# Kekule spin-orbit dimer phase and triplon dynamics

GiBaik Sim  Department of Physics TQM, Technische Universität München, $\&$
James-Franck-Straße 1, D-85748 Garching, Germany  Munich Center for Quantum
Science and Technology (MCQST), 80799 Munich, Germany

###### Abstract

We derive and study a spin-orbital model for ions with $d^{1}$ electronic
configuration on a honeycomb lattice. In this system, the directional
character of $t_{2g}$ orbital leads to extensively degenerate dimerized ground
states. We find that additional interactions from charge transfer processes
completely lift the degeneracy and stabilize the Kekule spin-orbit dimerized
phase where dimers form a kagome superlattice. For such phase, the triplon
band spectrum resembles the electronic band structure of the kagome lattice
and becomes topologically non-trivial in the presence of inter-dimer
Dzyaloshinskii-Moriya interactions. As an experimental verification of the
Kekule dimerized phase, we propose the thermal Hall experiment, which can
directly uncover the topological profile of the corresponding triplon band
spectrum.

Isotropic spin models on two (three) dimensional lattice often have long-range
ordered ground states with gapless collective excitation modes [1]. However,
this scenario fails for the geometrically frustrated cases whose classical
ground states are (sub-)extensively degenerate [2, 3, 4]. For such cases, the
ground state manifold responds to the following effects in various ways: It
can become a highly entangled quantum spin liquid state with quantum
fluctuations [5, 6], lift the degeneracy through the order by disorder
mechanism [7, 8], which often selects less entangled magnetically ordered
state, or encounter structural phase transition [9, 10], which reduces the
symmetry of the lattice.

The concept of classical degeneracy can be applied to a spin-orbital model
where the ground state manifold is formed by an extensive number of dimer
coverings [11]. Here, the degeneracy originates from the directional character
of $t_{2g}$ orbitals, which frustrate the spin interactions even on bipartite
lattices. Recently, the spin-orbital model has been applied to many materials
with dimerized ground state such as $\alpha-$RuCl3 in the presence of high
pressure [12], Ba2YMoO6 with magnetically active Mo5+ ($d^{1}$) ions on fcc
lattice [13, 14], and Li2RuO3 with Ru4+ ($d^{2}$) ions on honeycomb lattice
[15]. In fact, Li2RuO3 shows a specific crystallized dimer ground state with a
herringbone pattern which is well captured by the presence of magnetoelastic
coupling [16].

In this work, we point out an electronic degeneracy lifting mechanism that can
be relevant to $\alpha-$MX3 with M = Ti, Zr, Hf and X = F, Cl, Br where cation
ions (M) with $d^{1}$ electronic configuration occupy a honeycomb lattice
[17]. By introducing charge transfer contributions, we show that the
degeneracy of dimer covering is completely lifted and the system stabilizes
Kekule spin-orbit dimerized phase. To investigate the dynamics of the phase,
we introduce the triplon operator and investigate the corresponding triplon
band spectrum, which turns out to be topologically non-trivial even with
infinitesimal Dzyaloshinskii-Moriya interactions (DMI). We also investigate
the thermal response and find that the topological properties are directly
encoded in the thermal Hall conductivity.

Figure 1: (color online) Possible hopping processes in $\alpha-$MX3 which
includes direct hopping (green), charge transfer (blue), and cyclic exchange
(red) contribution. The magnetically active M sites with $d^{1}$ electronic
configuration are located at the center of octahedral cage. Each cage is
formed by X sites with $p^{6}$ electronic configuration. Indirect hopping
paths between M and X sites are marked as dotted curved lines.

We first introduce $\alpha-$MX3, a two dimensional edge-shared octahedral
system with two types of atoms. The honeycomb geometry is formed by
magnetically active cation ions (M) with $d^{1}$ electronic configuration.
Each M site is surrounded by an edge-shared octahedral cage, which is formed
by anion ions (X) with $p^{6}$ electronic configuration, as shown in Fig. 1.
We focus on a case with strong crystal field splitting where one electron
resides in threefold degenerate $t_{2g}$ orbitals at each M site. In the
$\alpha-$MX3 crystal structure, three distinct nearest-neighbor bonds are
formed in $yz$, $zx$, and $xy$ planes, respectively. We label such bonds and
the $t_{2g}$ orbitals with an axis $\gamma~{}(=a,b,c)$ normal to its plane:
$a$ indicates $yz$. We first consider the leading hopping process between
neighboring $t_{2g}$ orbitals due to direct hybridization which is illustrated
as a green bond in Fig. 1. In $\gamma$ plane, the hopping occurs between
$t_{2g}$ orbitals of same $\gamma$ type. The corresponding low energy
Hamiltonian for such process reads as the following form in the limit of zero
Hund’s coupling [11, 18].

$\displaystyle H_{d}\\!$ $\displaystyle=$ $\displaystyle\\!J_{d}\sum_{\langle
ij\rangle\parallel\gamma}\Big{(}\vec{S}_{i}\\!\cdot\\!\vec{S}_{j}+\frac{1}{4}\Big{)}n_{i}^{\gamma}n_{j}^{\gamma}$
(1)

where $\langle ij\rangle\\!\\!\parallel\\!\\!\gamma$ indicates $\gamma$ type
bond. Here, $\vec{S}_{i}$ are spin-$\frac{1}{2}$ operators, $n_{i}^{\gamma}$
is the number operator for orbital $\gamma$, and $J_{d}=4t_{d}^{2}/U$ where
$t_{d}$ is the amplitude of direct hopping and $U$ is the Coulomb repulsion at
a M site.

