Quantum spin Hall effect and spin-charge separation in kagom\'{e} lattice
Zhigang Wang
A two-dimensional kagom\'{e} lattice has been theoretically investigated within a simple tight-binding model, which includes the nearest neighbor hopping term and the intrinsic spin-orbit interaction between the next nearest neighbors. By using the topological winding properties of the spin-edge states on the complex-energy Riemann surface, the spin Hall conductance is obtained to be quantized as $-e/2\pi$ ($e/2\pi$) in insulating phases. This result keeps consistent with the numerical linear-response calculation and the Z$_{2}$ topological invariance analysis. When the sample boundaries are connected in twist, by which two defects with $\pi$ flux are introduced, we obtain the spin-charge separated solitons at $1/3$ (or $2/3$) filling.
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\begin{document}
\title{Quantum spin Hall effect and spin-charge separation in kagom\'{e} lattice}
\author{Zhigang Wang }
\affiliation{LCP, Institute of Applied Physics and Computational Mathematics, P.O. Box
8009, Beijing 100088, People's Republic of China}
\author{Ping Zhang}
\thanks{Corresponding author. zhang\_ping@iapcm.ac.cn}
\affiliation{LCP, Institute of Applied Physics and Computational
Mathematics, P.O. Box 8009, Beijing 100088, People's Republic of
China} \affiliation{Center for Applied Physics and Technology,
Peking University, Beijing 100871, People's Republic of China}

\pacs{73.43.-f, 71.10.Pm, 72.25.Hg}
\date{\today}

\begin{abstract}
A two-dimensional kagom\'{e} lattice has been theoretically investigated
within a simple tight-binding model, which includes the nearest neighbor
hopping term and the intrinsic spin-orbit interaction between the next nearest
neighbors. By using the topological winding properties of the spin-edge states
on the complex-energy Riemann surface, the spin Hall conductance is obtained
to be quantized as $-e/2\pi$ ($e/2\pi$) in insulating phases. This result
keeps consistent with the numerical linear-response calculation and the
\textbf{Z}$_{2}$ topological invariance analysis. When the sample boundaries
are connected in twist, by which two defects with $\pi$ flux are introduced,
we obtain the spin-charge separated solitons at $1/3$ (or $2/3$) filling.

\end{abstract}
\maketitle


Over the last two decades the topological band insulators (TBIs) have been a
subject of great interest in condensed matter field \cite{Thouless,Wenbook}.
Different from the normal band insulators, the TBIs have a prominent
characteristic, which is the necessary presence of gapless edge states on the
sample boundary \cite{Halperin,Hatsugai}. An early TBI model was proposed by
Haldane \cite{Haldane}. Therein it was shown that the gapless edge states
result in a remarkable feature of TBIs by showing a quantum Hall effect in
absence of a macroscopic magnetic field. Besides the Haldane model, several
other lattice models have been proposed to be the quantum Hall TBIs, which
include the two-dimensional (2D) spin-chiral kagom\'{e} lattice
\cite{Ohgushi,Wang2} and the three-dimensional (3D) distorted fcc lattice
\cite{Shindou, Wang3}. All the quantum Hall TBIs rely on the breaking of
time-reversal symmetry (TRS).

Recently, Kane and Mele generalized the spinless Haldane model to a spin one
by adding an intrinsic spin-orbit interaction (SOI) \cite{Kane1,Kane2}. The
TRS is conserved in the Kane-Mele model and the gapless spin edge states in
this model result in a quantum spin Hall effect. These TBIs like the Kane-Mele
model keeping TRS are different from the quantum Hall TBIs, and we call them
the quantum spin Hall TBIs. The quantum spin Hall TBIs have received
considerable attention. One impressive example is that only one year later
after its theoretical prediction to be a quantum spin Hall TBI in 2006
\cite{Bernevig}, it was proved in experiment that HgTe is an actual one
\cite{Koenig}.

