# Ordered states in the Kitaev-Heisenberg model: From 1D chains to 2D
honeycomb

Cliò Efthimia Agrapidis  IFW Dresden, Institute for Theoretical Solid State
Physics, Dresden, 01069, Germany<EMAIL_ADDRESS>Jeroen van den
Brink  IFW Dresden, Institute for Theoretical Solid State Physics, Dresden,
01069, Germany  Technical University, Department of Physics, Dresden, 01069,
Germany  Satoshi Nishimoto  IFW Dresden, Institute for Theoretical Solid State
Physics, Dresden, 01069, Germany  Technical University, Department of Physics,
Dresden, 01069, Germany

###### Abstract

We study the ground state of the 1D Kitaev-Heisenberg (KH) model using the
density-matrix renormalization group and Lanczos exact diagonalization
methods. We obtain a rich ground-state phase diagram as a function of the
ratio between Heisenberg ($J=\cos\phi)$ and Kitaev ($K=\sin\phi$)
interactions. Depending on the ratio, the system exhibits four long-range
ordered states: ferromagnetic-$z$ , ferromagnetic-$xy$, staggered-$xy$,
Néel-$z$, and two liquid states: Tomonaga-Luttinger liquid and spiral-$xy$.
The two Kitaev points $\phi=\frac{\pi}{2}$ and $\phi=\frac{3\pi}{2}$ are
singular. The $\phi$-dependent phase diagram is similar to that for the 2D
honeycomb-lattice KH model. Remarkably, all the ordered states of the
honeycomb-lattice KH model can be interpreted in terms of the coupled KH
chains. We also discuss the magnetic structure of the K-intercalated RuCl3, a
potential Kitaev material, in the framework of the 1D KH model. Furthermore,
we demonstrate that the low-lying excitations of the 1D KH Hamiltonian can be
explained within the combination of the known six-vertex model and spin-wave
theory.

## Introduction

It is well-known that magnetic ordering (e.g. the Néel state) is induced by
the spontaneous breaking of some sort of symmetry at low temperatures. If one
can prevent such a conventional magnetic ordering, an exotic type of quantum-
disordered state — the so-called quantum spin liquid (QSL) — may arise. In the
spin liquid phase, a macroscopic number of quasi-degenerate low-energy states
compete with each other. Since Anderson’s proposal of a resonating valence
bond state as a theoretical realization of a QSL in 1973 [1], geometrical
frustration [2], a situation where the network of magnetic interactions is
incompatible with the spatial symmetry, had been considered to be almost the
key to achieve such situation [3]. But in 2006, the advent of the Kitaev model
[4] shifted the focus on another way to realize a QSL — frustration due to
bond-dependent exchange interactions [5]. This model has nontrivial
topological phases with elementary excitations exhibiting a Majorana algebra.
After that, the microscopic origin of such Kitaev-type interactions in the
$d^{5}$ transition metal compounds was worked out [6, 7]. In the last decade,
there has been a growing number of researches on the Kitaev materials [8, 9].

The first Kitaev material candidates are 5$d^{5}$ honeycomb iridates
$A_{2}$IrO3 ($A$=Na or Li) [7, 10]. The Ir4+ ions centered in the edge-sharing
IrO6 octahedra form a planar hexagonal network. In each Ir4+ ion, the two
$e_{\rm g}$ levels are split off by the crystal field and five electrons in
the 5$d$ shell are put into the $t_{\rm 2g}$ orbitals with a finite effective
angular momentum $l=1$. Besides, the strong spin-orbit coupling (SOC) for 5$d$
electrons splits up the $t^{5}_{\rm 2g}$ manifold into a fully occupied
$j=\frac{3}{2}$ quartet band and a half-filled $j=\frac{1}{2}$ doublet band.
The latter half-filled band gives an effective $j=\frac{1}{2}$ pseudospin
model, in association with the opening of a Mott gap. In the edge-sharing IrO6
octahedra, there are two Ir-O-Ir exchange paths with a $\sim 90^{\circ}$ bond
angle. The anisotropic Kitaev-type coupling then arises from the particular
superexchange interactions between the $j=\frac{1}{2}$ pseudospins. A more
realistic spin model to describe the magnetic properties is the Kitaev-
Heisenberg (KH) model, that accounts for the residual Heisenberg-type
couplings. It is known that the Kitaev QSL is most likely preempted by a
certain level of the Heisenberg interaction. In fact, the above iridates
exhibit long-range magnetic order at low temperatures [11, 12, 13, 14],
possibly assisted by off-diagonal and/or long-range interactions.

At present, one of the most promising materials close to the Kitaev QSL is the
4$d^{5}$ ruthenium trichloride $\alpha$-RuCl3 in its honeycomb crystal phase
[15, 16, 17, 18, 19, 20, 21]. For this compound the effective $j=\frac{1}{2}$
description, as a Ru3+ 4$d$ analogue to the iridates, seems to still give a
good description of the magnetic properties, although some degree of admixture
between $j=\frac{1}{2}$ and $j=\frac{3}{2}$ in the $t^{5}_{\rm 2g}$ states may
arise from a weaker SOC for 4$d$ electrons than in the Ir 5$d$ orbitals.
Neutron and Raman scattering studies gave evidence for fractionalized
excitations typical of the Kitaev QSL [22, 23, 24], and both theoretical [25]
and experimental investigations [26, 27, 28, 29, 30, 31] indicated, for this
material, the existence of a transition into a QSL in the presence of external
magnetic field. Furthermore, a recent X-ray diffraction measurement suggested
the Kitaev QSL is induced by a small applied pressure of about 0.7 GPa [32]. A
possibility of pressure-induced QSL has been also reported for the other
Kitaev materials, honeycomb iridates $\beta$-Li2IrO3 [33] and $\gamma$-Li2IrO3
[34]. In both materials the low temperature phase is magnetically ordered
without field and under ambient pressure. Namely, the Kitaev QSL is derived
through a melting of the magnetic ordering by field or pressure. Therefore, it
is crucial to deeply understand the characteristics of the magnetically
ordered states.

Very recently, a temperature-dependent electron energy loss spectroscopy
(EELS) measurement was performed in a K-intercalated RuCl3, denoted as
K0.5RuCl3 (Knupfer M., private communications). The intercalated K+ ions
provide charge carriers, however, a sharp gap was observed at $\sim 0.9$ eV,
instead of the charge gap $E_{\rm g}=1.1-1.2$ eV for the undoped
$\alpha$-RuCl3 [35, 36, 37]. This indicates an insulating feature of K0.5RuCl3
and differs from the pseudogap behavior seen for charge localization in
disordered metals. This has been interpreted as half of the $j=\frac{1}{2}$
pseudospins being replaced by nonmagnetic $d^{6}$ ions. Therefore, this
insulating state can be pictured as a formation of superlattice by charge
disproportionation (charge ordering). Different possible charge ordering
patterns are shown in Fig. 1. Of particular interest in the present context is
the zigzag chain-type pattern exhibited in Fig. 1(c), where chains of
nonmagnetic ions are separated by chains of magnetic ions with KH
interactions.

The above motivations lead us to studying the 1D KH model not only for
fundamental theoretical reasons, but also as a possible minimal spin model to
describe the magnetic properties of the K-intercalated RuCl3. So far, there
are few studies on the 1D KH model [38, 39, 40, 41, 42, 43, 44, 45, 46]. Using
the density-matrix renormalization group (DMRG) method, we calculate total
spin, real-space spin-spin correlation functions, static spin structure
factor, central charge, and various order parameters in the ground state.
Based on the results, we obtain the ground-state phase diagram, including four
long-range ordered and two liquid phases, as a function of the ratio between
Heisenberg and Kitaev interactions. Moreover, the relevance of the phase
diagram to that of the honeycomb-lattice KH model is discussed. It is striking
that all the magnetically ordered states of the honeycomb-lattice KH model can
be interpreted in terms of coupled 1D KH chains. Furthermore, we calculate the
dynamical spin structure factors via the Lanczos exact diagonalization (ED)
technique. The basic low-lying excitations are considered by use of the known
six-vertex model and the spin-wave theory (SWT). The present investigation can
thus contribute to an elucidation of the fundamental properties of the
K-intercalated RuCl3 as well as a better insight in the understanding of the
physics of the honeycomb-lattice KH model.

## Model and Numerical Methods

### Pattern of charge ordering in the K-intercalated RuCl3

The insulating feature of K0.5RuCl3 with a sharp peak at $\sim 0.9$ eV in the
EELS spectrum can be explained by charge ordering associated with a
superlattice formation rather than a random distribution of $K^{+}$ ions
(Knupfer M., private communications). The superlattice structures are simply
constructed by replacing half of the $j_{\rm eff}=1/2$ pseudospins with
nonmagnetic sites in the original hexagonal cluster. In practice, there are
three possibilities for the charge ordering pattern, they shown in Fig
1(a)-(c). Recently, for exfoliated $\alpha$-RuCl3, a first-order structural
phase transition at $\sim 150$ K between low-temperature $C$2/m and high-
temperature $P$31 structures was reported. [47] At high temperature, the
emergence of charge ordering, originating from anisotropy in the charge
distribution along Ru-Cl-Ru hopping pathways, was observed. It suggests a
strong tendency to the bond-directional anisotropy along the Ru-Ru axes. In
this sense a realization of the zigzag-type charge ordering [Fig 1(c)] may be
most likely. Besides, the zigzag-type charge ordering can gain the largest
exchange energy (see below). To fix the detailed charge distribution,
experimental observation and/or microscopic analysis including structural
distortion by the K-intercalation are required in future.

Figure 1:  Possible charge ordering patterns of K0.5RuCl3; (a) triangular, (b)
dimer, and (c) zigzag. (d) Lattice structure of the 1D Heisenberg-Kitaev
model. The labels ‘$x$’ and ‘$y$’ indicate the $x$-bond and $y$-bond,
respectively (see text).