The antiferromagnetic term $H_{d}$ acts on the $\gamma$ type bonds only when
both sites involved are occupied by an electron with $\gamma$ orbital.
Accordingly, the antiferromagnetic bonds form non-intersecting linear chains
with specific orbital patterns. As the length of the chain grows, a negative
energy gain from the antiferromagnetic spin interactions increases. At the
same time, each bond gains a positive energy, $\frac{1}{4}$, from the second
term in Eq. 1. Under these two conditions, the minimum energy of the system is
achieved when all chains are dimers as described in Ref. [11]. Therefore, a
honeycomb lattice covered with hard-core dimers corresponds to an exact ground
state of $H_{d}$. Within a dimer, two electrons occupy the same orbital, which
matches the direction of the bond, and form a spin-singlet. The $H_{d}$ term
is inactive on every inter-dimer bond and thus the ground state manifold
becomes extensively degenerate: there are infinitely many different ways to
cover the lattice with dimers. In the following, we consider an additional
hopping process which results in inter-dimer interactions that completely lift
the degeneracy and stabilize a valence bond crystal state.

In addition to direct hopping between neighboring $t_{2g}$ orbitals, there
exist different exchange mechanisms: charge transfer and cyclic exchange
processes [19]. We first consider $pd$ charge transfer process,
$d^{1}_{i}\\!-\\!p^{6}\\!-\\!d^{1}_{j}\rightarrow
d^{2}_{i}\\!-\\!p^{4}\\!-\\!d^{2}_{j}$, where two holes are created on an X
site, which connects two neighboring M sites as illustrated in Fig. 1 with a
blue bond. The relative energy of a virtual state is $2U+2\Delta-9U_{p}$ where
$\Delta$ is $p\\!-\\!d$ charge transfer gap and $U_{p}$ is Coulomb repulsion
at an X site. Collecting possible charge transfer contributions, we obtain the
spin-orbital Hamiltonian

$\displaystyle H_{ct}\\!=\\!J_{ct}\sum_{\langle
ij\rangle\parallel\gamma}\\!\\!\Big{(}\vec{S}_{i}\\!\cdot\\!\vec{S}_{j}-\frac{1}{4}\Big{)}O^{\gamma}.$
(2)

where $O^{c}\\!=\\!n_{i}^{a}n_{j}^{b}\\!+\\!n_{i}^{b}n_{j}^{a}$ for a $c$ type
bond (See Section I of SM for details). Here,
$J_{ct}=2t_{s}^{2}/(2U+2\Delta-9U_{p})$ and $t_{s}$ is the amplitude of
hopping, which takes place via intermediate $p$ orbitals. Unlike the charge
transfer process, the cyclic exchange process,
$d^{1}\\!-\\!(p^{6},p^{6})\\!-\\!d^{1}\rightarrow
d^{2}\\!-\\!(p^{5},p^{5})\\!-\\!d^{2}$, involves two X sites where a hole is
created at each X site in the virtual state as shown in Fig. 1 with a red
bond. The cyclic exchange Hamiltonian is expressed as

$\displaystyle H_{ce}\\!=\\!-J_{ce}\sum_{\langle
ij\rangle\parallel\gamma}\Big{(}\vec{S}_{i}\\!\cdot\\!\vec{S}_{j}+\frac{1}{4}\Big{)}\tilde{O}^{\gamma}$
(3)

where
$\tilde{O}^{c}=\Big{(}d^{\dagger,b}_{i}d_{i}^{a}d^{\dagger,b}_{j}d_{j}^{a}+d^{\dagger,a}_{i}d_{i}^{b}d^{\dagger,a}_{j}d_{j}^{b}\Big{)}$
for a $c$ type bond and $J_{ce}=2t_{s}^{2}/(2U+2\Delta-10U_{p})$.

Figure 2: (color online) (a,b) Two different types of inter-dimer bonds,
$b_{1}$ and $b_{2}$, which are shown as green dashed lines. Each spin-orbital
dimer is represented as a thick line and is colored according to the orbital
it occupies. $b_{1}$ couple two dimers with the same orbital. Meanwhile, the
bond $b_{2}$ couple dimers with different orbitals. (c) Kekule dimerized
pattern which shows the modification of particular bonds of the honeycomb
lattice. (d) Plot of the per site energy $E_{0}$, spin correlation
$\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle$, and electron density, $\langle
n_{i}^{a}\rangle$ and $\langle n_{j}^{a}\rangle$ for the ground state on YC-6
cylinder. Here, $i$ and $j$ are nearest neighboring sites and connected
through $yz$ type bond as shown in (c).

$H_{ct}$ generates interactions between dimers and, as it follows, selects a
particular superstructure of dimers, a Kekule dimerization pattern. We assume
that the charge transfer term is small and treat $H_{ct}$ perturbatively.
Figs. 2(a) and 2(b) contain two different green dotted inter-dimer bonds,
$b_{1}$ and $b_{2}$, which can be formed in the ground state of $H_{d}$.
$H_{ct}$ is active only on inter-dimer bonds whose two associated dimers have
different orbital indices as $b_{2}$ shown in Fig. 2(b). At the same time,
such bonds gain negative energy, $-\frac{1}{4}$, from the last term in Eq.
(2). Accordingly, the system prefers a specific type of dimer orientation,
resulting in Kekule dimerized ground state as shown in Fig. 2(c).