One special character of the quantum spin Hall TBIs was recently attributed to
their spin-charge separated excitations \cite{Ran, Qi} in the presence of a
$\pi$ flux. This attribution is motivated by the recent advance in studying 2D
fractionalized quasiparticles \cite{Lee,Hou}, and is a straightforward result
when, like what Kane and Mele \cite{Kane1} have dealt with Haldane's TBI,
considering spins of the 2D edge soliton. The separate spinon, holon, and
chargeon obey Bose statistics and the experimental measurement of these
soliton excitations would provide an undoubted verification of the
\textbf{Z}$_{2}$ topological properties of the quantum spin Hall TBIs. At
present, besides the necessity for further studies to gain more insights into
the nature of spin-charge separation and its connection to the other
topological phenomena, obviously, it is also important to identify and study
various model systems that exhibit the phenomenon of spin-charge separation.
Motivated by this observation, as well as by the recent attention on the
layered metal oxides as possible candidates for the quantum spin Hall TBI
\cite{Shitade}, in this paper, we study the quantum spin Hall effect and
construct spin-charge separated edge solitons in 2D kagom\'{e} lattice.
Different from the previously studied kagom\'{e} TBI \cite{Ohgushi,Wang2,
Zhang} whearein the presence of ferromagnetic spin chirality breaks TRS, here
TRS persists and the quantum spin Hall effect occurs due to the intrinsic SOI.
By using the bulk linear-response theory, as well as the topological winding
numbers of the spin-edge states on the complex-energy Riemann surface, we
obtain the spin Hall conductance (SHC) $\sigma_{xy}^{s}$. The quantized value
of $\sigma_{xy}^{s}$ is $-e/2\pi$ ($e/2\pi$) at 1/3 (2/3) filling. Then, we
construct spin-charge separated edge solitons by introducing $\pi$ fluxes with
a method similar to that in Ref. \cite{Lee}. The quantum statistics of these
solitons is also discussed. \begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.6\linewidth]{Fig1.eps}
\end{center}
\caption{(Color online). Schematic picture of the 2D kagom\'{e} lattice. The
dashed lines represent the Wigner-Seitz unit cell, which contains three
independent sites (A, B, C).}%
\label{f1}%
\end{figure}

Consider the tight-binding model for independent electrons on the 2D
kagom\'{e} lattice (Fig. \ref{f1}). The spin-independent part of the
Hamiltonian is given by
\begin{equation}
\mathcal{H}_{0}=t\sum_{\langle ij\rangle\sigma}c_{i\sigma}^{\dag}c_{j\sigma},
\label{1}%
\end{equation}
where $t_{ij}$=$t$ is the hopping amplitude between the nearest-neighbor link
$\langle ij\rangle$ and $c_{i\sigma}^{\dag}$ ($c_{i\sigma}$) is the creation
(annihilation) operator of an electron with spin $\sigma$ (up or down) on
lattice site $i$. For simplicity, we choose $t$=$-1$ as the energy unit and
the distance $a$ between the nearest sites as the length unit throughout this paper.

The Hamiltonian (\ref{1}) can be diagonalized in the momentum space as%
\begin{equation}
\mathcal{H}_{0}=\sum_{\mathbf{k}}\psi_{\mathbf{k}\sigma}^{+}H_{0}%
(\mathbf{k})\psi_{\mathbf{k}\sigma}, \label{Hk}%
\end{equation}
where the electron field operator $\psi_{\mathbf{k}\sigma}$=$(c_{A\mathbf{k}%
\sigma},c_{B\mathbf{k}\sigma},c_{C\mathbf{k}\sigma})^{\text{T}}$ includes the
three lattice sites ($A,B,C$) in the Wigner-Seitz unit cell shown in Fig. 1.
$H_{0}(\mathbf{k})$ is a $3\mathtt{\times}3$ spinless matrix given by
\begin{equation}
H_{0}(\mathbf{k})=-t\left(
\begin{array}
[c]{ccc}%
0 & 2\cos\left(  \mathbf{k}\mathtt{\cdot}\mathbf{a}_{1}\right)  & 2\cos\left(
\mathbf{k}\mathtt{\cdot}\mathbf{a}_{3}\right) \\
2\cos\left(  \mathbf{k}\mathtt{\cdot}\mathbf{a}_{1}\right)  & 0 & 2\cos\left(
\mathbf{k}\mathtt{\cdot}\mathbf{a}_{2}\right) \\
2\cos\left(  \mathbf{k}\mathtt{\cdot}\mathbf{a}_{3}\right)  & 2\cos\left(
\mathbf{k}\mathtt{\cdot}\mathbf{a}_{2}\right)  & 0
\end{array}
\right)  , \label{Hamilton}%
\end{equation}
where $\mathbf{a}_{1}$=$(-1/2,-\sqrt{3}/2)$, $\mathbf{a}_{2}$=$(1,0)$, and
$\mathbf{a}_{3}$=$(-1/2,\sqrt{3}/2)$ represent the displacements in a unit
cell from A to B site, from B to C site, and from C to A site, respectively.
In this notation, the first Brillouin zone (BZ) is a hexagon with the corners
of $\mathbf{K}$=$\pm\left(  2\pi/3\right)  \mathbf{a}_{1}$, $\pm\left(
2\pi/3\right)  \mathbf{a}_{2}$, $\pm\left(  2\pi/3\right)  \mathbf{a}_{3}$.