### 1D Kitaev-Heisenberg Hamiltonian

At present, it is widely believed that the magnetic properties of the undoped
$\alpha$-RuCl3 are well described by the KH model on a honeycomb lattice. If
we assume the zigzag-type charge ordering in Fig 1(c), the zigzag chains are
well separated by the nonmagnetic ions. Then, each chain is considered to be a
1D KH model, which is equivalent to a system obtained by removing the
$z$-bonds from the honeycomb-lattice KH model. The Hamiltonian of 1D KH model
is written as

$\mathcal{H}=K\sum_{i=1}^{L/2}(S^{x}_{2i-1}S^{x}_{2i}+S_{2i}^{y}S^{y}_{2i+1})+J\sum_{i=1}^{L}\vec{S}_{i}\cdot\vec{S}_{i+1}$
(1)

for a system with $L$ sites, where
$\vec{S}_{i}=(S_{i}^{x},S_{i}^{y},S_{i}^{z})$ is a spin-$\frac{1}{2}$ operator
at site $i$, and the Kitaev and exchange couplings are defined as $K=\sin\phi$
and $J=\cos\phi$ via a phase parameter $\phi$. Throughout the paper, we take
$\sqrt{K^{2}+J^{2}}=1$ as the energy unit. The system has two kinds of
neighboring links and they appear alternately along the chain. Hence, the
structural unit cell contains two lattice sites. Hereafter, we call the links
($2i-1$,$2i$) and ($i$,$2i+1$) “$x$-link” and “$y$-link”, respectively. By
rewriting Eq. (1) as

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{2J+K}{4}\sum_{i=1}^{L}(S^{+}_{i}S^{-}_{i+1}+S^{-}_{i}S^{+}_{i+1})+J\sum_{i=1}^{L}S^{z}_{i}S^{z}_{i+1}+\frac{K}{4}\sum_{i}^{L}(-1)^{i}(S^{+}_{i}S^{+}_{i+1}+S^{-}_{i}S^{-}_{i+1})$
$\displaystyle\equiv$ $\displaystyle{\cal H}_{\rm ex}+{\cal H}_{\rm
Ising}+{\cal H}_{\rm dsf},$

we can easily notice that a XXZ Heisenberg chain containing exchange (${\cal
H}_{\rm ex}$) and Ising (${\cal H}_{\rm Ising}$) terms is disturbed by sign-
alternating double-spin-flip (${\cal H}_{\rm dsf}$) fluctuations [38].

### Numerical methods

We employ the density-matrix renormalization group (DMRG) method, which is one
of the most powerful numerical techniques for studying various quasi 1D
quantum systems [48]. Open boundary conditions are applied unless stated
otherwise. This enables us to calculate ground-state and low-lying excited-
state energies, as well as static quantities, quite accurately for very large
finite-size systems. We are thus allowed to carry out an accurate finite-size-
scaling analysis to obtain energies and quantities in the thermodynamic limit
$L\to\infty$. We hence study chains with several lengths up to $L=200$ sites
for a given $\phi$. Since, in hindsight, the system (1) exhibits only
commensurate phases in the ground state and the largest magnetic unit cell
contains four lattice sites, its size is taken as $L=4n$ ($n$: integer). For
each calculation, we keep up to $m=1200$ density-matrix eigenstates in the
renormalization procedure and extrapolate the calculated quantities to the
limit $m\to\infty$ if needed. Since the SU(2) symmetry is broken in system (1)
and total $S^{z}$ is no longer a good quantum number except at $\phi=0$,
$\pi$, one may have some difficulty in obtaining accurate results in
comparison to usual DMRG calculations. Nevertheless, in this way, the maximum
truncation error, i.e., the discarded weight, is less than $1\times 10^{-10}$
while the maximum error in the ground-state is less than $1\times 10^{-8}$.

For the dynamical properties calculation, we used the Lanczos exact
diagonalization (ED) method. To examine the low-energy excitations for each
phase, we calculate the dynamical spin structure factor, defined as

$\displaystyle S_{\gamma}(q,\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}{\rm
Im}\langle\psi_{0}|(S^{\gamma}_{q})^{\dagger}\frac{1}{\hat{H}+\omega-
E_{0}-{\rm i}\eta}S^{\gamma}_{q}|\psi_{0}\rangle$ (2) $\displaystyle=$
$\displaystyle\sum_{\nu}|\langle\psi_{\nu}|S^{\gamma}_{q}|\psi_{0}\rangle|^{2}\delta(\omega-
E_{\nu}+E_{0}),$

where $\gamma$ is $z$ or $-(+)$, $|\psi_{\nu}\rangle$ and $E_{\nu}$ are the
$\nu$-th eingenstate and the eigenenergy of the system, respectively ($\nu=0$
corresponds to the ground state). Under periodic boundary conditions, the spin
operators $S^{\gamma}_{q}$ can be precisely defined by

$S^{\gamma}_{q}=\sqrt{\frac{2}{L}}\sum_{i}S^{\gamma}_{i}\exp(iqr_{i})$ (3)

where $r_{i}$ is the position of site $i$ and the sum runs over either $i$
even or $i$ odd sites. They provide the same results. The momentum is taken as
$q=\frac{4\pi}{L}n$ ($n=0,\pm 1,\dots,\pm\frac{L}{4}$) since the lattice unit
cell includes two sites and the number of unit cells is $\frac{L}{2}$ in a
system with $L$ sites. We calculate both spectral functions
$S_{\pm}(q,\omega)$ and $S_{z}(q,\omega)$ as they are different due to broken
SU(2) symmetry except at $\phi=0$ and $\pi$. We study chains with $L=24$,
namely, 12 unit cells, by the Lanczos ED method. As shown below, system (1)
contains only commensurate phases with unit cell containing one, two, or four
sites. Therefore, a quantitative discussion of the low-lying excitations is
possible even within the $L=24$ chain.

## Results

### Ground-state properties

#### Quantum phase transitions

Lets first look at the ground-state energy and total spin with respect to
$\phi$ in order to capture the overall appearance of quantum phase
transitions. The results, using a periodic 24-site KH chain, are shown in Fig.
2. We clearly see discontinuities in the first derivative of the ground-state
energy at four values: $\phi=\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and
$\approx 1.65\pi$, which indicate the first-order phase transitions. The
second derivative $-\partial^{2}E_{0}/\partial\phi^{2}$ is, in a precise
sense, continuous except for the above four $\phi$ points, nonetheless, there
exists a distinguishable peak around $\phi\approx 0.65\pi$, which may
corresponds to a second-order (or continuous) phase transition. Furthermore,
as shown below, we find another phase transitions at $\phi=0$. Therefore, we
suggest that the simple 1D model (1) exhibits a variety of phases including
six quantum phase transitions. It will be confirmed by studying various
corresponding order parameters or spin-spin correlation functions. We also
confirm that the ground-state energy of the 1D KH chain is always lower than
that given by the dimer-type charge ordering [see Fig. 2(a)].

Figure 2:  (a) Ground-state energy $E_{0}$, (b) the second derivative of
$E_{0}$ with respect to $\phi$, and (c) total spin $S_{tot}$ as a function of
$\phi$, obtaind with a 24-site periodic Kitaev-Heisenberg chain. Dotted line
in (a) indicates ground-state energy for the dimer-type charge ordering.
Figure 3:  Schematic pictures of five states realized in the 1D KH model,
except the TLL state at $0<\phi<\frac{\pi}{2}$. In the phases (a)(c)(d), the
spins lie mostly on the xy-plane and these states are rotational invariant
around the $z$-axis. In phases (b)(e), the spins almost align along the
$z$-direction and a state having opposite spin directions is degenerate.

#### Ferromagnetic-$xy$ phase ($\pi<\phi<\frac{3\pi}{2}$)

Since both $K$ and $J$ are ferromagnetic (FM), a long-range FM ordered state
is naively expected in the range $\pi<\phi<\frac{3\pi}{2}$. At the spin
isotropic point $\phi=\pi$, the spins can align along any arbitrary spatial
direction due to SU(2) spin rotation invariance and the total spin takes the
maximum value $\frac{S_{tot}}{L}=\frac{1}{2}$ [see Fig. 2(c)]. Away from the
isotropic point, the SU(2) symmetry is broken to U(1) and the configurations
for higher $|S^{z}|$ sectors are projected out [see Fig. 3(c)]. When $\phi$ is
still close to $\pi$ ($-J\gg-K>0$), the ground state is approximately
expressed as

$|\Psi_{0}\rangle=\frac{1}{\sqrt{\cal N}}\sum_{m}|\psi_{m}\rangle,$ (4)

where $|\psi_{m}\rangle$ is a basis in real space, namely each site state is
represented by either spin-up ($\uparrow$) or spin-down ($\downarrow$). The
basis is here restricted to a $S^{z}_{\rm tot}=\sum_{i=1}^{L}\langle
S^{z}_{i}\rangle=0$ subspace, so that $m$ is summed over all possible
combinations of spin configuration with $\frac{L}{2}$ up and $\frac{L}{2}$
down spins in a $L$ lattice sites. ${\cal N}$ is the total number of the spin
configurations, i.e., $\frac{L!}{(L/2)!(L/2)!}$. The wave function (4) becomes
exact in the isotropic spin limit $\phi=\pi+$. Accordingly, the spin-spin
correlations have long-range FM ordering for all three spin components:
$\langle S^{x}_{i}S^{x}_{j}\rangle=\langle
S^{y}_{i}S^{y}_{j}\rangle=\frac{1}{6}$ and $\langle
S^{z}_{i}S^{z}_{j}\rangle=-\frac{1}{12}$ for any $i\neq j$. Taking the spin
isotropic Hamiltonian at $\phi=\pi+$ as an unperturbed one, the unperturbed
ground state is given by Eq. (4). When $\phi-\pi\ll 1$, the perturbed
Hamiltonian can be written as ${\cal
H}^{\prime}\approx\frac{\phi-\pi}{4}\sum_{i}(S_{i}^{+}S_{i+1}^{-}+S_{i}^{-}S_{i+1}^{+})$
and the lowest-order energy correction is $E^{\prime}=\frac{\phi-\pi}{4}$.
Therefore, with increasing $\phi$ from $\pi$, the antiferromagnetic (AFM)
fluctuations increase and the long-range FM ordering is weakened. Nonetheless,
the correlations $\langle S^{x}_{i}S^{x}_{j}\rangle$ and $\langle
S^{y}_{i}S^{y}_{j}\rangle$ retain the same asymptotic behaviors indicating the
long-range FM ordering until $\phi=\frac{3\pi}{2}$, characterized by the
saturation to a finite negative value in the limit $|i-j|\to\infty$; whereas,
$\langle S^{z}_{i}S^{z}_{j}\rangle$ decays in a power law with $|i-j|$. This
is because the AFM fluctuations are mainly introduced along the $z$-direction.
It is confirmed by a slow decrease of the total spin as a function of $\phi$,
as seen in Fig. 2(c). We thus call this state FM-$xy$ state. A collapse of the
long-range FM ordering is detected by a drop-off of the total spin at
$\phi=\frac{3\pi}{2}$, suggesting a first-order transition.