To check the stability of the Kekule dimerized state, we have performed
infinite density matrix renormalization group (iDMRG) simulation for the
Hamiltonian, $H_{t}\equiv H_{d}+H_{ct}+H_{ce}$ with $J_{d}\\!=\\!1$ and
$\tilde{J}\\!\equiv\\!J_{ct}\\!=\\!2J_{ce}/3$, on YC-2$L_{y}$ cylinders with
circumference $L_{y}\\!=\\!2$ and 3, where the periodic boundary condition
(PBC) is applied along the $y$ direction (See Section II of SM for details).
To find a ground state for each $\tilde{J}$, we compared the per site ground
state energy of YC-4 cylinder with the one of YC-6 cylinder. We find that the
latter is energetically more stable at least up to
$\tilde{J}\\!\approx\\!1/2$. In Fig. 2(d), we plot the per site energy
$E_{0}$, nearest-neighbor spin correlation
$\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle$, and electron density in the $yz$
orbital, $\langle n_{i}^{a}\rangle$ and $\langle n_{j}^{a}\rangle$, for a
given dimerized $ij$ bond, which is $yz$ type. It indicates the stabilization
of spin-orbit Kekule dimer state and small hybridization of spin-singlet and
spin-triplet state within the dimer. Note that the geometry of YC-4 cylinder
cannot support the Kekule dimerized state. We also performed exact
diagonalization (ED) on a cluster of six sites using PBC. The result of ED
also shows stabilization of the Kekule dimer state at least up to
$\tilde{J}\\!\approx\\!1/2$ (See Section III of SM for details). Below, we
focus on a regime where direct hopping contributions dominate the system,
$\tilde{J}/J_{d}=1/8$.

Having realized the crystallized dimer ground state, we now find the
excitation spectrum by introducing auxiliary bosons on each dimer [20, 21,
22]. We define the operator $s^{\dagger,a}$ which creates the spin-singlet
state
$|s\rangle=(|\uparrow_{L}\downarrow_{R}\rangle-|\downarrow_{L}\uparrow_{R}\rangle)/\sqrt{2}$
and $t^{\dagger,a}_{1}$, $t^{\dagger,a}_{0}$, and $t^{\dagger,a}_{-1}$ which
create the spin-triplet states
$|t_{1}\rangle=|\uparrow_{L}\uparrow_{R}\rangle$,
$|t_{0}\rangle=(|\uparrow_{L}\downarrow_{R}\rangle+|\downarrow_{L}\uparrow_{R}\rangle)/\sqrt{2}$,
and $|t_{-1}\rangle=|\downarrow_{L}\downarrow_{R}\rangle$, respectively, where
$a$ labels the orbital index of dimer. Then, spin operators, which act on an
electron with orbital $a$, read:

$\displaystyle S^{+}_{i}n_{i}^{a}$ $\displaystyle=$
$\displaystyle\frac{t^{\dagger,a}_{1,i}t^{a}_{0,i}+t^{\dagger,a}_{0,i}t^{a}_{-1,i}}{\sqrt{2}}\pm\frac{s^{\dagger,a}_{i}t^{a}_{-1,i}-t^{\dagger,a}_{1,i}s^{a}_{i}}{\sqrt{2}}\;,$
$\displaystyle S^{-}_{i}n_{i}^{a}$ $\displaystyle=$
$\displaystyle\frac{t^{\dagger,a}_{-1,i}t^{a}_{0,i}+t^{\dagger,a}_{0,i}t^{a}_{1,i}}{\sqrt{2}}\mp\frac{s^{\dagger,a}_{i}t^{a}_{1,i}-t^{\dagger,a}_{-1,i}s^{a}_{i}}{\sqrt{2}}\;,$
$\displaystyle S^{z}_{i}n_{i}^{a}$ $\displaystyle=$
$\displaystyle\frac{t^{\dagger,a}_{1,i}t^{a}_{1,i}-t^{\dagger,a}_{-1,i}t^{a}_{-1,i}}{2}\pm\frac{s^{\dagger,a}_{i}t^{a}_{0,i}+t^{\dagger,a}_{0,i}s^{a}_{i}}{2}$
(4)

where $i$ indicates the position of the dimer and the upper (lower) sign is
for the left (right) spin within the dimer.

Kekule state is a product of spin-singlet at each dimer, and thus, the density
of triplon is zero. So the mean-field theory, which abandons dynamics of
singlet operators, is applicable. By replacing these operators with
expectation value, $\langle s^{\dagger}_{i}\rangle=\langle s_{i}\rangle=1$,
and neglecting high order terms, $H_{t}$ is rewritten in terms of triplon
operators in momentum space as

$\displaystyle\mathcal{H}_{\mathbf{k}}=\sum_{\mathbf{k}}\left(\\!\\!\begin{array}[]{c}\mathbf{t}_{1,{\bf
k}}^{\dagger}\\\ \mathbf{t}_{1,-\\!{\bf k}}^{\phantom{\dagger}}\\\
\mathbf{t}_{0,{\bf k}}^{\dagger}\\\ \mathbf{t}_{0,-\\!{\bf
k}}^{\phantom{\dagger}}\end{array}\\!\right)\\!\\!\left(\\!\\!\begin{array}[]{cccc}M_{1,\mathbf{k}}&N_{1,\mathbf{k}}&\mathbf{0}&\mathbf{0}\\\
N_{1,\mathbf{k}}&M_{1,\mathbf{k}}&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&M_{0,\mathbf{k}}&N_{0,\mathbf{k}}\\\
\mathbf{0}&\mathbf{0}&N_{0,\mathbf{k}}&M_{0,\mathbf{k}}\end{array}\\!\\!\right)\\!\left(\\!\\!\begin{array}[]{c}\mathbf{t}_{1,{\bf
k}}^{\phantom{\dagger}}\\\ \mathbf{t}_{1,-\\!{\bf k}}^{\dagger}\\\
\mathbf{t}_{0,{\bf k}}^{\phantom{\dagger}}\\\ \mathbf{t}_{0,-\\!{\bf
k}}^{\dagger}\end{array}\\!\right)$ (17) (18)

