there is a disordered phase
denoted by A$\Gamma$ (AFM $\Gamma$) in Fig. 10 (c). This phase shows
similarities to the nematic paramagnet that has been predicted by iDMRG [170]
for the 2D model, see Fig. 5 (e).

iv) There is a spin-nematic phase denoted by SN in Fig. 10 (c), which is
sandwiched between the RS and the AFM Kitaev phase. This phases features local
quadrupolar order parameters and is likely gapless [213].

v) Finally, there is a narrow phase, denoted by X in Fig. 10 (c), which is
sandwiched between the FM and A$\Gamma$ phases. A recent DMRG study has
reported two symmetry-protected topological (SPT) phases inside X [253].

###  VII.3 Other developments and phenomena in 1D

Given the numerical accessibility of 1D models, a variety of modified 1D
models have been explored. Examples include an extension of the pure Kitaev
chain to bond-alternating Kitaev-$\Gamma$ chain for $S=1/2$ and $S=1$ [212,
244]. Similar to the spin-1/2 and spin-1 chains, these studies show clear
differences in the ground states between half-integer and integer spin chain
systems. Another line of investigations concerns the response of these 1D
models to various perturbations, such as external magnetic fields [254, 255,
256, 257, 213, 258]. In the following, we briefly review two notable examples.

####  VII.3.1 Field-induced chiral solitons in Kitaev chains

The influence of magnetic fields on the spin-1/2 Kitaev chain has been
explored in several works [254, 255, 256, 257, 258]. Here we shall briefly
expand on the field-induced chiral soliton phase reported in Ref. [258].
Starting from the Kitaev limit at zero field, a phase with finite vector spin
chirality can be induced by an in-plane magnetic field along the [110]
direction. For $S\\!=\\!1/2$, the ground state evolves continuously and
remains gapless up to a critical field $h_{c_{1}}$, but enters into the
polarized phase only above a critical field $h_{c_{2}}$, leaving an
intermediate phase in between. This phase is characterised by a nonzero vector
chirality of the staggered form $(-1)^{j}\langle\left({\bf S}_{j}\times{\bf
S}_{j+1}\right)^{z}\rangle$, similar to the phase found in the context of the
anisotropic $J_{1}$-$J_{2}$ model with $J_{1}\\!<\\!0$ and $J_{2}\\!>\\!0$
[259, 260, 261] and observed recently in the spin-$1/2$ chain compound LiCuVO4
[262]. For periodic boundary conditions, this phase features a finite
excitation gap above the two-fold degenerate ground state. For open boundary
conditions, the ground state exhibits a single soliton, lowering the energy,
and in-gap excitations. Such a soliton can also be revealed by a twist in the
very middle of the magnetization [258].

It is noteworthy that the chiral soliton phase can also appear in the integer-
spin Kitaev chains [263]. The ground states of the integer-spin Kitaev chains,
at least for $S\leq 4$, are gapped. By adding an in-plane magnetic field,
these excitation gaps vanish at certain magnetic fields that are smaller than
$\sqrt{2}SK$. Hence, the chiral soliton phase is stabilized in these regions
around $h=\sqrt{2}SK$. Finally, we mention that although the chiral soliton
phase is easily accessible when the magnetic field is along the [110]
direction, it can also exist if the in-plane azimuth angle is slightly away
from $45^{\circ}$, and/or the field slightly deviate from the $xy$ plane
[263].

Figure 11: Field-induced phase diagram of the spin-1/2 two-leg $K\Gamma$
ladder versus ${\bf h}\parallel[111]$ (where $h=H/(g\mu_{B})$ and $H$ is the
applied field) and $\phi/\pi$, where $K\\!=\\!\cos\phi$ and
$\Gamma\\!=\\!\sin\phi$, as obtained from iDMRG calculations [213]. The
colour-coded intensity corresponds to the bipartite von Neumann entanglement
entropy $S_{\text{rung}}$ on the rung. Adapted from Ref. [213].