#### Ferromagnetic-$z$ phase ($0.65\pi\lesssim\phi<\pi$)

Figure 4:  (a) Finite-size scaling analyses of the FM-$z$ order parameter with
polynomial fitting functions. (b) Extrapolated values of the FM-$z$ order
parameter ${\cal O}_{{\rm FM}-z}$ to the thermodynamic limit $L\to\infty$ as a
function of $\phi$.

A long-range FM ordered state stabilizes also at $\frac{3\pi}{4}<\phi<\pi$,
where all the exchange tensors are FM since $J+K$ is negative despite positive
$K$. However, in contrast to the FM-$xy$ state, spin alignment along the
$z$-direction is favored because of the easy-axis-XXZ-like interactions
$|\frac{2J+K}{2}|<|J|$. Therefore, the most dominant spin configurations are
given by the highest $|S^{z}|$ sectors, i.e., the ground state is expressed as

$|\Psi_{0}\rangle=\frac{1}{\sqrt{2}}(|\Uparrow\rangle+|\Downarrow\rangle),$
(5)

with
$|\Uparrow\rangle=|\cdots\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\cdots\rangle$
and
$|\Downarrow\rangle=|\cdots\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\cdots\rangle$.
This wave function is exact in the isotropic limit of $\phi=\pi-$. We call
this type of long-range FM ordering FM-$z$ state. Let us take Eq. (5) as the
unperturbed ground state. For $\pi-\phi\ll 1$, the perturbed Hamiltonian can
be written as ${\cal
H}^{\prime}\approx\frac{(\pi-\phi)^{2}}{2}\sum_{i}S_{i}^{z}S_{i+1}^{z}$. This
is an Ising-like AFM correlation and it clearly disturbs the FM-$z$ state.
However, the lowest-order correction to the ground-state energy is
$E^{\prime}=\frac{(\pi-\phi)^{2}}{8}$. It means that the perturbation acts
only gradually with being away from the isotropic point $\phi=\pi$. As a
result, the total spin is not that much reduced around $\phi=\pi$ in the
FM-$z$ phase. To estimate the lower bound of the FM-$z$ phase, we shall define
the FM-$z$ order parameter. A state with long-range FM ordering is a state
with broken spin symmetry along the $z$-direction; macroscopically, there are
two degenerate ground states $|\Psi_{0}\rangle\approx|\Uparrow\rangle$ and
$|\Psi_{0}\rangle\approx|\Downarrow\rangle$. Applying open boundary
conditions, one of the two ground states is picked imposing initial conditions
on the calculation. The long-range ordered state is thus directly observable
as a symmetry-broken state in our DMRG calculations. The order parameter is
defined as

${\cal O}_{{\rm FM}-z}=2\lim_{L\to\infty}\langle S^{z}_{L/2}\rangle$ (6)

The finite-size scaling analysis is performed. The extrapolated values in the
thermodynamic limit are shown in Fig. 4. Notably, the FM-$z$ state survives
even at $\phi<\frac{3\pi}{4}$, where a part of the interactions is AFM, i.e.,
$J+K>0$. Nevertheless, the long-ranged FM order is drastically suppressed by
the AFM fluctuations at $\phi<\frac{3\pi}{4}$ and completely destroyed at
$\phi\approx 0.65\pi$. The order parameter has no jump at the transition
point, suggesting a second-order phase transition.

#### Spiral-$xy$ phase ($\frac{\pi}{2}<\phi\lesssim 0.65\pi$)

Figure 5:  Static spin structure factors for (a) $\phi=0.51\pi$, (b)
$\phi=0.6\pi$, (c) $\phi=0.65\pi$, and (d) $\phi=0.75\pi$. (e) Log-log plot of
the spin-spin correlation functions as a function of distance at
$\phi=0.6\pi$.

As shown above, the FM-$z$ phase remains down to $\phi\approx 0.65\pi$. Then,
we ask which kind of phase comes next at $\frac{\pi}{2}<\phi\lesssim 0.65\pi$.
In this range, $K$ is AFM, $J$ is FM, and $K\gg-J$. On the $x$-links, the
$x$-components of spins tend to be antiparallel due to the strong AFM
interaction along the $x$ direction; whereas, their $y$-components tend to be
parallel due to FM $J$ term. One can think about the $y$-links in the same
way. Eventually, the spins lie on the $xy$ plane and rotate by 90∘ from one
site to the next. The magnetic unit cell is twice as large as the structural
unit cell, i.e., including four lattice sites [see Fig. 3(a)]. We call this
state spiral-$xy$ state. To confirm this magnetic structure, we calculate the
static spin structure factor, defined as

$S^{\gamma}(q)=\frac{2}{L}\sum_{k,l}\langle
S_{k}^{\gamma}S_{l}^{\gamma}\rangle e^{-iq(k-l)},$ (7)

where the length of the structural unit cell, i.e., two lattice spacings, is
taken to be unity. The DMRG results with $L=60$ cluster are shown in Fig. 5.
Near the AFM Kitaev point $\phi=\frac{\pi}{2}$, $S^{x}(q)+S^{y}(q)$ has a
large $q=\pi$ peak indicating a periodicity of four lattice sites on the
$xy$-plain. On the other hand, $S_{z}(q)$ is almost zero for all $q$ since the
FM $z$ interaction $J$ is tiny compared to the dominant AFM $x$ or $y$
interactions $J+K$. With increasing $\phi$, the $q=\pi$ peak becomes lower and
a $q=0$ peak in $S^{z}(q)$ develops. Basically, the spins are tilted in one
direction along the $z$-axis with keeping the periodicity on the $xy$-plain.
Those peak heights are reversed around the transition point $\phi=0.65\pi$
from the spiral-$xy$ to the FM-$z$ phases. We note that the height of the
$q=0$ peak in $S^{z}(q)$ coincides with the FM-$z$ order parameter. As shown
in Fig. 5(e), the spin-spin correlations decay as a power law for all the spin
components in the spiral-$xy$ phase. It means that the $q=\pi$ peak in
$S^{x}(q)+S^{y}(q)$ disappears in the thermodynamic limit: the spiral-$xy$
structure is not long-range ordered.

#### Néel-$z$ ordered phase ($1.65\pi\lesssim\phi<2\pi$)

Figure 6:  (a) Finite-size scaling analyses for the Néel-$z$ order parameter
${\cal O}_{\textrm{N\'{e}el}}$ with polynomial fitting functions. (b) The
extrapolated values of ${\cal O}_{\textrm{N\'{e}el}}$ to the themodynamic
limit as a function of $\phi$. (c) Central charge, calculated with periodic
chains, as a function of $\phi$ for several chain lengths.

Next, we turn to the parameter region $\frac{3\pi}{2}<\phi<2\pi$ ($J>0$,
$K<0$). We start from the SU(2) symmetric limit $\phi=2\pi$, where the system
is the original 1D AFM Heisenberg model. The ground-state wave function can be
exactly obtained and it is know to have $S_{\rm tot}=0$. So, the first
perturbative correction is given by ${\cal
H}^{\prime}\approx(\phi-2\pi)\sum_{i}(S^{+}_{i}S^{-}_{i+1}+S^{-}_{i}S^{+}_{i+1})$,
which provides an easy-axis anisotropy to the system. Therefore, the
possibility of a continuous Néel-$z$ order transition is conceived by analogy
to the easy-axis anisotropic XXZ chain. For small $2\pi-\phi$, the
magnetization is expect to grow gradually with

$M\simeq\frac{\pi}{2(2\pi-\phi)}\exp\left\\{-\frac{\pi^{2}}{2\sqrt{2(2\pi-\phi)}}\right\\}.$
(8)

Let us confirm it numerically. In the long-range Néel-$z$ ordered state, the
translational symmetry is broken in a finite system due to the Friedel
oscillation under the open boundary conditions, so that the Néel-$z$ state can
be directly observed by extracting one of the degenerate states, like in the
FM-$z$ state. Generally, the Friedel oscillations in the center of the system
decay as a function of the system length. If the amplitude at the center of
the system persists for arbitrary system lengths, it corresponds to a long-
range ordering. Here, we are interested in the formation of alternating spin
flip along the $z$-direction. Thus, the Néel-$z$ order parameter is defined as

${\cal O}_{\textrm{N\'{e}el}}=\lim_{L\to\infty}|\langle
S_{L/2}^{z}\rangle-\langle S_{L/2+1}^{z}\rangle|.$ (9)

This quantity is equivalent to the magnetization $M$ in the thermodynamic
limit. The finite-size scaling analyses of ${\cal O}_{\textrm{N\'{e}el}}$ was
performed using the results for systems with up to $L=120$ and the
extrapolated values to the thermodynamic limit were obtained. They are shown
in Fig. 6. We indeed see a slow increase of ${\cal O}_{\textrm{N\'{e}el}}$
near the SU(2) symmetric limit $\phi=2\pi$. Further with decreasing $\phi$,
the order parameter develops up to ${\cal O}_{\textrm{N\'{e}el}}=1$ and drops
down to $0$ at $\phi\approx 1.65\pi$.