where $\mathbf{t}_{1,{\bf k}}^{\dagger}=(t_{1,{\bf k}}^{\dagger,a},t_{{1,{\bf
k}}}^{\dagger,b},t_{{1,{\bf k}}}^{\dagger,c},t_{{-1,{\bf
k}}}^{\dagger,a},t_{{-1,{\bf k}}}^{\dagger,b},t_{{-1,{\bf k}}}^{\dagger,c})$
and $\mathbf{t}_{0,{\bf k}}^{\dagger}=(t_{{0,{\bf k}}}^{\dagger,a},t_{{0,{\bf
k}}}^{\dagger,b},t_{{0,{\bf k}}}^{\dagger,c})$. The Bogoliubov–de Gennes
Hamiltonian, $\mathcal{H}_{\mathbf{k}}$, is separated into two blocks which
make manifest the absence of terms mixing spinful $|t_{\pm 1}\rangle$ triplon
and spinless $|t_{0}\rangle$ triplon. $\mathcal{H}_{\mathbf{k}}$ can be
conveniently expressed by using 8 Gell-Mann matrices, $\lambda_{i}$, as the
basis for orbital indices $a,b$, and $c$ (See Section IV of SM for the Gell-
Mann matrix). The hopping matrices are written as

$\displaystyle M_{1,\mathbf{k}}$ $\displaystyle\\!=\\!$ $\displaystyle
J_{d}[I_{2}\\!\otimes\\!I_{3}]+J_{ct}\big{[}\cos\frac{\bm{\delta}_{1}\cdot\mathbf{k}}{2}I_{2}\\!\otimes\\!\lambda_{4}$
$\displaystyle+$
$\displaystyle\cos\frac{\bm{\delta}_{2}\cdot\mathbf{k}}{2}I_{2}\\!\otimes\\!\lambda_{1}+\cos\frac{\bm{\delta}_{3}\cdot\mathbf{k}}{2}I_{2}\\!\otimes\\!\lambda_{6}\big{]},$
$\displaystyle M_{0,\mathbf{k}}$ $\displaystyle\\!=\\!$ $\displaystyle
J_{d}[I_{3}]+J_{ct}[\cos\frac{\bm{\delta}_{1}\cdot\mathbf{k}}{2}\lambda_{4}$
(19) $\displaystyle+$
$\displaystyle\cos\frac{\bm{\delta}_{2}\cdot\mathbf{k}}{2}\lambda_{1}+\cos\frac{\bm{\delta}_{3}\cdot\mathbf{k}}{2}\lambda_{6}]$

where $\bm{\delta}_{1}$ and $\bm{\delta}_{2}$ are two primitive vectors of the
kagome superlattice and
$\bm{\delta}_{3}\\!=\\!-\bm{\delta}_{1}-\bm{\delta}_{2}$. The pairing matrices
are given as

$\displaystyle N_{1,\mathbf{k}}$ $\displaystyle=$ $\displaystyle-
M_{1,\mathbf{k}}+J_{d}\left[I_{2}\\!\otimes\\!I_{3}\right],$ $\displaystyle
N_{0,\mathbf{k}}$ $\displaystyle=$ $\displaystyle
M_{0,\mathbf{k}}-J_{d}\left[I_{3}\right].$ (20)

The dispersion can be obtained by Bogoliubov transformation and the resulting
band structure with nine triplon bands is shown in Fig. 3(a). As a result of
SU(2) spin rotation symmetry, three bands are all three-fold degenerate. At
the same time, discrete lattice symmetries give rise to the emergence of
quadratic band touching at $\Gamma$ point and linear band crossings at $K$
point resembling the electronic band structure of kagome lattice [23].

Figure 3: (color online) (a) Complete ($|t_{\pm 1}\rangle$ and
$|t_{0}\rangle$) triplon band spectrum of Kekule dimerized state without DMI
for $J_{d}=3$meV. Because of SU(2) spin rotation symmetry, each band is triply
degenerate. (b) The band structure of $|t_{1}\rangle$ triplon with $D=0.1$meV.
The out–of–plane DMI induce gap and topological property to the band
structure. The sign of Chern number, $C$, for each band is opposite for
$|t_{-1}\rangle$ triplon. (c,d) Complete triplon band spectrum for
$J_{d}\\!=\\!3$meV, $D\\!=\\!0.1$meV, and $h_{z}\\!=\\!\pm 1$T. The sign of
the Chern number for the lowest-dotted band depends on the sign of the
external magnetic field $h_{z}$. (e) Thermal Hall conductivity $\kappa_{xy}$
versus external magnetic field $h_{z}$ at three distinct temperatures. In the
calculation, we include very small in-plane DMI to gap out band touching
points so that Berry curvature $F^{(n)}_{xy}(\mathbf{k})$ is well-defined.