####  VII.3.2 Field-induced scalar chiral orders in $K\Gamma$ ladders

The entire phase diagram of the spin-1/2 two-leg $K\Gamma$ ladder in the [111]
magnetic field has been mapped out by the iDMRG and DMRG methods [213],
motivated by the rich magnetic field effects in Kitaev materials. The system
features a multitude of distinct orders with multicritical points,
incommensurate phases, as well as disordered states characterized by various
scalar chiralities, see Fig. 11 and Figs. 4 and 5 of Ref. [213]. Throughout
the phase diagram, the most exciting parts are the dominant positive $\Gamma$
regime (A$\Gamma$) perturbed by FM Kitaev, and the dominant AFM Kitaev (AK)
regime perturbed by a negative $\Gamma$ interaction. The competition between
FK and A$\Gamma$ under the [111] magnetic field leads to FK phase sitting
above A$\Gamma$ phase leading to a field induced FK phase. However, such
field-induced FK phase region shrinks as the number of legs grows [170]. While
A$\Gamma$ is robust under the [111] field, it is replaced by the ZZ order when
a small negative $\Gamma^{\prime}$ is introduced, which has been also found in
the honeycomb model [51].

Among the phases identified in Ref. [213], two scalar chirality orders are
intriguing, as they may be connected to the magnetic vortex liquid-like phases
found in Ref. [264] around the similar phase space in 2D classical
Kitaev-$\Gamma$ model. This is near the AFM Kitaev regime, where the gapless
quantum spin liquid under the magnetic field was also suggested [186]. By
increasing the magnetic field, there appear two scalar chirality orders. The
scalar spin chirality is defined on triangular plaquettes in anticlockwise
direction, as shown in Fig. 12. More explicitly, for the lower and upper
triangles involving the rungs $r$ and $r+1$, the corresponding chiralities
$\kappa_{1}(r)$ and $\kappa_{2}(r)$ take the form

$\begin{array}[]{l}\kappa_{1}(r)=\bm{\hat{\sigma}}_{r,2}\cdot\big{(}\bm{\hat{\sigma}}_{r,1}\times\bm{\hat{\sigma}}_{r+1,1}\big{)},\\\
\kappa_{2}(r)=\bm{\hat{\sigma}}_{r,2}\cdot\big{(}\bm{\hat{\sigma}}_{r+1,1}\times\bm{\hat{\sigma}}_{r+1,2}\big{)}\,,\end{array}$
(83)

where $\mathbf{\hat{S}}_{r,\ell}\\!=\\!\bm{\hat{\sigma}}_{r,\ell}/2$
corresponds to the spin residing at the $r$-th rung and $\ell$-th leg, and
where $\ell\\!=\\!1$ (2) denoting the lower (upper) leg. To measure the
intensity of the chirality, one introduces the chiral-chiral correlation
function between two triangles separated by distance $d$,
$C_{\kappa_{\upsilon}}(d)=\langle\kappa_{\upsilon}(r)\kappa_{\upsilon}(r+d)\rangle$,
such that $C_{\kappa_{\upsilon}}(d)\\!\to\\!\kappa_{\upsilon}^{2}$ when
$|d|\\!\gg\\!1$. The magnitude of the chirality $\kappa_{\upsilon}$ is
extracted by calculating $C_{\kappa_{\upsilon}}(d)$ in the large distance
limit and its sign is given by the corresponding local expectation value. The
resulting pattern of $\kappa_{\upsilon}$ at two representative points of the
phase diagram, $(\phi,h)=(1.73\pi,0.125)$ and $(1.92\pi,0.42)$, are shown in
Fig. 12. In the two panels, the width of the arrows in each triangle indicates
the magnitude of the chirality, and the color of the triangle represents its
sign. In the upper panel, the scalar chirality is negative in each triangle,
and the magnitude shows period-2 modulation, reflecting the period of the
Hamiltonian. By contrast, in the lower panel, the sign of chirality shows an
alternating staggered pattern but the magnitude is uniform. Since the
summation of the chirality within each unit cell is zero, the net flux in the
ladder is zero, despite the presence of the external field.