In fact, this lower $\phi$-boundary of the Néel-$z$ ordered phase can be
estimated analytically. At $\phi=\tan^{-1}(-2)\approx 1.6476\pi$, the exchange
term ${\cal H}_{\rm ex}$ disappears in the Hamiltonian (1) and the system is
just written as a sum of double-spin-flip and Ising parts

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle{\cal H}_{\rm
dsf}+{\cal H}_{\rm Ising}$ (10) $\displaystyle=$
$\displaystyle\frac{K}{4}\sum_{i}^{L}(-1)^{i}(S^{+}_{i}S^{+}_{i+1}+S^{-}_{i}S^{-}_{i+1})+\frac{J}{4}\sum_{i}^{L}S^{z}_{i}S^{z}_{i+1}.$

Each of the partition Hamiltonians ${\cal H}_{\rm dsf}$ and ${\cal H}_{\rm
Ising}$ is exactly solvable. For ${\cal H}_{\rm dsf}$, the system is regarded
as noninteracting fermions with “pair hopping‘’ and the ground-state wave
function is

$|\Psi_{0}\rangle=\frac{1}{\sqrt{\cal N}}\sum_{\cal C}\prod_{i({\cal
C}),\gamma({\cal C})}(-1)^{i-1}S_{i}^{\gamma}S_{i+1}^{\gamma}\ket{\Uparrow},$
(11)

where ${\cal C}$ is summed over all possible spin configurations created by
the double spin flips starting from $\ket{\Uparrow}$ or $\ket{\Downarrow}$
(${\cal N}=_{L}C_{L/2}$), $i$ and $\gamma$(=+ or -) are taken to create the
configuration ${\cal C}$. While, for ${\cal H}_{\rm Ising}$, the system is a
simple Ising one and the ground-state wave function is

$|\Psi_{0}\rangle=\frac{1}{\sqrt{2}}\sum_{\sigma}\prod_{i=1}^{L/2}c_{2i-1,\sigma}^{\dagger}c_{2i,\bar{\sigma}}^{\dagger}\ket{0},$
(12)

where $\ket{0}$ is the vacuum state. The ground-state wave functions (11) and
(12) are orthogonal since they do not share the same spin configurations. It
means the ground state is two-fold degenerate at the critical point
$\phi=\tan^{-1}(-2)$. The ground-state energy is $E_{0}=-\frac{L\cos\phi}{4}$.
This degeneracy is lifted away from $\phi=\tan^{-1}(-2)$. Eq. (12) is the
ground state for larger-$\phi$ side, namely, in the Néel-$z$ phase; while, Eq.
(11) for lower-$\phi$ side (see Sec. Staggered-$xy$ ordered phase
($\frac{3\pi}{2}<\phi\lesssim 1.65\pi$)). This is consistent with the fact
that ${\cal O}_{\textrm{N\'{e}el}}$ reaches exactly 0.5 at
$\phi=\tan^{-1}(-2)$, as shown in Fig. 6(b). Therefore, the ground-state wave
function is completely changed at $\phi=\tan^{-1}(-2)$ and it clearly suggests
a first-order transition. This is also confirmed by the jump of ${\cal
O}_{\textrm{N\'{e}el}}$ as well as by the singularity in
$-\partial^{2}E_{0}/\partial\phi^{2}$ at $\phi\approx 1.65\pi$.

#### Staggered-$xy$ ordered phase ($\frac{3\pi}{2}<\phi\lesssim 1.65\pi$)

Figure 7:  (a) The $x$-component of local spin, where opposite magnetic fields
$\pm 10$ are applied at either end of the system. (b) Finite-size scaling
analyses for the staggered-$xy$ order parameter with polynomial fitting
functions. (c) The extrapolated values of ${\cal O}_{xy}$ to the themodynamic
limit as a function of $\phi$.

Let us then consider a region at $\frac{3\pi}{2}<\phi\lesssim 1.65\pi$, where
the signs of Heisenberg and Kitaev terms counterchange from the spiral-$xy$
state. This may mean the local spin structure is similar to the spiral-$xy$
state. As discussed above, the system has a first-order transition at
$\phi=\tan^{-1}(-2)\approx 1.65\pi$. In the lower-$\phi$ vicinity
($\phi\lesssim 1.65$) the ground state is exactly given by Eq. (11), which
derives asymptotic behaviors of the spin-spin correlations: $\langle
S^{x}_{0}S^{x}_{j}\rangle=\frac{\alpha^{2}}{4\sqrt{2}}\cos\frac{\pi}{2}[(j+\frac{1}{2})]$,
$\langle
S^{y}_{0}S^{y}_{j}\rangle=\frac{\beta^{2}}{4\sqrt{2}}\cos\frac{\pi}{2}[(j-\frac{1}{2})]$,
$\langle S^{z}_{0}S^{z}_{j}\rangle=0$, and
$\langle\vec{S}_{0}\vec{S}_{j}\rangle=\frac{1}{4}\cos\left(\frac{\pi}{2}j\right)+\frac{\alpha^{2}-\beta^{2}}{4}\sin\left(\frac{\pi}{2}j\right)$.
Since the system is rotation invariant around the $z$-axis, the coefficients
$\alpha$ and $\beta$ can take arbitrary real numbers under a condition
$\alpha^{2}+\beta^{2}=1$. As illustrated in Fig. 3(d), all the spins lie on
the $xy$-plane and the magnetic unit cell contains four lattice sites as in
the spiral-$xy$ state. However, the crucial difference from the spiral-$xy$
state is that this $xy$ state exhibits a long-range ordering, which is clearly
indicated by the correlation functions. We call this state staggered-$xy$
state, since it resembles a Néel-like state with a period of four sites.

We then investigate the $\phi$-dependence of this $xy$ state. Applying a
staggered field along the $x$-direction on both open system edge sites, the
presence or absence of the long-range staggered-$xy$ order can be determined
by studying the decay of the $x$-component of local spin $\langle
S^{x}_{i}\rangle$ as the Friedel oscillation. Thus, the staggered-$xy$ order
parameter is defined at the center of the system as

${\cal O}_{xy}=\frac{1}{2}\lim_{L\to\infty}|\langle S_{L/2}^{x}\rangle-\langle
S_{L/2+1}^{x}\rangle|.$ (13)

In the upper and lower vicinity of $\phi\approx 1.65\pi$, ${\cal O}_{xy}=0$
and $\frac{1}{2}$ are obtained from Eqs. (11) and (12), respectively. For the
other $\phi$ values the profile of the $x$-component of local spin and the
finite-size scaling of the order parameter are shown in Fig. 7(a)(b). The
extrapolated values of ${\cal O}_{xy}$ to the thermodynamic limit are plotted
in Fig. 7(c). We can see that the order parameter jumps at the both phase
boundaries suggesting first-order transitions. The collapse of the
staggered-$xy$ ordering at the lower boundary $\phi=\frac{3\pi}{2}$ is clearly
confirmed by the profile of $\langle S^{x}_{i}\rangle$, as shown in Fig. 7(a).

Figure 8:  Central charge, calculated with periodic $L=10$, $20$, and $30$
chains, as a function of $\phi$ .

#### Tomonaga-Luttinger-liquid phase ($0\leq\phi<\frac{\pi}{2}$)

The remaining region is $0\leq\phi<\frac{\pi}{2}$. At $\phi=0$ the system is
equivalent to the spin-isotropic AFM Heisenberg chain, which is a gapless spin
liquid. It is known that the low-energy physics is described by a Tomonaga-
Luttinger (TL) model with a boson field, which is equivalent to the unity
central charge ($c=1$) conformal field theory (CFT) [49]. Let us now consider
what happens when we move away from $\phi=0$. At $0<\phi<\frac{\pi}{2}$ the
deviating interactions from the spin-isotropic Hamiltonian are written as

$\mathcal{H}_{\rm
aniso}=\frac{K}{4}\sum_{i=1}^{L}[S^{+}_{i}S^{-}_{i+1}+S_{i}^{-}S^{+}_{i+1}+(-1)^{i}(S^{+}_{i}S^{+}_{i+1}+S_{i}^{-}S^{-}_{i+1})].$
(14)

Obviously, these interactions cannot produce any explicit magnetic order,
since they only enhance the quantum fluctuations. However, it is a nontrivial
question whether the $c=1$ CFT is conserved. To examine it, we directly
calculate the central charge, which can be obtained from the von Neumann
entanglement entropy in the DMRG procedure as follows: Let us consider a
quantum 1D periodic system with length $L$. The von Neumann entanglement
entropy of its subsystem with length $l$ is given as $S_{L}(l)=-{\rm
Tr}_{l}\rho_{l}\log\rho_{l}$, where $\rho_{l}={\rm Tr}_{L-l}\rho$ is the
reduced density matrix of the subsystem and $\rho$ is the full density matrix
of the whole system. Using the CFT, the entropy of the subsystem with length
$l$ for a fixed system length $L$ has been derived: [50, 51, 52]

$\displaystyle S_{L}(l)=\frac{c}{3}\ln\left[\frac{L}{\pi}\sin\left(\frac{\pi
l}{L}\right)\right]+s_{1}$ (15)

where $s_{1}$ is a non-universal constant. In the DMRG calculations this
quantity can be accurately estimated by the second derivative of Eq. (15) with
respect to $l$, namely,

$\displaystyle c=-\frac{3L^{2}}{\pi^{2}}\frac{\partial^{2}S_{L}(l)}{\partial
l^{2}}\bigg{|}_{l=\frac{L}{2}}.$ (16)

The results for periodic chains with L=10, 20, and 30 are plotted in Fig. 8.
We see a quick convergence to $c=1$ with increasing system size in the whole
range of $0\leq\phi<\frac{\pi}{2}$. Hence, we confirm the system remains in
the TL liquid state as far as both $J$ and $K$ are AFM. Similarly, we studied
the central charge in the Néel-$z$ phase. The results for periodic chains with
$L=16$, $24$, and $32$ are plotted in Fig. 6(c). Although $c=0$ should be
retained in the Néel-$z$ phase, we see a very slow converge to $c=0$ with
increasing $L$ near $\phi=2\pi$, reflecting the tiny magnetization.