To investigate the effect of anisotropic terms on the triplon band structure,
we include small inter-dimer Dzyaloshinskii-Moriya interactions (DMI),
$\sum_{\langle
ij\rangle\parallel\gamma}D(S^{x}_{i}S^{y}_{j}-S^{y}_{i}S^{x}_{j})O^{\gamma}$,
which are allowed by the lattice symmetry. The effect of DMI can be encoded by
including
$M_{1,\mathbf{k}}^{D}\\!=\\!D[\cos\frac{\bm{\delta}_{1}\cdot\mathbf{k}}{2}S_{z}\\!\otimes\\!\lambda_{5}+\cos\frac{\bm{\delta}_{2}\cdot\mathbf{k}}{2}S_{z}\\!\otimes\\!\lambda_{2}+\cos\frac{\bm{\delta}_{3}\cdot\mathbf{k}}{2}S_{z}\\!\otimes\\!\lambda_{7}]$
and $N_{1,\mathbf{k}}^{D}\\!=\\!-M_{1,\mathbf{k}}^{D}$ to the spinful hopping
matrix $M_{1,\mathbf{k}}$ and pairing matrix $N_{1,\mathbf{k}}$ in Eq. (18),
respectively. The intra-dimer DMI are forbidden since the center of each
intra-dimer bond is an inversion center. The out–of–plane DMI still preserve
the U(1) spin rotation symmetry and the excitation modes of different triplon,
$|t_{1}\rangle$, $|t_{-1}\rangle$, and $|t_{0}\rangle$, completely decouple
from each other. At the same time, it gap out all the band touching points in
$|t_{\pm 1}\rangle$ triplon band structure separately and leave non-trivial
topological bands. In Fig. 3(b), we plot $|t_{1}\rangle$ triplon band spectrum
in the presence of DMI where the bottom (top) band carries the finite Chern
number 1 (-1). Although the band spectrum of $|t_{-1}\rangle$ triplon is
exactly the same as the one of $|t_{1}\rangle$ tripon, it contains opposite
topological property: bottom (top) band carries the Chern number -1 (1). As
pointed out in Ref.[24], such an opposite pattern could lead to the triplon
analog of $Z_{2}$ spin-hall insulator state with the corresponding helical
edge states at the boundary of the system. However, the $Z_{2}$ topological
phase is destroyed by infinitesimal inter-dimer in-plane DMI, which break the
U(1) and consequently pseudo time-reversal symmetry. The pseudo time-reversal
operator is defined as U(1)$\times\mathcal{T}$ with physical time reversal
operator $\mathcal{T}$. The in-plane DMI is symmetrically allowed in
$\alpha-$MX3 where a kagome plane is no longer a mirror plane since the
octahedral cage surrounding M ion is tilted [25]. Below, we neglect the in-
plane DMI and keep U(1) symmetry intact.

The external magnetic field gives a knob to induce topological transitions
through the Zeeman term, $\sum_{i,a}g_{z}\mu_{B}h^{z}S^{z}_{i}n_{i}^{a}$.
Here, $g_{z}\\!\approx\\!2$ is the Landé $g$-factor, $\mu_{B}$ the Bohr
magneton, and $h_{z}$ an applied magnetic field perpendicular to the plane. In
Figs. 3(c) and 3(d), we plot the complete triplon band spectrum with finite
DMI at two different fields. With $h_{z}\\!>\\!0$, the well-separated lowest
band is associated to $|t_{-1}\rangle$ triplon and carries the Chern number 1.
On the other hand, the lowest band is formed by $|t_{1}\rangle$ triplon with
the Chern number -1 when $h_{z}\\!<\\!0$.

The Chern number in electronic systems is directly connected to physical
observable quantities such as quantized Hall conductance. In bosonic systems
where it is not possible, the thermal Hall effect gives an alternative [26,
27]. It depends on the Berry curvature in which the bosons are thermally
populated, rather than solely depending on the Chern number. In our case,
triplons carry a thermal current perpendicular to the temperature gradient
when the temperature is above the triplon gap. The thermal Hall conductivity
of triplons is written as

$\displaystyle\kappa_{xy}=-\frac{k_{\rm B}^{2}T}{\hbar
L}\sum_{\mathbf{k},n}c_{2}[\rho(\omega_{n,\mathbf{k}})]F^{(n)}_{xy}(\mathbf{k})$
(21)

where $F^{(n)}_{xy}(\mathbf{k})$ is the Berry curvature of $n$th band at
$\mathbf{k}$, $c_{2}(\rho)\\!=\\!\int_{0}^{\rho}\ln^{2}(1+t^{-1})dt$, and
$\rho(\omega_{n,\mathbf{k}})\\!=\\!(\exp(\omega_{n,\mathbf{k}}/k_{\rm
B}T)-1)^{-1}$ is the Bose distribution function. Since $\alpha-$MX3 is quasi
two-dimensional material and the thermal conductivity of each layer is in the
same direction, we sum the contribution of each layer to calculate the
conductivity for a three-dimensional sample with a single layer thickness
$L=0.5$nm [28]. Fig. 3(e) shows the thermal Hall conductivity $\kappa_{xy}$ as
a function of the magnetic field $h_{z}$. As $h_{z}\\!>\\!0$ is turned on,
$\kappa_{xy}$ starts to increase. The increase originates from the fact that
the well-seperated lowest band, which is formed by $|t_{-1}\rangle$ triplon
and most thermally populated, carries the Chern number -1 and dominates the
Hall response. On the contrary, $\kappa_{xy}$ decreases when the field
$h_{z}<0$ is included which can be interpreted by the dominance of the lowest
$|t_{1}\rangle$ triplon band with the Chern number 1. Such an opposite
response can be a direct signature of Kekule dimerized phase in $\alpha-$MX3.
In our calculation, we focus on a low temperature regime where triplon bands
are weakly populated and neglect interaction effects.