Figure 12: Sketch of the scalar chirality in the (a) uniform chirality (UC)
case and the (b) staggered chirality (SC) case. The width of the line
indicates the stength of the scalar chirality, while the color represents the
sign.

In summary, the broader picture emerging from the above body of results on
quasi-1D geometries is that the interplay of Kitaev and $\Gamma$ interactions
on these geometries gives rise to a wide range of new possibilities. Some of
the phases are generic to 1D, like the critical Luttinger-Liquids for
spin-1/2, or the Haldane-like phases for spin-1. Other phases, seem to be the
precursors of particular orders in the 2D model. Examples include the FM and
RS phases of Fig. 10 (c), which give way to the FM and Néel phases of the 2D
model; the disordered A$\Gamma$ phase of Fig. 10 (c) which shows some
similarities to the nematic paramagnet of Fig. 5 (e) [170]; and the two field-
induced scalar chirality orders of Fig. 12 (denoted by UC (uniformed
chirality) and SC (staggered chirality) in Fig. 11), which seem to be
connected to the magnetic vortex liquid-like phases found in Ref. [264] in the
2D model. Finally, there are phases whose connection to the 2D lattice model
is not clear, including the SN phase of Fig. 10 (c), and the two SPT phases
inside the region denoted by X in Fig. 10 (c). Further systematic work on
$n$-leg ladders with $n>2$ can shed more light into this possible
correspondence and eventually to the nature of the enigmatic phases of the 2D
$K\Gamma$ model. In terms of materials, 1D Kitaev systems are still lacking,
apart from the recently suggested twisted Kitaev chain [265]. Developing
material platforms to explore the physics of quasi-1D models is a promising
endeavour.

##  VIII Summary and discussion

The present review aims to complement previous review articles on various
aspects of the broader activity in the field of Kitaev materials [5, 7, 8, 9,
10, 11, 12, 100], and focuses primarily on the theoretical understanding of
how other interactions intertwine with the Kitaev coupling $K$ and what
physics to expect beyond the Kitaev limit.

To expose the relevance of these additional interactions, we begun with a
comprehensive overview of the microscopic exchange mechanisms that are
generically present in strong spin-orbit coupled honeycomb Mott insulators.
This led us to the $J$-$K$-$\Gamma$-$\Gamma^{\prime}$ model as a minimal
description of many Kitaev materials, and the identification of the $\Gamma$
coupling as the main source of additional frustration besides the Kitaev
coupling.

We then set out to explore the various phases in this extended four-
dimensional parameter space. To that end, we provided a detailed discussion of
a number of special symmetries and duality transformations, which play a
crucial role in mapping out the most interesting (and oftentimes hidden)
regions of the parameter space. We have organized our exploration of this
space by reviewing the physics along cuts that connect these special regions.
This led us to a number of ‘simpler’ models that have attracted a lot of
interest in the community, either due to their close relevance to existing
materials or due to their rich physics.

One such region that stands out is the line in the parameter space connecting
a pure Kitaev point to a pure $\Gamma$ point, where one encounters a novel
interplay between the Kitaev quantum spin liquid and the classical $\Gamma$
spin liquid. In particular, we highlighted the region corresponding to
negative $K$ and positive $\Gamma$ as the one that is most relevant to
existing materials, but also the one that is most challenging to tackle from
the theory and numerical side, as this is where the two types of spin liquids
compete the most. We reviewed the current debate on the nature of the ground
state and the phases proposed by various state-of-the-art methods. Altogether,
despite intensive studies, the ground state for $K\\!<\\!0$ and
$\Gamma\\!>\\!0$ is currently unknown and remains a challenging question.
Although several numerical studies have suggested a magnetically disordered
state, most of them are limited by small cluster sizes or wave function
ansätze. Developing new numerical techniques could potentially expand our
current understanding of this region.