#### Kitaev points

As discussed above, the Kitaev points are singular, not adiabatically
connected to the neighboring phases. At these points $\phi=\pm\frac{\pi}{2}$,
the Heisenberg interaction $J$ vanishes in our Hamiltonian Eq. (1) and it is
reduced to

$\mathcal{H}=\pm K\sum_{x-links}S^{x}_{i}S^{x}_{i+1}\pm
K\sum_{y-links}S_{i}^{y}S^{y}_{i+1}$ (17)

In order to make the notation easier, we here focus on the case of
$\phi=-\pi/2$. Note that the case of $\phi=\frac{\pi}{2}$ can be similarly
considered. It is convenient to fermionize the Hamiltonian (17). By applying
the Jordan-Wigner transformation, we obtain

$\displaystyle\mathcal{H}=\sum_{x-links}(c_{{\rm b},i}^{\dagger}-c_{{\rm
b},i})(c_{{\rm w},i}^{\dagger}+c_{{\rm w},i})$ (18)
$\displaystyle-\sum_{y-links}(c_{{\rm w},i}^{\dagger}+c_{{\rm w},i})(c_{{\rm
b},i+1}^{\dagger}-c_{{\rm b},i+1}),$ (19)

where $c_{{\rm b},i}$ and $c_{{\rm w},i}$ are the fermion annihilation
operators at the black and white labeling sites in the $i$-th unit cell,
illustrated in Fig. 1(d), respectively. Generally, it is possible to write
fermion operators in terms of Majorana fermions defined as:

$\displaystyle c_{{\rm b},i}^{\dagger}$
$\displaystyle=\frac{b_{1,i}-ib_{2,i}}{2}$ (20) $\displaystyle c_{{\rm
w},i}^{\dagger}$ $\displaystyle=\frac{w_{1,i}-iw_{2,i}}{2}$ (21)

With these operators Eq. (19) is transformed to

$\mathcal{H}=-i\frac{K}{4}\sum_{x-links}b_{2,i}w_{1,i}+i\frac{K}{4}\sum_{y-links}w_{1,i}b_{2,i+1}.$
(22)

We immediately notice that the Majoranas $b_{1},\,w_{2}$ are not included in
the Hamiltonian (22). We then define new, non local, fermion operators as

$\displaystyle d_{i}^{\dagger}$ $\displaystyle=\frac{w_{1,i}-ib_{2,i}}{2}$
(23) $\displaystyle d_{i}$ $\displaystyle=\frac{w_{1,i}+ib_{2,i}}{2},$ (24)

where the indices $i$ run over the unit cells instead of over the lattice
sites [see Fig. (1)(d)]. Using these operators, (22) can be expressed as

$\displaystyle\mathcal{H}=\frac{K}{2}\sum_{i}d_{i}^{\dagger}d_{i}-\frac{K}{4}\sum_{i}(d_{i}^{\dagger}d_{i+1}+d_{i}d_{i+1}+\text{h.c.}).$
(25)

This describes a $p$-wave paired superconductor [53]. The exact solution for
the ground state is easily obtained by a Fourier transformation:

$\mathcal{H}=\frac{1}{2}\sum_{q}\left[\varepsilon_{q}d^{\dagger}_{q}d_{q}+i\frac{\Delta_{q}}{2}(d^{\dagger}_{q}d^{\dagger}_{-q}+\text{h.c.})\right]$
(26)

where $\varepsilon_{q}=-\frac{K}{2}(\cos q+1)$ and $\Delta_{q}=\frac{K}{2}\sin
q$. Using a Bogoliubov transformation, this can be diagonalized and the
quasiparticle excitation is given by

$E_{q}=\pm\frac{1}{4}\sqrt{\varepsilon_{q}^{2}+\Delta_{q}^{2}}.$ (27)

Thus, we can confirm the system is still in the gapless region. The ground-
state energy is calculated as

$\frac{E_{0}}{L}=\frac{1}{L}\sum_{q}(-E_{q})=\frac{1}{2\pi}\int_{-\pi}^{\pi}dq(-E_{q})=-\frac{K}{2\pi}.$
(28)

To consider the meaning of the Majoranas $b_{1},\,w_{2}$ which do not appear
explicitly in the Hamiltonian (22), we take a possible recombination of them
into new fermion operators:

$\displaystyle\widetilde{d}_{i}^{\,\dagger}$ $\displaystyle=$
$\displaystyle\frac{b_{1,i}-iw_{2,i}}{2}$ (29)
$\displaystyle\widetilde{d_{i}}$ $\displaystyle=$
$\displaystyle\frac{b_{1,i}+iw_{2,i}}{2}$ (30)

We can now trace these back to the initial spin operators. Using Eqs. (20) and
(21), we get

$\widetilde{d}_{i}^{\,\dagger}=\frac{1}{2}\left(c_{b,i}^{\dagger}+c_{b,i}+c_{w,i}^{\dagger}-c_{w,i}\right).$
(31)

Moreover, the inverse of Jordan-Wigner transformation used to obtain Eq. (19)
derives

$\displaystyle\widetilde{d}_{i}^{\,\dagger}=\prod_{k=1}^{i-1}\prod_{k^{\prime}=b,w}(S^{z}_{k,k^{\prime}})(S_{b,i}^{x}-iS_{b,i}^{z}S_{w,i}^{y})$
(32)

Thus, the Majorana operators can be expressed by spin operators, like

$\displaystyle b_{1,i}$ $\displaystyle=$
$\displaystyle\widetilde{d}_{i}^{\dagger}+\widetilde{d}_{i}=2\prod_{k=1}^{i-1}(-S_{k,k^{\prime}}^{z})S_{b,i}^{x}$
(33) $\displaystyle w_{2,i}$ $\displaystyle=$ $\displaystyle
i(\widetilde{d}_{i}^{\dagger}-\widetilde{d}_{i})=2\prod_{k^{\prime}=w,b}(-S_{k^{\prime},k}^{z})(-S_{b,i}^{z})(-S_{w,i}^{y})$
(34)

We now confirm that the system has a free spin per unit cell at the Kitaev
points. Therefore, the dimensionality of the ground-state manifold, or number
of zero Majorana modes, is $2^{\frac{L}{2}-1}$ for periodic chain and
$2^{\frac{L}{2}}$ for open chain. Some more details are given in the Appendix.

### Phase diagram

Figure 9:  (a) Ground-state phase diagram of the 1D KH model. (b) Ground-state
phase diagram of the honeycomb-lattice KH model, where the same notations as
in Eq. (1) are used for $K$ and $J$. (c) Brick-wall lattice created by
coupling the 1D KH chains by the $z$-bond. This is topologically equivalent to
the honeycomb lattice.

In Fig. 9(a) we summarize the ground-state phase diagram as a function of
$\phi$ in a pie chart form. Notably, each phase has another exactly-
symmetrically-placed phase in the pie chart, i.e, TLL and FM-$xy$, Néel-$z$
and FM-$z$, spiral-$xy$ and staggered-$xy$. The paired phases have the common
features. The TLL and FM-$xy$ states, where the exchange components are either
all AFM or all FM, are basically understood within the framework of the
isotropic Heisenberg chain; the FM-$xy$ state is long-range ordered and the
TLL state is critical. The Néel-$z$ and FM-$z$ states are described by the
easy-axis XXZ Heisenberg chain. The dominant features are determined by the
Ising term. Depending on the sign of the Ising term, the spins are aligned
ferromagnetically or antiferromagnetically along the $z$ axis. The spiral-$xy$
and staggered-$xy$ states can be basically interpreted as the easy-plane XXZ
Heisenberg chain affected by the double-spin-flip term. The system has a four-
site periodicity, and exhibits a critical behavior and a long-range ordering
for the former and latter states, respectively, in analogy to the relation
between the TLL and FM-$xy$ states.

It is now possible to establish a connection between our ground-state phase
diagram and that of the two-dimensional KH model on a honeycomb lattice [Fig.
9(b), normalized to the current notation from [10]]. To consider the phase-to-
phase correspondence, it is helpful to regard the honeycomb lattice as coupled
KH chains, as shown in Fig. 9(c). Here, the interchain coupling is equivalent
to the $z$-bond in the honeycomb-lattice KH model, written as

${\cal H}_{z-{\rm
bond}}=\frac{J}{2}\sum_{k,l}(S^{+}_{k}S^{-}_{l}+S^{-}_{k}S^{+}_{l})+(J+K)\sum_{k,l}S^{z}_{k}S^{z}_{l},$
(35)

where $k$, $l$ are summed over all connected sites between the KH chains.

Hereafter, we report how the introduction of the $z$-bonds coupling affects
the (quasi-)ordered states present in the one-dimensional case, mapping them
to those of the honeycomb system:

(i) At $0\leq\phi<\frac{\pi}{2}$ the interchain coupling is AFM. Our TLL state
has no long-range order but strong AFM fluctuations. Therefore, a Néel order
is intuitevely expected once the chains are antiferromagnetically coupled.
Hence, our TLL phase corresponds to the Néel phase in the honeycomb case.

(ii) At $0.65\pi\lesssim\phi<\pi$ our system is in the FM-$z$ state. While
considering this interval, we need to examine two different cases as the sign
of the interaction on the $z$-bonds changes from AFM to FM. We obtain a zigzag
state in the honeycomb case when the FM-$z$ chain state is coupled by AFM
interchain couplings; whereas, a FM state in the case of FM interchain
couplings. The change in sign of the interchain coupling is assumed to happen
at $\phi=\frac{3\pi}{4}$, where $J+K=0$. This $\phi$ value is reasonably close
to the transition point $\phi\approx 0.81\pi$ between the zigzag and FM states
in the honeycomb case. Thus, our FM-$z$ phase is distributed to either zigzag
or FM phases of the honeycomb KH model depending on the AFM or FM nature on
the $z$-bond.

(iii) At $1.65\pi\lesssim\phi<2\pi$ our system is in the Néel-$z$ state. In
the same way as above, we need to consider the cases of AFM and FM $z$-bonds.
We easily find that our Néel-$z$ state is base for the Néel and stripy states
of the honeycomb KH model as also discussed in previous work [38]. Namely, our
Néel-$z$ phase is distributed to either Néel phase in the presence of AFM
interchains coupling or the stripy phase in the presence of FM interchains
coupling of the honeycomb KH model. The value of $\phi=\frac{7\pi}{4}$ giving
$J+K=0$ is again near the transition point $\phi\approx 1.65\pi$ between the
Néel and stripy states in the honeycomb case.

Therefore, it is possible to understand all the long-range ordered phases of
the honeycomb-lattice KH model in the framework of the coupled KH chains. In
brief, an extracted zigzag lattice line from any ordered state of the
honeycomb-lattice KH model corresponds to one of the phases in the 1D KH
chain; and the remaining degrees of freedom derive from the fact that the
zigzag lines are coupled either ferromagnetically or antiferromagnetically by
${\cal H}_{z-{\rm bond}}$.

(iv) Two $xy$ phases in our phase diagram are left. Since these states are
stabilized by the dominant $xy$-Kitaev term, they are connected to the Kitaev
spin liquid states in the honeycomb case. In hindsight, this means that our
staggered-$xy$ ordered state can collapse to the spin liquid state due to
strong FM fluctuations along the $z$-direction.