In the present work, we demonstrate that the neglected charge transfer
contribution crystalizes the system into Kekule spin-orbit dimerized state. To
check the stability, we perform iDMRG simulation which indicates a small
hybridization of spin-singlet and triplet within each dimer. We then introduce
triplon operators and investigate the triplon band structure which is
topologically non-trivial in the presence of inter-dimer DMI. For the
experimental verification of Kekule dimer phase, we calculate the thermal Hall
conductivity, which directly encodes the topological profile of the triplon
band spectrum.

As a future direction, it would be desirable to apply the degeneracy lifting
mechanism to many valence bond crystals with $d^{1}$ ions which include
MgTi2O4 with Ti3+ on a pyrochlore lattice [29]. An essential question regards
then whether it can verify the experimentally observed helical dimerization
pattern [30]. Regarding the fact that $d^{5}$ ions give the exactly same form
of spin-orbital Hamiltonian, it would be interesting to study the effect of
charge-transfer processes for such systems. Another challenging but
interesting avenue is to take into account the triplon interaction effects
which might stabilize the topological gap [31] or induce the emergence of
triplon bound states [32].

##  I Acknowledgments

The author thanks S. B. Lee, M. J. Park, H. Kee, K. Penc, J. Romhányi, and J.
Knolle for stimulating conversations, and the latter also for detailed
comments on the experimental identification. G.B.S. is funded by the European
Research Council (ERC) under the European Unions Horizon 2020 research and
innovation program (grant agreement No. 771537). The research is part of the
Munich Quantum Valley, which is supported by the Bavarian state government
with funds from the Hightech Agenda Bayern Plus. Tensor network calculations
were performed using the TeNPy Library [33].

## References

  * Auerbach [2012] A. Auerbach,  _Interacting electrons and quantum magnetism_ (Springer Science & Business Media, 2012).
  * Henley [1989] C. L. Henley, Physical review letters 62, 2056 (1989).
  * Baskaran  _et al._ [2008] G. Baskaran, D. Sen, and R. Shankar, Physical Review B 78, 115116 (2008).
  * Rousochatzakis and Perkins [2017] I. Rousochatzakis and N. B. Perkins, Physical review letters 118, 147204 (2017).
  * Canals and Lacroix [1998] B. Canals and C. Lacroix, Physical Review Letters 80, 2933 (1998).
  * Baskaran  _et al._ [2007] G. Baskaran, S. Mandal, and R. Shankar, Physical review letters 98, 247201 (2007).
  * Bergman  _et al._ [2007] D. Bergman, J. Alicea, E. Gull, S. Trebst, and L. Balents, Nature Physics 3, 487 (2007).
  * Lee  _et al._ [2014] S. Lee, E. K.-H. Lee, A. Paramekanti, and Y. B. Kim, Physical Review B 89, 014424 (2014).
  * Yamashita and Ueda [2000] Y. Yamashita and K. Ueda, Physical review letters 85, 4960 (2000).
  * Tchernyshyov  _et al._ [2002] O. Tchernyshyov, R. Moessner, and S. Sondhi, Physical review letters 88, 067203 (2002).
  * Jackeli and Ivanov [2007] G. Jackeli and D. Ivanov, Physical Review B 76, 132407 (2007).
  * Bastien  _et al._ [2018] G. Bastien, G. Garbarino, R. Yadav, F. J. Martinez-Casado, R. B. Rodríguez, Q. Stahl, M. Kusch, S. Limandri, R. Ray, P. Lampen-Kelley, _et al._ , Physical Review B 97, 241108 (2018).
  * de Vries  _et al._ [2010] M. A. de Vries, A. Mclaughlin, and J.-W. Bos, Physical review letters 104, 177202 (2010).
  * Romhányi  _et al._ [2017] J. Romhányi, L. Balents, and G. Jackeli, Physical review letters 118, 217202 (2017).
  * Miura  _et al._ [2007] Y. Miura, Y. Yasui, M. Sato, N. Igawa, and K. Kakurai, Journal of the Physical Society of Japan 76, 033705 (2007).
  * Jackeli and Khomskii [2008] G. Jackeli and D. Khomskii, Physical review letters 100, 147203 (2008).
  * B. and Flengas [1964] B. and S. Flengas, Canadian Journal of Chemistry 42, 1495 (1964).
  * [18] W. M. H. Natori,  _j= 3/2 Quantum spin-orbital liquids_ , Ph.D. thesis, Universidade de São Paulo.
  * Liu  _et al._ [2022] H. Liu, J. Chaloupka, and G. Khaliullin, Physical Review B 105, 214411 (2022).
  * Sachdev and Bhatt [1990] S. Sachdev and R. Bhatt, Physical Review B 41, 9323 (1990).
  * Romhányi  _et al._ [2011] J. Romhányi, K. Totsuka, and K. Penc, Physical Review B 83, 024413 (2011).
  * Nawa  _et al._ [2019] K. Nawa, K. Tanaka, N. Kurita, T. J. Sato, H. Sugiyama, H. Uekusa, S. Ohira-Kawamura, K. Nakajima, and H. Tanaka, Nature communications 10, 1 (2019).
  * Bergman  _et al._ [2008] D. L. Bergman, C. Wu, and L. Balents, Physical Review B 78, 125104 (2008).
  * Thomasen  _et al._ [2021] A. Thomasen, K. Penc, N. Shannon, and J. Romhányi, Physical Review B 104, 104412 (2021).
  * Cheong and Mostovoy [2007] S.-W. Cheong and M. Mostovoy, Nature materials 6, 13 (2007).
  * Onose  _et al._ [2010] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura, Science 329, 297 (2010).
  * Romhányi  _et al._ [2015] J. Romhányi, K. Penc, and R. Ganesh, Nature communications 6, 1 (2015).
  * Jain  _et al._ [2013] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, _et al._ , APL materials 1, 011002 (2013).
  * Di Matteo  _et al._ [2004] S. Di Matteo, G. Jackeli, C. Lacroix, and N. Perkins, Physical review letters 93, 077208 (2004).
  * Schmidt  _et al._ [2004] M. Schmidt, I. W Ratcliff, P. Radaelli, K. Refson, N. Harrison, and S.-W. Cheong, Physical review letters 92, 056402 (2004).
  * Mook  _et al._ [2021] A. Mook, K. Plekhanov, J. Klinovaja, and D. Loss, Physical Review X 11, 021061 (2021).
  * McClarty  _et al._ [2017] P. A. McClarty, F. Krüger, T. Guidi, S. Parker, K. Refson, A. Parker, D. Prabhakaran, and R. Coldea, Nature Physics 13, 736 (2017).
  * Hauschild and Pollmann [2018] J. Hauschild and F. Pollmann, SciPost Physics Lecture Notes , 005 (2018).