We have also collected complementary insights from studies of the classical
$K\Gamma$ model and highlighted the emergence of ‘period-three chain’ orders
with counter-rotating spin sublattices, or long-distance modulations thereof,
as one of the key ramifications of the interplay between $K$ and $\Gamma$.
These orders have been observed in the three members of the Li2IrO3 family of
3D polytopes, for which we have provided a brief overview of the most relevant
results and phenomenology.

Finally, we have briefly reviewed some of the recent exciting results on
quasi-1D versions of Kitaev models, which are motivated by the idea that one
can approach the 2D lattice by bridging together 1D chains or ladders. Part of
the emerging phenomenology is specific to 1D, such as the presence of critical
Luttinger-Liquid phases in spin-1/2 chains, Haldane-like phases in spin-1
chains, or the rung singlet phase in spin-1/2 ladders. Other exciting results
in these quasi-1D models include the stabilization of field-induced chiral
soliton phases in spin-1/2 Kitaev chains and the scalar chiral orders reported
for the spin-1/2 $K$-$\Gamma$ ladder.

The theoretical exploration of the phase diagram of Kitaev materials continues
to be strongly driven by discoveries of new materials and the application of
state-of-the-art experimental techniques that have been designed for detecting
signatures of Kitaev physics. On the material front, $\alpha$-RuCl3 stands
center stage due to the report of the half-integer thermal Hall conductivity
under an in-plane magnetic field [266, 267, 268, 269, 270, 271]. However,
there are debates on whether the measured thermal Hall conductivity can be
attributed to chiral magnons [272, 273] or phonons [274, 275]. The experiments
of the longitudinal thermal conductivity, which have shown oscillations as a
function of in-plane magnetic field have generated new excitement on
$\alpha$-RuCl3 [276]. While these oscillations were initially interpreted as
quantum oscillations of a gapless spin liquid, akin to Landau levels of
gapless metals [276], later experiments attributed the oscillations to
anomalies associated with magnetic transitions [277]. Various numerical
studies have uncovered an intermediate phase between the magnetic ordered
ground state and the polarized paramagnetic state when the field is out of the
honeycomb plane [43, 49]. However, when the field is within the plane, a
direct transition from the magnetically ordered phase to the polarized state
was reported [278, 43]. Further experimental and theoretical studies are
needed to confirm or disprove the existence of a field-induced spin liquid in
$\alpha$-RuCl3.

Beyond $\alpha$-RuCl3 and $d^{5}$ Iridates, there have been intense studies on
$d^{7}$ honeycomb cobaltates [79, 81, 80, 89, 279, 90, 93, 83, 84, 280, 85,
281, 282, 240, 283, 284, 285]. There are also several other candidate
materials proposed, ranging from quasi-1D systems and twisted Kitaev chains
[265] to f-electron honeycomb materials[286, 287, 288]. Further experiments
and microscopic theories on these candidates need to be explored.

In parallel to the rapid developments in bulk materials, the experimental
community has been actively pursuing the development of various two-
dimensional (2D) single layer compounds. Among the Kitaev candidate materials,
in Na2IrO3 and Li2IrO3, the presence of Na+ and Li+ ions in the interlayer
space introduced unwanted interlayer interactions, which could potentially
disturb the desired Kitaev physics in 2D systems. In the case of RuCl3, the
interlayer interactions are relatively smaller compared to honeycomb iridates
due to the absence of charged ions between the layers, but they are still
present and not negligible. Consequently, significant efforts have been made
to control and mitigate these interlayer couplings. One approach to achieve a
2D system involves the exfoliation of 2D honeycomb compounds into monolayers
[289], allowing for better isolation and manipulation of the desired
properties. Among several measurements, Raman spectroscopy performed to study
the evolution of structure with thickness and temperature revealed a symmetry-
forbidden mode appearing in only the thinnest flakes at low temperature [290].
This evidences a possible structural transformation. It is likely that the
exfoliation process or substrates generate significant strain leading to the
distorted structure. To advance our understanding of the effects of interlayer
couplings and various substrates in such van der Waals Kitaev candidates,
further studies the role of the structure and the substrate remain to be
carried out in future.