The energy spectrum for spinless Hamiltonian $H_{0}(\mathbf{k})$ is
characterized by one dispersionless flat band ($\epsilon_{1\mathbf{k}}^{(0)}%
$=$2$), which reflects the fact that the 2D kagom\'{e} lattice is a line graph
of the honeycomb structure \cite{Mie}, and two dispersive bands,
$\epsilon_{2(3)\mathbf{k}}^{(0)}=-1\mathtt{\mp}\sqrt{4b_{\mathbf{k}}-3}$ with
$b_{\mathbf{k}}$=$\sum_{i=1}^{3}\cos^{2}\left(  \mathbf{k\cdot a}_{i}\right)
$. These two dispersive bands touch at the corners (\textbf{K}-points) of the
BZ and exhibit Dirac-type energy spectra, $\epsilon_{2(3)\mathbf{k}}%
^{(0)}\mathtt{=}(-1\mathtt{\mp}\sqrt{3}|\mathbf{k}\mathtt{-}\mathbf{K}|)$,
which implies a \textquotedblleft particle-hole\textquotedblright\ symmetry
with respect to the Fermi energy $\epsilon_{F}\mathtt{=}-1$. The corresponding
eigenstates of $H_{0}(\mathbf{k})$ are given by%
\begin{equation}
\left\vert u_{n\mathbf{k}}^{(0)}\right\rangle =G_{n\mathbf{k}}\left(
q_{1\mathbf{k}},q_{2\mathbf{k}},q_{3\mathbf{k}}\right)  ^{\text{T}},
\label{wave}%
\end{equation}
where the expressions of the components $q_{i\mathbf{k}}$ and the normalized
factor $G_{n}(\mathbf{k})$ for each band are given in Table I.

\begin{table}[th]
\caption{The expressions for the coefficients in Eq. (\ref{wave}) with $x_{i}%
$=$\mathbf{k\mathtt{\cdot}a}_{i}$.}
%\label{specs}%
\begin{tabular}
[c]{cc}\hline\hline
$q_{1k}$ & $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{2}[\epsilon
_{nk}^{(0)2}-4\cos^{2}x_{2}]$\\
$q_{2k}$ & $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \epsilon_{nk}^{(0)}\cos
x_{1}+2\cos x_{2}\cos x_{3}$\\
$q_{3k}$ & $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \epsilon_{nk}^{(0)}\cos
x_{3}+2\cos x_{2}\cos x_{1}$\\
$G_{nk}^{-2}$ & $\ \ \ \ \ 2b_{k}\epsilon_{nk}^{(0)2}+[4b_{k}-3\epsilon
_{nk}^{(0)2}]\cos^{2}x_{2}+6(b_{k}-1)\epsilon_{nk}^{(0)}\ \ $\\\hline
\end{tabular}
\end{table}