##  II Supplementary Material

###  II.1 I. Derivation of spin-orbital model for charge-transfer processes

Here, we consider the charge-transfer processes of the type
$d^{1}_{i}\\!-\\!p^{6}\\!-\\!d^{1}_{j}\rightarrow
d^{2}_{i}\\!-\\!p^{4}\\!-\\!d^{2}_{j}$ where two holes are created on an anion
ion connecting nearest-neighbor cation ions $i$ and $j$. The virtual-state
energy is $E_{v}\equiv 2U+2\Delta-9U_{p}$. Here we give details about the
derivation for bonds on the $xy$ planes for concreteness. There are four
possible electron hopping processes for this case :

$\displaystyle d_{yz}^{1}\\!-\\!p_{z}^{2}\\!-\\!d_{zx}^{1}\rightarrow
d_{yz}^{2}\\!-\\!p_{z}^{0}\\!-\\!d_{zx}^{2},\text{ as given in
Fig.~{}\ref{fig:ct3}}$ (S1) $\displaystyle
d_{yz}^{1}\\!-\\!p_{z}^{2}p_{y}^{2}\\!-\\!d_{xy}^{1}\rightarrow
d_{yz}^{2}\\!-\\!p_{z}^{1}p_{y}^{1}\\!-\\!d_{xy}^{2},$ (S2) $\displaystyle
d_{xy}^{1}\\!-\\!p_{x}^{2}p_{z}^{2}\\!-\\!d_{zx}^{1}\rightarrow
d_{xy}^{2}\\!-\\!p_{z}^{1}p_{x}^{1}\\!-\\!d_{zx}^{2},$ (S3) $\displaystyle
d_{xy}^{1}\\!-\\!p_{x}^{2}p_{y}^{2}\\!-\\!d_{xy}^{1}\rightarrow
d_{xy}^{2}\\!-\\!p_{x}^{1}p_{y}^{1}\\!-\\!d_{xy}^{2}.$ (S4)

Figure S1: A 90∘ Mi-X-Mj geometry where magnetic ions Mi and Mj interact via
an ion X. Here, two holes with opposite spins arrive at the same $p_{z}$
orbital in an ion X.

First, we focus on a case with Eq. (S1) and give details about the derivation
of effective low-energy spin-orbital Hamiltonian within the subspace where an
initial state occupies an electron at site $i(j)$ with $yz(zx)$ orbital. The
basis for the subspace is written as

$\displaystyle
S_{yz,zx}=\\{|yz_{\uparrow},zx_{\uparrow}\rangle,|yz_{\uparrow},zx_{\downarrow}\rangle,|yz_{\downarrow},zx_{\uparrow}\rangle,|yz_{\downarrow},zx_{\downarrow}\rangle\\}.$
(S5)

There are two possible hopping processes: the spin-conserving process and the
spin-flipping process. The contribution from the first one can be represented
in $S_{yz,zx}$ subspace as

$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,zx}=-\frac{t_{s}^{2}}{E_{v}}\left(\\!\begin{array}[]{cccc}0&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&0\\\ \end{array}\\!\right).$ (S10)

On the other hand, the spin-flipping process can be represented in $S_{yz,zx}$
subspace as

$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,zx}=-\frac{t_{s}^{2}}{E_{v}}\left(\\!\begin{array}[]{cccc}0&0&0&0\\\
0&0&-1&0\\\ 0&-1&0&0\\\ 0&0&0&0\\\ \end{array}\\!\right).$ (S15)

Here, initial states $|yz_{\uparrow},zx_{\uparrow}\rangle$ and
$|yz_{\downarrow},zx_{\downarrow}\rangle$ cannot give any contribution since
two holes with same spins cannot occupy the same orbital at an anion ion.
Next, we focus on a case where an initial state occupies an electron at site
$i(j)$ with $xy(xy)$ orbital. The basis for such subspace is written as