On the computational side, exploring the ‘physics beyond the Kitaev point’
that we discuss in this review have been a testing ground for a broad range of
state-of-the-art computational techniques, including ab initio methods.
Indeed, the inherent complexity of the materials, the interplay of several
degrees of freedom (including correlations, crystal field, spin-orbit
coupling, and their sensitivity to structural details), as well as the
presence of strong frustration, and the necessity to treat quantum and thermal
fluctuations on equal footing renders the modelling of these materials a
challenging task.

As discussed in the introduction, the determination of the right starting
minimal model requires using ab initio methods in tandem with careful analysis
of the experimental data based on models, which is quite nontrivial given the
large number of microscopic parameters.

Characterizing the physics of simpler generic models (such as the $K\Gamma$
model) plays a crucial role in mapping out distinctive regions of the broader
multi-dimensional phase diagram and building a phenomenology that can then be
contrasted with experiment. To that end, a number of many-body techniques have
been employed over the years. Large-scale exact diagonalizations have the
advantage of exploiting symmetries to analyze the low-energy part of the many-
body spectrum, but the exponential growth of the basis with system size (and
the generic lack of continuous spin rotational symmetry in spin-orbit models)
imposes severe constraints on the size of finite clusters, typically up to 36
sites. The DMRG method is a powerful tool to target the ground state and the
low-lying excited states by mapping a long cylinder into a snake-like chain
with more than two hundred sites. However, the width of the cylinder is
typically less than ten lattice sites, and the convergence is slow when the
ground state is gapless or approaching a quantum critical point. Although the
VMC method can achieve a balance of the lattice sites along two sides, it is
intrinsically a variational iteration over a mean-field Hamiltonian and relies
highly on the proposed trial wave functions. In contrast to these methods
mentioned above, the tensor network approach is claimed to solve the model
with an infinite number of lattice sites via wave function ansatz like
infinite projected entangled pair state. Nevertheless, it may suffer from an
artificial optimization route towards the ground state and a fast-growing
computational cost when considering further-neighbor interactions and/or large
unit cell. Finally, classical minimizations and semiclassical $1/S$ expansions
are often the natural starting point for the description of the ground state
properties and low-energy excitations, as most (if not all) materials exhibit
magnetic ordering at low enough temperatures. However, these approaches may
break down at intermediate and higher energy scales due to strong magnon-
magnon interactions and decay processes, which can be strongly amplified when
these orders are selected by weak perturbations out of a nearby highly
frustrated point. While every numerical technique comes with their own
weakness and limitations, a convincing conclusion can be reached when results
from different methods are contrasted and analyzed.

We hope this review will provide a valuable contribution to the current
literature on Kitaev materials, and set the basic phenomenology and some
useful baselines for researchers working in this flourishing field. We have
also exposed the current challenges in the field but also the rich
opportunities for exploring novel physics in these materials.