### Low-lying excitations

Figure 10:  Dynamical structure factors calculated with a periodic 24-site
chain for (a) TLL [$\phi=\frac{\pi}{3}$], (b) spiral-$xy$
[$\phi=\frac{5\pi}{8}]$, and (c) FM-$z$ [$\phi=\frac{7\pi}{8}$] states. The
left and right panels show $S^{z}(q,\omega)$ and $S^{-}(q,\omega)$,
respectively. The red lines are analytical dispersions: Eq. (38) for
$\phi=\frac{\pi}{3}$ and Eq. (41) for $\phi=\frac{7\pi}{8}$. The arrow
indicates an appearance of the spiral-$xy$ fluctuation in the FM-$z$ state.
Figure 11:  Dynamical structure factors calculated with a periodic 24-site
chain for (a) FM-$xy$ [$\phi=\frac{5\pi}{4}$], (b) staggered-$xy$
[$\phi=\frac{19\pi}{12}$], and (c) Néel-$z$ [$\phi=\frac{17\pi}{10}$] states.
The left and right panels show $S^{z}(q,\omega)$ and $S^{-}(q,\omega)$,
respectively. The red lines are analytical dispersions: Eq. (42) for
$\phi=\frac{5\pi}{4}$, Eq. (43) in $S^{z}(q,\omega)$ and Eq. (44) in
$S^{-}(q,\omega)$ for $\phi=\frac{19\pi}{12}$, and Eq. (45) for
$\phi=\frac{17\pi}{10}$.

To examine the low-energy excitations for each phase, we calculated the
dynamical spin structure factors $S^{z,\pm}(q,\omega)$, defined in Eq. (2).
Under periodic boundary conditions the momentum is taken as
$q=\frac{4\pi}{L}n$ ($n=0,\pm 1,\dots,\pm\frac{L}{4}$) since the unit cell
contains two lattice sites and the number of unit cells is $\frac{L}{2}$ in a
system with $L$ sites. We study chains with $L=24$ using the Lanczos ED method
and the results are shown only for $q\geq 0$ as
$S^{z,\pm}(q,\omega)=S^{z,\pm}(-q,\omega)$.

#### Tomonaga-Luttinger-liquid phase ($0\leq\phi<\pi/2$)

As confirmed above, the low-energy physics at $0\leq\phi<\frac{\pi}{2}$ is
described by a gapless TL model. The low-lying excitations are expected to be
basically equivalent to those of the easy-plane XXZ chain because
$(2J+K)/2>J>0$ [see Eq. (1D Kitaev-Heisenberg Hamiltonian)]. If we ignore the
double-spin-flip term ${\cal H}_{\rm dsf}$, the main dispersion (lower bound
of the spectrum) is given by

$\omega(q)=\frac{\pi(2J+K)}{4}\frac{\sin\mu}{\mu}\sin\frac{q}{2}$ (36)

with $\cos\mu=\left(1+\frac{K}{2J}\right)^{-1}$. This was obtained by using
the transfer matrix of the six-vertex model [54, 55]. Nevertheless, the effect
of ${\cal H}_{\rm dsf}$ is unknown. Therefore, to investigate the effect of
${\cal H}_{\rm dsf}$ we employ a standard SWT for the bipartite system. Using
the Holstein-Primakovff representation the spin operators of ${\cal H}_{\rm
dsf}$ for the black labeling sites in Fig.1(d) are replaced as
$S^{+}_{i}=\sqrt{2S-a^{\dagger}_{i}a_{i}}a_{i}$ and
$S^{-}_{i}=a_{i}^{\dagger}\sqrt{2S-a^{\dagger}_{i}a_{i}}$; similarly, for the
white labeling sites $S^{+}_{i}=b_{i}^{\dagger}\sqrt{2S-b^{\dagger}_{i}b_{i}}$
and $S^{-}_{i}=\sqrt{2S-b^{\dagger}_{i}b_{i}}b_{i}$, where $a_{i}$ and $b_{i}$
are canonical boson annihilation operators at site $i$. Then, applying the
Fourier transform we obtain

${\cal H}_{\rm
dsf}=i\frac{K}{2}\sum_{q}\sin\frac{q}{2}(a_{\frac{q}{2}}^{\dagger}b_{\frac{q}{2}}-a_{\frac{q}{2}}b_{\frac{q}{2}}^{\dagger}).$
(37)

This off-diagonal term gives a splitting in the main dispersion (36). Then, we
can speculate the total dispersions as

$\omega(q)=\left(\frac{\pi(2J+K)}{4}\frac{\sin\mu}{\mu}\pm\frac{K}{2}\right)\sin\frac{q}{2}.$
(38)

Fig. 10(a) shows the dynamical structure factors $S^{z}(q,\omega)$ and
$S^{-}(q,\omega)$ for $\phi=\frac{\pi}{3}$. The largest peak appears in
$S^{-}(q=0,\omega=0)$ reflecting the AFM fluctuations. The intensities in
$S^{z}(q,\omega)$ are weaker than those in $S^{-}(q,\omega)$ due to the easy-
plane $xy$ anisotropy. The split dispersions are well described by Eq. (38).
The splitting is largest in the vicinity of the Kitaev point
$\phi=\frac{\pi}{2}-$ ($J\to 0+$, $K\to 1$):

$\omega(q)=\left(\frac{K}{2}\pm\frac{K}{2}\right)\sin\frac{q}{2}.$ (39)

The main dispersion (lower bound of the spectrum) of the isotropic Heisenberg
model at $\phi=0$ ($K=0$) is also reproduced by Eq. (38).

#### Spiral-$xy$ phase ($\frac{\pi}{2}<\phi\lesssim 0.65\pi$)

Across the Kitaev point from the TL to spiral-$xy$ phases the momentum of the
Fermi point is shifted from $q=\pi$ to $q=0$. Thus, in the vicinity of the
Kitaev point $\phi=\frac{\pi}{2}+$ ($J\to 0-$, $K\to 1$), the momentum of the
excitation dispersion is shifted from Eq. (39) by $\pi$:

$\omega(q)=\left(\frac{K}{2}\pm\frac{K}{2}\right)\left(1-\sin\frac{q}{2}\right)$
(40)

for $S^{-}(q,\omega)$. In contrast, $S^{z}(q,\omega)$ exhibits a
$q$-independent continuum between $\omega=0$ and $\omega=K$ since the
$z$-component of exchange interaction is much smaller than the other
interactions. With increasing $\phi$ from $\frac{\pi}{2}$, the FM Ising
interaction and double-spin-flip fluctuations increase and the exchange
interaction decreases becoming zero at the critical boundary to the FM-$z$
phase. Fig. 10(b) shows the dynamical structure factors $S^{z}(q,\omega)$ and
$S^{-}(q,\omega)$ for $\phi=\frac{5}{8}\pi$. The main dispersion in
$S^{-}(q,\omega)$ is scaled as $\omega(q)\propto 1-\sin\frac{q}{2}$. Although
the largest peak at $(q,\omega)=(\pi,0)$ indicates a four-site periodicity in
the spiral-$xy$ state, it diminishes with increasing system size because of no
long-range ordering. While in $S^{z}(q,\omega)$, a dispersion
$\omega(q)\propto\sin\frac{q}{2}$ appears and a peak toward the FM-$z$ state
develops at $(q,\omega)\approx(0,0)$. Both the dispersion widths are roughly
scaled by $|K|$.

#### Ferromagnetic-$z$ phase ($0.65\pi\lesssim\phi<\pi$)

Fig. 10(c) shows the dynamical structure factors $S^{z}(q,\omega)$ and
$S^{-}(q,\omega)$ for $\phi=\frac{7}{8}\pi$. Near $\phi=\pi$ ($K\approx 0$) in
the FM-$z$ phase, the system is effectively described by a FM XXZ Heisenberg
chain with easy-axis anisotropy $J<\frac{2J+K}{2}<0$ and the ground state is
approximately expressed by Eq. (5). An applied operator $S^{z}_{q}$ does not
change the ground state $|\psi_{0}\rangle$ in Eq. (2) and the final state
$|\psi_{\nu}\rangle$ has only a zero energy excitation from
$|\psi_{0}\rangle$. Therefore, $S^{z}(q,\omega)$ has a very sharp peak at
$(q,\omega)=(0,0)$ and almost no spectral weight appears at the other momenta.
This peak keeps its weight constant with increasing system length, indicating
the long-range FM-$z$ ordering. On the other hand, in Eq. (2) the operator
$S^{-}_{q}$ dopes one magnon into the FM-$z$ alignment so that the SWT is
expected to give a good approximation for the excitation dispersion of
$S^{-}(q,\omega)$:

$\omega(q)=-J\pm\frac{2J+K}{2}\cos\frac{q}{2}.$ (41)

We can confirm that a gap $\Delta=\frac{K}{2}$ opens at $q=0$ reflecting the
easy-axis anisotropy. With approaching the neighboring spiral-$xy$ phase, the
double-spin-flip fluctuations grow gradually in influence; accordingly, the
$q=0$ peak in $S^{z}(q,\omega)$ shrinks and a peak develops at $q=\pi$,
$\omega\approx 0$ in $S^{-}(q,\omega)$.

#### Ferromagnetic-$xy$ phase ($\pi<\phi<\frac{3\pi}{2}$)

Fig. 11(a) shows the dynamical structure factors $S^{z}(q,\omega)$ and
$S^{-}(q,\omega)$ for $\phi=\frac{5}{4}\pi$. Near $\phi=\pi$ ($K\approx 0$) in
the FM-$xy$ phase, the system is effectively described by a FM XXZ Heisenberg
chain with easy-plane anisotropy $\frac{2J+K}{2}<J<0$ and the ground state is
approximately expressed by Eq. (4). Thus, the excitation spectrum is expected
to be gapless. Since the total $S^{z}$ of the ground state is zero, unlike the
case of FM-$z$ state both $S^{z}(q,\omega)$ and $S^{-}(q,\omega)$ have the
same excitation dispersion as

$\omega(q)=-\frac{2J+K}{2}\left(1\pm\cos\frac{q}{2}\right).$ (42)

Note that a sharp peak at $(q,\omega)=(0,0)$ in $S^{-}(q,\omega)$ keeps its
weight constant with increasing system length, indicating the FM-$xy$ long-
range ordering. On the other hand, in the vicinity of the Kitaev point
$\phi\approx\frac{3\pi}{2}-$, the excitation dispersion of $S^{-}(q,\omega)$
is well described by Eq. (39).