$\displaystyle
S_{xy,xy}=\\{|xy_{\uparrow},xy_{\uparrow}\rangle,|xy_{\uparrow},xy_{\downarrow}\rangle,|xy_{\downarrow},xy_{\uparrow}\rangle,|xy_{\downarrow},xy_{\downarrow}\rangle\\}.$
(S16)

Here, the spin-flipping process is not possible and the contribution can be
represented in $S_{xy,xy}$ subspace as

$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,xy}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4}.$ (S17)

The contributions associated to Eq. (S1) in other subspaces are written as

$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,yz}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,xy}=-\frac{t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,yz}=-\frac{t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,yz}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4}.$ (S18)

Now we consider terms associated with Eq. (S2). In each subspace, the
contribution is given as

$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,zx}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,xy}$ $\displaystyle=$
$\displaystyle-\frac{t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,yz}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,zx}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4}.$ (S19)

The contribution from Eq. (S3) in each subspace is given as

$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,yz}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,zx}=-\frac{t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,xy}$ $\displaystyle=$
$\displaystyle-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,xy}$ $\displaystyle=$
$\displaystyle-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4}.$ (S20)

The contribution from Eq. (S4) in each subspace is given as

$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,xy}$ $\displaystyle=$
$\displaystyle-\frac{t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,yz}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,zx}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,zx}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4}.$ (S21)

Figure S2:

One also needs to take into account symmetric processes $(yz\leftrightarrow
zx)$ with another anion as illustrated in Fig. S2. Collecting all possible
charge-transfer contributions, we obtain the effective spin-orbital
Hamiltonian,

$\displaystyle H_{ct}\\!=\\!\frac{2t_{s}^{2}}{2U+2\Delta-9U_{p}}\sum_{\langle
ij\rangle\parallel\gamma}\\!\\!\Big{(}\vec{S}_{i}\\!\cdot\\!\vec{S}_{j}-\frac{1}{4}\Big{)}(n_{i}^{a}n_{j}^{b}\\!+\\!n_{i}^{b}n_{j}^{a}).$
(S22)

###  II.2 II. Detail of infinite density matrix renormalization group
simulation

To perform the infinite matrix product state related simulation, we first
clarify the ordering of lattice sites, which is implemented by labeling an
integer $i$ to each lattice site. We follow a site labeling method for
honeycomb lattice on the YC-2$L_{y}$ cylinders as shown in Fig. S3. Note that
YC-4 cylinder cannot support the Kekule spin-orbit dimerized phase.

Figure S3: (color online) (a,b) Site-labeling scheme for a honeycomb lattice
on a YC-4 or YC-6 cylinder with $x$ direction length $L_{x}=2$ or 3,
respectively. The total number of sites is $2L_{x}L_{y}=$ 8 or 18,
respectively. We wrap the honeycomb lattice onto a cylinder which is periodic
along $y$ direction and infinite along $x$ direction.

To check the convergence of the ground state obtained by iDMRG, we investigate
the scaling behavior of the ground state energy per site as a function of bond
dimension $\chi$ on the YC-6 cylinder with $L_{x}=3$. Fig. S4 shows that the
per site energy $E_{0}/J_{d}$ scales linearly with $1/\chi$ indicating a
reliable convergence.

Figure S4: (color online) Per site ground state energy $E_{0}/J_{d}$ as a
function of bond dimension $\chi$ on a YC-6 cylinder with length $L_{x}$ = 3.

###  II.3 III. Detail of exact diagonalization simulation

In this section, we give a detail of ED simulation on a cluster of six sites
with periodic boundary conditions as shown in Fig. S5(a). The cluster can
support the Kekule as well as the columnar spin-orbit dimerized phase and the
ground state turns out to be the Kekule phase. In Fig. S5(b), we plot the
energy per site $E_{0}$ as a function of $\tilde{J}$ for the ground state,
Kekule phase, and the first excited state. The result indicates the
stabilization of the Kekule phase up to $\tilde{J}=0.5$.

Figure S5: (color online) (a) Six sites cluster with periodic boundary
conditions which can support the Kekule phase. (b) Energy per site $E_{0}$ for
the ground state, which is the Kekule phase, and the first excited state.

###  II.4 IV. Gell-Mann matrices

In this section, we give the explicit expressions of Gell-Mann matrices
$\lambda_{i}$.

$\displaystyle\lambda_{1}\\!=\\!\\!\\!\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\
0&0&0\end{array}\right)\\!\\!,~{}\lambda_{2}\\!=\\!\\!\\!\left(\begin{array}[]{ccc}0&-i&0\\\
i&0&0\\\
0&0&0\end{array}\right)\\!\\!,~{}\lambda_{3}=\\!\\!\\!\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\
0&0&0\end{array}\right)\\!\\!,~{}\lambda_{4}\\!=\\!\\!\\!\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ 1&0&0\end{array}\right)\\!\\!,$ (S35)
$\displaystyle\lambda_{5}\\!=\\!\\!\\!\left(\begin{array}[]{ccc}0&0&i\\\
0&0&0\\\
-i&0&0\end{array}\right)\\!\\!,~{}\lambda_{6}=\\!\\!\\!\left(\begin{array}[]{ccc}0&0&0\\\
0&0&1\\\
0&1&0\end{array}\right)\\!\\!,~{}\lambda_{7}\\!=\\!\\!\\!\left(\begin{array}[]{ccc}0&0&0\\\
0&0&-i\\\
0&i&0\end{array}\right)\\!\\!,~{}\lambda_{8}\\!=\\!\frac{1}{\sqrt{3}}\\!\\!\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&-2\end{array}\right)\\!\\!.$ (S48)