###### Acknowledgements.

We would like to acknowledge our theory and experimental colleagues from the
broader field of Kitaev magnetism all over the world, whose work and insights
have been inspirational over the years. In particular, we would like to
acknowledge J. G. Analytis, C. Batista, D. Bogdanov, J. van den Brink, C.
Broholm, B. Büchner, K. S. Burch, A. Catuneanu, G.-W. Chern, A. L. Chernyshev,
M. Daghofer, S. Ducatman, A. Frano, J. Gordon, T. Halloran, L. Hozoi, G.
Jackeli, V. Katukuri, G. Khaliullin, Y. B. Kim, Y. J. Kim, J. Knolle, S.
Kourtis, M. Li, R. Moessner, S. Rachel, J. Rau, J. Reuther, U. K. Rößler, A.
Ruiz, Y. Sizyuk, E. Sørensen, P. P. Stavropoulos, H. Takagi, R. Thomale, A. A.
Tsirlin, M. Vojta, X. Wang, Y. Yang, and J. Zhao. Finally, we also like to
thank our funding agencies: IR acknowledges the support by the Engineering and
Physical Sciences Research Council, Grant No. EP/V038281/1. NBP acknowledges
the support by the U.S. Department of Energy, Office of Science, Basic Energy
Sciences under Award No. DE-SC0018056. NBP also acknowledge the support of the
Alexander von Humboldt Foundation and the hospitality of the Technical
University of Munich – Institute for Advanced Study. Additionally, IR and NBP
acknowledge the support by the National Science Foundation under Grant No. NSF
PHY-1748958 and the hospitality of KITP, UC Santa Barbara, where this work has
been finalized. QL acknowledges the support from the Natural Science
Foundation of Jiangsu Province Grant No. BK20220876 and the National Natural
Science Foundation of China Grant No. 12247183. HYK acknowledges the support
from the Natural Sciences and Engineering Research Council of Canada Discovery
Grant No. 2022-04601, the Canadian Institute for Advanced Research, and the
Canada Research Chairs Program. HYK also acknowledges the Aspen Center for
Physics, supported by National Science Foundation grant PHY-2210452, where
part of this work was carried out.

## Appendix

###  .1 Sign structure of off-diagonal couplings

Here we return to the issue raised at the end of Sec. III.2.3 regarding the
sign structures of $\Gamma$ and $\Gamma^{\prime}$ interactions and their
dependence on the convention used for the directions of the cubic axes $x$,
$y$ and $z$. We know that these axes are perpendicular to the corresponding
type of bonds, however their orientation with respect to the crystallographic
axes is defined only up to an overall $\pm 1$ sign. In Fig. 2, for example,
once we have identified the $x$ and $y$ type of bonds with respect to the
crystallographic axes $(a,b,c^{\ast})$ of the monoclinic cell, we can choose
$(\tilde{x},\tilde{y},\tilde{z})\\!=\\!(\epsilon_{x}x,\epsilon_{y}y,\epsilon_{x}\epsilon_{y}z)$,
where $\epsilon_{x,y}\\!=\\!\pm 1$ are arbitrary, as the cubic frame instead
of the $(xyz)$ frame shown.

While the choice of $\epsilon_{x}$ and $\epsilon_{y}$ may seem innocuous, it
can lead to quite different sign structure for the $\Gamma$ and
$\Gamma^{\prime}$ couplings across different bonds. Taking the three-fold
symmetric honeycomb case as an example, the cubic frame shown in Fig. 2
($\epsilon_{x}\\!=\\!\epsilon_{y}\\!=\\!1$) features uniform signs of $\Gamma$
across all bonds,

$\Gamma_{x}\\!=\\!\Gamma_{y}\\!=\Gamma_{z}\\!\equiv\\!\Gamma,$ (84)

which is quite naturally desirable for this three-fold symmetric case.
However, the choice $(\tilde{x},\tilde{y},\tilde{z})\\!=\\!(-x,y,-z)$, which
corresponds to $\epsilon_{x}\\!=\\!-1$ and $\epsilon_{y}\\!=\\!1$, features

$\Gamma_{\tilde{x}}\\!=\\!-\Gamma_{\tilde{y}}\\!=\\!\Gamma_{\tilde{z}}\\!=\\!-\Gamma,$
(85)

i.e., a modulation of the sign of $\Gamma$ across different types of bonds,
despite the fact that the model is threefold symmetric. (Incidentally, this
example also shows that such a sign modulation in $\Gamma_{\gamma}$, that
could appear in some model, can be ‘gauged away’ by a twofold rotation around
one of the cubic axes.)