Then, we introduce the intrinsic SOI term, which, according to the symmetry of
the kagom\'{e} lattice, takes the form \cite{Kane1,Guo}
\begin{equation}
\mathcal{H}_{\text{SO}}=i\frac{2\lambda_{\text{SO}}}{\sqrt{3}}\sum
_{\langle\langle ij\rangle\rangle\sigma_{1}\sigma_{2}}\left(  \mathbf{d}%
_{ij}^{1}\times\mathbf{d}_{ij}^{2}\right)  \cdot\mathbf{s}_{\sigma_{1}%
\sigma_{2}}c_{i\sigma_{1}}^{\dag}c_{j\sigma_{2}}. \label{Hso}%
\end{equation}
Here $\lambda_{\text{SO}}$ represents the SOI strength, $\mathbf{s}$ is the
vector of Pauli spin matrices, $i$ and $j$ are next-nearest neighbors, and
$\mathbf{d}_{ij}^{1}$ and $\mathbf{d}_{ij}^{2}$ are the vectors along the two
bonds that connect $i$ to $j$. Taking the Fourier transform, we have
\[
\mathcal{H}_{\text{SO}}\mathtt{=}\sum_{\mathbf{k}\sigma}\psi_{\mathbf{k}%
\sigma}^{+}H_{\text{SO}}(\mathbf{k})\psi_{\mathbf{k}\sigma}%
\]
with%
\begin{equation}
H_{\text{SO}}(\mathbf{k})=\pm2\lambda_{\text{SO}}\left(
\begin{array}
[c]{ccc}%
0 & i\cos(\mathbf{k}\mathtt{\cdot}\mathbf{b}_{1}) & -i\cos(\mathbf{k}%
\mathtt{\cdot}\mathbf{b}_{3})\\
-i\cos(\mathbf{k}\mathtt{\cdot}\mathbf{b}_{1}) & 0 & i\cos(\mathbf{k}%
\mathtt{\cdot}\mathbf{b}_{2})\\
i\cos(\mathbf{k}\mathtt{\cdot}\mathbf{b}_{3}) & -i\cos(\mathbf{k}%
\mathtt{\cdot}\mathbf{b}_{2}) & 0
\end{array}
\right)  , \label{HsoK}%
\end{equation}
where $\mathbf{b}_{1}$=$\mathbf{a}_{3}-\mathbf{a}_{2}$, $\mathbf{b}_{2}%
$=$\mathbf{a}_{1}-\mathbf{a}_{3}$, $\mathbf{b}_{3}$=$\mathbf{a}_{2}%
-\mathbf{a}_{1}$, and the $+$($-$) sign refers to spin up (down) electrons.

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.8\linewidth]{Fig2.eps}
\end{center}
\caption{(Color online). (a) Energy spectrum (solid curves) of the 2D
kagom\'{e} lattice along the high-symmetry lines in the BZ with intrinsic
spin-orbit couplings $\lambda_{\text{SO}}$=$0.1t$. There are two band gaps
appearing with gap width $\Delta$ and $\delta$. For comparison, we also draw
the energy spectrum without intrinsic spin-orbit couplings (dotted lines). (b)
The corresponding BZ of the 2D kagom\'{e} lattice. (c) The SHC $\sigma
_{xy}^{s}$ as a function of the Fermi energy $\epsilon_{F}$. The shaded areas
correspond to the bulk gaps.}%
\label{f2}%
\end{figure}