#### Staggered-$xy$ ordered phase ($\frac{3\pi}{2}<\phi\lesssim 1.65\pi$)

In the vicinity of the Kitaev point $\phi=\frac{3}{2}\pi+$, the excitation
spectra are the same as in the other Kitaev point $\phi=\frac{1}{2}\pi+$,
namely, Eq. (40) for $S^{-}(q,\omega)$ and a $q$-independent continuum for
$S^{z}(q,\omega)$. Whereas in the vicinity of the Néel-$z$ ordered phase
$\phi=\tan^{-1}(-2)-$, the ground state is expressed by Eq. (11). Thus, the
dispersions are described by a single magnon excitation:

$\omega(q)=-\frac{K}{2}\left(1\pm\cos\frac{q}{2}\right)$ (43)

for $S^{z}(q,\omega)$, and

$\omega(q)=-\frac{K}{2}\left(1\pm\sin\frac{q}{2}\right)$ (44)

for $S^{-}(q,\omega)$. Fig. 11(b) shows the dynamical structure factors
$S^{z}(q,\omega)$ and $S^{-}(q,\omega)$ for $\phi=\frac{19\pi}{12}$. The
excitation dispersions are well reproduced by Eqs. (43) and (44). We also find
a sharp peak at $(q,\omega)=(\pi,0)$ in $S_{-}(q,\omega)$, which indicates the
staggered-$xy$ long-range ordering. This peak keeps its weight constant with
increasing the system length, in contrast to the similar peak for the
spiral-$xy$ state.

#### Néel-$z$ ordered phase ($1.65\pi\lesssim\phi<2\pi$)

In the vicinity of the staggered-$xy$ ordered phase $\phi=\tan^{-1}(-2)+$, the
ground state is expressed by Eq. (12). Accordingly, $S^{z}(q,\omega)$ has only
a delta peak at $(q,\omega)=(0,0)$. Whereas, $S^{-}(q,\omega)$ is exactly
explained by a single magnon dispersion. It is obtained by the SWT as

$\omega(q)=J\pm\frac{K}{2}\sin\frac{q}{2}.$ (45)

Although this is equivalent to Eq. (43), the spectral weight is uniform for
all $q$ values. Since the transition at $\phi=\tan^{-1}(-2)$ is first ordered,
there is no peak indicating a connection to the staggered-$xy$ ordered phase.
On the other hand, near $\phi=2\pi$, the system can be basically regarded as
an easy-axis AFM XXZ Heisenberg chain so that the excitation dispersion is
described by Eq. (38) for both $S^{z}(q,\omega)$ and $S^{-}(q,\omega)$. Fig.
11(c) shows the dynamical structure factors $S^{z}(q,\omega)$ and
$S^{-}(q,\omega)$ for the intermediate region $\phi=\frac{17}{10}\pi$. The
main dispersion of $S^{-}(q,\omega)$ is basically described by Eq. (45) but
some features from Eq. (38) seem to be somewhat mixed.

## Discussion

We have established the presence of a variety of phases in the 1D KH system.
Especially, it is surprising that most of the $\phi$ ranges are covered by
long-range ordered phases despite considering a pure 1D system. In this
context, we now consider the K-intercalated $\alpha$-RuCl3, namely, K0.5RuCl3.
One should be aware of the fact that several different parameter sets have
been suggested for undoped $\alpha$-RuCl3: (i) $K=-5.6$, $J=1.2$ ($\phi\approx
1.57\pi$) [25], (ii) $K=7.0$ ,$J=4.6$ ($\phi\approx 0.68\pi$) [22], (iii)
$K=8.1$, $J=2.9$ ($\phi\approx 0.61\pi$) [22], (iv) $K=-6.8$, $J=0$
($\phi=1.5\pi$) [56] in unit of meV. If we assume that the charge ordering
pattern in K0.5RuCl3 is that illustrated in Fig. 1(c), the parameter sets
(i)-(iv) correspond to the staggered-$xy$, FM-$z$, spiral-$xy$, and FM Kitaev
point, respectively. In practice, the charge ordering could cause a
significant change of the parameter since they are very sensitive to the Ru-
Cl-Ru bond angle [25]. In other words, once the magnetic properties of
K0.5RuCl3 are observed, we may easily speculate the possible parameter set of
K0.5RuCl3 and even the charge ordering pattern by comparing then to our rich
phase diagram. To gain deeper insights, theoretical and experimental studies
under magnetic fields are also required.

It is relevant to seek other possible realizations of 1D KH system. Even if
the Kitaev interaction in 1D systems is present, it is considerede to be very
small. However, as shown above, even a tiny Kitaev interaction can stabilize
the ordered state. It might also be intriguing to reconsider quasi-1D
materials having two sublattices like Ni2(EDTA)(H2O)4, [Ni(f-rac-L)(CN)2],
LiCuSbO4, and Rb2Cu2Mo3O12 from the point of view of the 1D Kitaev system.

The $\phi$-dependent phase diagram of our model is similar to that for the
honeycomb-lattice KH model. Remarkably, all the magnetically ordered states of
the honeycomb-lattice KH model can be interpreted in terms of the coupled 1D
KH chains. In other words, the key elements to derive the magnetic ordering in
the 2D honeycomb-lattice KH model are already contained in the 1D KH chain.
This will give us a deeper insight in the understanding of transition from an
ordered state to a disordered state such as the Kitaev QSL.

## Methods

In this section, we present a detailed derivation of (33), (34), and the
related additional information. We used a Jordan-Wigner transformation

$\displaystyle S_{i}^{z}$ $\displaystyle=2c_{i}^{\dagger}c_{i}-1$ (46)
$\displaystyle S_{i}^{+}$
$\displaystyle=\prod_{k=1}^{i-1}(-S^{z}_{k})c_{i}^{\dagger}$ (47)
$\displaystyle S_{i}^{-}$ $\displaystyle=\prod_{k=1}^{i-1}(-S^{z}_{k})c_{i}$
(48)

to rewrite the spin operators in Eq. (17) by fermion operators, and we
obtained Eq. (19).

By applying the inverse of the Jordan-Wigner transformation

$\displaystyle c_{w,i}^{\dagger}$
$\displaystyle=\prod_{k=1}^{i-1}\prod_{k^{\prime}=b,w}(-S^{z}_{k,k^{\prime}})(-S^{z}_{b,i})S_{w,i}^{+}$
(49) $\displaystyle c_{w,i}$
$\displaystyle=\prod_{k=1}^{i-1}\prod_{k^{\prime}=b,w}(-S^{z}_{k,k^{\prime}})(-S^{z}_{b,i})S_{w,i}^{-}$
(50) $\displaystyle c_{b,i}^{\dagger}$
$\displaystyle=\prod_{k=1}^{i-1}\prod_{k^{\prime}=b,w}(-S^{z}_{k,k^{\prime}})S_{b,i}^{+}$
(51) $\displaystyle c_{b,i}$
$\displaystyle=\prod_{k=1}^{i-1}\prod_{k^{\prime}=b,w}(-S^{z}_{k,k^{\prime}})S_{b,i}^{-},$
(52)

the fermion operator $\widetilde{d}_{i}^{\,\dagger}$ was traced back to the
initial spin operators, and we obtained Eq. (32) from Eq. (31).

We can also derive the same results as Eqs. (33) and (34) by taking a
different combination of the Majoranas from Eqs. (29) and (30):

$\displaystyle B_{i}^{\dagger}$ $\displaystyle=\frac{b_{1,i}-ib_{1,i+1}}{2}$
(53) $\displaystyle W_{i}^{\dagger}$
$\displaystyle=\frac{w_{2,i}-iw_{2,i+1}}{2}$ (54)

From Eqs. (20) and (21), we have

$\displaystyle B_{i}^{\dagger}$
$\displaystyle=\frac{1}{2}(c_{b,i}^{\dagger}+c_{b,i}-ic_{b,i+1}^{\dagger}-ic_{b,i+1})$
(55) $\displaystyle W_{i}^{\dagger}$
$\displaystyle=\frac{1}{2}(c_{w,i+1}^{\dagger}-c_{w,i+1}+ic_{w,i}^{\dagger}-ic_{w,i})$
(56)

Using (50)-(52), we obtain

$\displaystyle B_{i}^{\dagger}$
$\displaystyle=\prod_{k=1}^{i-1}\prod_{k^{\prime}=b,w}(-S_{k^{\prime},k})\left[S_{b,i}^{x}-i(-S_{k^{\prime},i}^{z})S_{b,i+1}^{x}\right]$
(57) $\displaystyle B_{i}$
$\displaystyle=\prod_{k^{\prime}=b,w}(-S_{k^{\prime},k})(S_{b,i}^{x}-i(-S_{k^{\prime},i}^{z})S_{b,i+1}^{x})$
(58)

Then, we obtain

$b_{1,i}=B_{i}^{\dagger}+B_{i}=2\prod_{k=1}^{i-1}(-S_{k,k^{\prime}}^{z})S_{b,i}^{x}.$
(59)

In a similar way, we find

$w_{2,i}=W_{i}^{\dagger}+W_{i}=2\prod_{k=1}^{i-1}\prod_{k^{\prime}=w,b}(-S_{k^{\prime},k}^{z})(-S_{b,i}^{z})(-S_{w,i}^{y}).$
(60)

Actually, Eqs. (59) and (60) are identical to Eqs. (33) and (34).