The case of $\beta$-Li2IrO3 offers another striking example. Here, there exist
two different sign structures of $\Gamma$, depending on the convention used
for the cubic frame. Specifically:

i) the cubic frame of Ref. [45],

$\hat{\bf x}=\frac{\hat{\bf c}_{o}+\hat{\bf a}_{o}}{\sqrt{2}},~{}~{}\hat{\bf
y}=\frac{\hat{\bf c}_{o}-\hat{\bf a}_{o}}{\sqrt{2}},~{}~{}\hat{\bf
z}=-\hat{\bf b}_{o},$ (86)

where $({\bf a}_{o},{\bf b}_{o},{\bf c}_{o})$ are the orthorhombic axes, see
Fig. 1 (b-c), gives, by virtue of the crystal symmetries $C_{2{\bf c}}$ and
$C_{2{\bf a}}$,

$\Gamma_{x}=-\Gamma_{y}=-\Gamma_{x^{\prime}}=\Gamma_{y^{\prime}},$ (87)

where the bonds with $\gamma\\!=\\!x$ and $y$ are the ones on the $xy$-chains
running along ${\bf a}_{o}$+${\bf b}_{o}$, whereas $\gamma\\!=\\!x^{\prime}$
and $y^{\prime}$ correspond to the $xy$-chains along ${\bf a}_{o}$-${\bf
b}_{o}$. We note in passing that flipping the directions of two of the
orthorhombic axes (without altering the actual directions of $x$ and $y$),
does not change this sign structure. For example, flipping $\hat{\bf
a}_{o}\\!\to\\!-\hat{\bf a}_{o}$ and $\hat{\bf b}_{o}\\!\to\\!-\hat{\bf
b}_{o}$, would change Eq. (86) to $\hat{\bf x}\\!=\\!(\hat{\bf c}_{o}-\hat{\bf
a}_{o})/\sqrt{2}$, $\hat{\bf y}\\!=\\!(\hat{\bf c}_{o}+\hat{\bf
a}_{o})/\sqrt{2}$, and $\hat{\bf z}\\!=\\!\hat{\bf b}_{o}$ (which is the frame
used in Refs. [238, 159]), but would not alter Eq. (87), because the actual
directions of $x$ and $y$ are the same as before.

ii) By contrast, the frame used in Ref. [31],

$\hat{\tilde{\bf x}}=\frac{\hat{\bf c}_{o}-\hat{\bf
a}_{o}}{\sqrt{2}},~{}~{}\hat{\tilde{\bf y}}=\frac{\hat{\bf c}_{o}+\hat{\bf
a}_{o}}{\sqrt{2}},~{}~{}\hat{\tilde{\bf z}}=\hat{\bf b}_{o},$ (88)

which corresponds to the choice $\epsilon_{x}\\!=\\!1$ and
$\epsilon_{y}\\!=\\!-1$, gives, again by virtue of the symmetries $C_{2{\bf
c}}$ and $C_{2{\bf a}}$,

$\Gamma_{\tilde{x}}=\Gamma_{\tilde{y}}=-\Gamma_{\tilde{x}^{\prime}}=-\Gamma_{\tilde{y}^{\prime}},~{}~{}~{}\Gamma_{\tilde{z}}=-\Gamma_{z}\,,$
(89)

which is a drastically different modulation of the sign of $\Gamma$ compared
to that of Eq. (87). In particular, the $xy$\- and
$x^{\prime}y^{\prime}$-chains now each feature a uniform (instead of
alternating) sign of $\Gamma$, opposite to each other. Importantly, the sign
of $\Gamma$ on the $z$-bond also depends on the choice of the cubic frame.
Similar modifications in the sign structure of $\Gamma^{\prime}$ exist as
well, and can be found in a straightforward manner.

The above examples demonstrate that it does not make sense of stating the sign
of the off-diagonal couplings without specifying the choice made for the
directions of the cubic axes with respect to the crystalographic frame. This
is important both when comparing experimental data to theoretical models, but
also when extracting numerical values of the couplings from ab initio
calculations.

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