## References

  * [1] Anderson, P. W.  Resonating valence bonds: A new kind of insulator?  _Materials Research Bulletin_ 8, 153–160 (1973).
  * [2] Moessner, R. & Ramirez, A. P.  Geometrical frustration.  _Physics Today_ 59, 24–29 (2006).
  * [3] Balents, L.  Spin liquids in frustrated magnets.  _Nature_ 464, 199–208 (2010).
  * [4] Kitaev, A.  Anyons in an exactly solved model and beyond.  _Annals of Physics_ 321, 2–111 (2006).
  * [5] Nussinov, Z. & van den Brink, J.  Compass models: Theory and physical motivations.  _Reviews of Modern Physics_ 87, 1–59 (2015).
  * [6] Khaliullin, G.  Orbital order and fluctuations in Mott insulators.  _Progress of Theoretical Physics Supplement_ 160, 155–202 (2005).
  * [7] Jackeli, G. & Khaliullin, G.  Mott insulators in the strong spin-orbit coupling limit: From Heisenberg to a quantum compass and Kitaev models.  _Physical Review Letters_ 102 (2009).
  * [8] Trebst, S.  Kitaev materials.  _Arxiv eprint_ (2017).  arxiv:1701.07056.
  * [9] Savary, L. & Balents, L.  Quantum spin liquids: a review.  _Reports on Progress in Physics_ 80, 016502 (2016).
  * [10] Chaloupka, J., Jackeli, G. & Khaliullin, G.  Kitaev-Heisenberg model on a honeycomb lattice: Possible exotic phases in iridium oxides A2IrO3.  _Physical Review Letters_ 105 (2010).
  * [11] Singh, Y. & Gegenwart, P.  Antiferromagnetic Mott insulating state in single crystals of the honeycomb lattice material Na2IrO3.  _Physical Review B_ 82 (2010).
  * [12] Ye, F. _et al._ Direct evidence of a zigzag spin-chain structure in the honeycomb lattice: A neutron and x-ray diffraction investigation of single-crystal Na2IrO3.  _Physical Review B_ 85 (2012).
  * [13] Choi, S. K. _et al._ Spin waves and revised crystal structure of honeycomb iridate Na2IrO3.  _Physical Review Letters_ 108 (2012).
  * [14] Takayama, T. _et al._ Hyperhoneycomb Iridate $\beta$-Li2IrO3 as a platform for Kitaev magnetism.  _Physical Review Letters_ 114 (2015).
  * [15] Plumb, K. W. _et al._ $\alpha$-RuCl3: A spin-orbit assisted Mott insulator on a honeycomb lattice.  _Physical Review B_ 90 (2014).
  * [16] Rousochatzakis, I., Reuther, J., Thomale, R., Rachel, S. & Perkins, N. B.  Phase diagram and quantum order by disorder in the Kitaev ${K}_{1}-{K}_{2}$ honeycomb magnet.  _Phys. Rev. X_ 5, 041035 (2015).
  * [17] Kim, H.-S., V., V. S., Catuneanu, A. & Kee, H.-Y.  Kitaev magnetism in honeycomb RuCl3 with intermediate spin-orbit coupling.  _Physical Review B_ 91 (2015).
  * [18] Sears, J. A. _et al._ Magnetic order in $\alpha$-RuCl3: A honeycomb-lattice quantum magnet with strong spin-orbit coupling.  _Physical Review B_ 91 (2015).
  * [19] Majumder, M. _et al._ Anisotropic Ru3+ 4d5 magnetism in the $\alpha$-RuCl3 honeycomb system: Susceptibility, specific heat, and zero-field NMR.  _Physical Review B_ 91 (2015).
  * [20] Kubota, Y., Tanaka, H., Ono, T., Narumi, Y. & Kindo, K.  Successive magnetic phase transitions in $\alpha$-RuCl3: XY-like frustrated magnet on the honeycomb lattice.  _Physical Review B_ 91 (2015).
  * [21] Nasu, J., Knolle, J., Kovrizhin, D. L., Motome, Y. & Moessner, R.  Fermionic response from fractionalization in an insulating two-dimensional magnet.  _Nature Physics_ 12, 912–915 (2016).
  * [22] Banerjee, A. _et al._ Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet.  _Nature Materials_ 15, 733–740 (2016).
  * [23] Sandilands, L. J., Tian, Y., Plumb, K. W., Kim, Y.-J. & Burch, K. S.  Scattering continuum and possible fractionalized excitations in $\alpha$-RuCl3.  _Physical Review Letters_ 114 (2015).
  * [24] Nasu, J., Motome, Y. & Yoshitake, J.  Thermal transport in the Kitaev model.  _ArXiv e-prints_ (2017).  arXiv:1703.10395.
  * [25] Yadav, R. _et al._ Kitaev exchange and field-induced quantum spin-liquid states in honeycomb $\alpha$-RuCl3.  _Scientific Reports_ 6 (2016).
  * [26] Baek, S.-H. _et al._ Evidence for a field-induced quantum spin liquid in $\alpha$-RuCl3.  _Physical Review Letters_ 119 (2017).
  * [27] Wolter, A. U. B. _et al._ Field-induced quantum criticality in the Kitaev system $\alpha$-RuCl3.  _Phys. Rev. B_ 96, 041405 (2017).
  * [28] Hentrich, R. _et al._ Large field-induced gap of Kitaev-Heisenberg paramagnons in $\alpha$-RuCl3.  _ArXiv e-prints_ (2017).  arXiv:1703.08623.
  * [29] Zheng, J. _et al._ Gapless spin excitations in the field-induced quantum spin liquid phase of $\alpha$-RuCl3.  _ArXiv e-prints_ (2017).  arXiv:1703.08474.
  * [30] Leahy, I. A. _et al._ Anomalous thermal conductivity and magnetic torque response in the honeycomb magnet $\alpha$-RuCl3.  _Phys. Rev. Lett._ 118, 187203 (2017).
  * [31] Zhou, X. _et al._ Angle-resolved photoemission study of the Kitaev candidate $\alpha$-RuCl3.  _Phys. Rev. B_ 94, 161106 (2016).
  * [32] Wang, Z. _et al._ Observation of the quantum spin liquid state in pressurized $\alpha$-RuCl3.  _ArXiv e-prints_ (2017).  arXiv:1705.06139.
  * [33] Veiga, L. S. I. _et al._ Pressure-tuning of bond-directional exchange interactions and magnetic frustration in hyperhoneycomb iridate $\beta$-Li2IrO3.  _ArXiv e-prints_ (2017).  arXiv:1705.07059.
  * [34] Breznay, N. P. _et al._ Resonant x-ray scattering reveals possible disappearance of magnetic order under hydrostatic pressure in the Kitaev candidate $\gamma$-Li2IrO3.  _Phys. Rev. B_ 96, 020402 (2017).
  * [35] Koitzsch, A. _et al._ ${J}_{\mathrm{eff}}$ description of the honeycomb Mott insulator $\alpha$-RuCl3.  _Phys. Rev. Lett._ 117, 126403 (2016).
  * [36] Sandilands, L. J. _et al._ Spin-orbit excitations and electronic structure of the putative Kitaev magnet $\alpha$-RuCl3.  _Phys. Rev. B_ 93, 075144 (2016).
  * [37] Sandilands, L. J. _et al._ Optical probe of Heisenberg-Kitaev magnetism in $\alpha$-RuCl3.  _Phys. Rev. B_ 94, 195156 (2016).
  * [38] Sela, E., Jiang, H.-C., Gerlach, M. H. & Trebst, S.  Order-by-disorder and spin-orbital liquids in a distorted heisenberg-kitaev model.  _Phys. Rev. B_ 90, 035113 (2014).
  * [39] Brzezicki, W., Dziarmaga, J. & Oleś, A. M.  Quantum phase transition in the one-dimensional compass model.  _Phys. Rev. B_ 75, 134415 (2007).
  * [40] Mondal, S., Sen, D. & Sengupta, K.  Quench dynamics and defect production in the Kitaev and extended Kitaev models.  _Phys. Rev. B_ 78, 045101 (2008).
  * [41] Divakaran, U. & Dutta, A.  Reverse quenching in a one-dimensional Kitaev model.  _Phys. Rev. B_ 79, 224408 (2009).
  * [42] Eriksson, E. & Johannesson, H.  Multicriticality and entanglement in the one-dimensional quantum compass model.  _Phys. Rev. B_ 79, 224424 (2009).
  * [43] Mahdavifar, S.  Numerical study of the one-dimensional quantum compass model.  _The European Physical Journal B_ 77, 77–82 (2010).
  * [44] Subrahmanyam, V.  Block entropy for Kitaev-type spin chains in a transverse field.  _Phys. Rev. A_ 88, 032315 (2013).
  * [45] Katsura, H., Schuricht, D. & Takahashi, M.  Exact ground states and topological order in interacting Kitaev/Majorana chains.  _Phys. Rev. B_ 92, 115137 (2015).
  * [46] Steinigeweg, R. & Brenig, W.  Energy dynamics in the Heisenberg-Kitaev spin chain.  _Phys. Rev. B_ 93, 214425 (2016).
  * [47] Ziatdinov, M. _et al._ Atomic-scale observation of structural and electronic orders in the layered compound $\alpha$-RuCl3.  _Nature Communications_ 7, 13774 (2016).
  * [48] White, S. R.  Density matrix formulation for quantum renormalization groups.  _Phys. Rev. Lett._ 69, 2863–2866 (1992).
  * [49] A.O., G., A.A., N. & Tsvelik, A.  _Bosonization and Strongly Correlated Systems_ (Cambridge University Press, Cambridge, 1998).
  * [50] Affleck, I. & Ludwig, A. W. W.  Universal noninteger “ground-state degeneracy” in critical quantum systems.  _Phys. Rev. Lett._ 67, 161–164 (1991).
  * [51] Holzhey, C., Larsen, F. & Wilczek, F.  Geometric and renormalized entropy in conformal field theory.  _Nuclear Physics B_ 424, 443–467 (1994).
  * [52] Calabrese, P. & Cardy, J.  Entanglement entropy and quantum field theory.  _Journal of Statistical Mechanics: Theory and Experiment_ 2004, P06002 (2004).
  * [53] Kitaev, A. Y.  Unpaired Majorana fermions in quantum wires.  _Physics-Uspekhi_ 44, 131–136 (2001).
  * [54] Johnson, J. D., Krinsky, S. & McCoy, B. M.  Vertical-arrow correlation length in the eight-vertex model and the low-lying excitations of the $x-y-z$ Hamiltonian.  _Phys. Rev. A_ 8, 2526–2547 (1973).
  * [55] Schneider, T., Stoll, E. & Glaus, U.  Excitation spectrum of planar spin-1/2 Heisenberg $\mathrm{XXZ}$ chains.  _Phys. Rev. B_ 26, 1321–1330 (1982).
  * [56] Ran, K. _et al._ Spin-wave excitations evidencing the Kitaev interaction in single crystalline ${\alpha}$-RuCl3.  _Phys. Rev. Lett._ 118, 107203 (2017).

## Acknowledgements

We thank U. Nitzsche for technical assistance. C.E.A. thanks A. Lau for useful
discussion. This work is supported by SFB 1143 of the Deutsche
Forschungsgemeinschaft.