Inclusion of the intrinsic SOI in the Hamiltonian makes the appearance of the
eigenstates $\left\vert u_{n\mathbf{k}}\right\rangle $ and eigenenergies
$\epsilon_{n\mathbf{k}}$ very tedious except for at some high-symmetry
$\mathbf{k}$ points. Instead of writing their explicit forms, here we show in
Fig. \ref{f2}(a) (solid curves) the numerically calculated energy spectrum for
the total Hamiltonian $\mathcal{H}\mathtt{=}\mathcal{H}_{0}\mathtt{+}%
\mathcal{H}_{\text{SO}}$ along the high-symmetry lines ($\mathbf{\Gamma
}\mathtt{\rightarrow}\mathbf{K}$, $\mathbf{K}\mathtt{\rightarrow}\mathbf{M}$,
and $\mathbf{M}\mathtt{\rightarrow}\mathbf{\Gamma}$) in the BZ. The
SOI\ coefficient is chosen to be $\lambda_{\text{SO}}$=$0.1t$. For comparison
we also plot in Fig. \ref{f2}(a) (dashed curves) the energy spectrum in the
absence of SOI ($\lambda_{\text{SO}}$=$0$). One can see that while the spin
degeneracies is not lifted by the presence of the intrinsic SOI, nevertheless,
the Dirac contacts of the lower and middle bands at the $\mathbf{K}$ point are
removed and there evolves a direct gap between these two bands. The amplitude
of this gap turns out to be $\Delta$=$4\sqrt{3}|\lambda_{\text{SO}}|$.
Similarly, the original contact at the $\mathbf{\Gamma}$ point between the
middle and upper (flat) bands is also lifted by the presence of SOI and a gap
with amplitude $\delta$ is opened. However, this gap is an indirect one, i.e.,
the middle-band maximum and upper-band minimum are not at the same $k$-point.

To see the behavior of the system in the insulating state, we have calculated
the SHC using the following Kubo formula \cite{Sinova}%
\begin{align}
\sigma_{xy}^{s}  &  =-e\hslash\sum_{n\neq n^{\prime},\mathbf{k}}\left[
f\left(  \epsilon_{n\mathbf{k}}\right)  -f\left(  \epsilon_{n^{\prime
}\mathbf{k}}\right)  \right] \label{kubo}\\
&  \times\frac{\operatorname{Im}\langle u_{n\mathbf{k}}|\frac{1}{2}\left\{
\hat{v}_{x},\hat{s}_{z}\right\}  |u_{n^{\prime}\mathbf{k}}\rangle\langle
u_{n^{\prime}\mathbf{k}}|\hat{v}_{y}|u_{n\mathbf{k}}\rangle}{\left(
\epsilon_{n\mathbf{k}}-\epsilon_{n^{\prime}\mathbf{k}}\right)  ^{2}+\eta^{2}%
},\nonumber
\end{align}
where $\mathbf{\hat{v}}\left(  \mathbf{k}\right)  $=$\partial H(\mathbf{k}%
)/\hslash\partial\mathbf{k}$ and $H(\mathbf{k})$=$H_{0}(\mathbf{k}%
)+H_{\text{SO}}(\mathbf{k})$. The calculated result at zero temperature is
shown in Fig. \ref{f2}(c) by varying the Fermi energy $\epsilon_{F}$ with the
SOI\ coefficient $\lambda_{\text{SO}}$=$0.1t$. From Fig. 2(c), one can see
that initially the SHC $\sigma_{xy}^{s}$ decreases as the filling factors of
the (spin-degenerate) lower band increases, arriving at the minimum value
$-(e/2\pi)$ at $\epsilon_{F}$=$-1.35$ (=$-1\mathtt{-}\Delta/2$), a value
corresponding to the top of the lower band. Then, as the Fermi energy
$\epsilon_{F}$\ continues to vary in the first gap region (shaded area wherein
$-1.35\mathtt{\leqslant}\epsilon_{F}\mathtt{\leqslant}-0.65$), the SHC keeps
this minimum value unchanged. As shown in the following, this quantized SHC
can be understood by the \textbf{Z}$_{2}$-valued topological invariant
associated with this quantum spin Hall phase \cite{Kane2}. When the Fermi
energy increases to touch the bottom of the middle band at $\epsilon_{F}%
$=$-0.65$(=$-1+\Delta/2$), then the SHC suddenly switches up and rapidly
increases when the Fermi energy goes through the middle two bands. When
$\epsilon_{F}$ increases at the top of the middle two bands ($\epsilon_{F}%
$=$1.65$), $\sigma_{xy}^{s}$ arrives at the maximum value $e/2\pi$. Then as
the Fermi energy $\epsilon_{F}$\ continues to vary in the second gap region
(shaded area $1.65\mathtt{\leqslant}\epsilon_{F}\mathtt{\leqslant}2$), the SHC
$\sigma_{xy}^{s}$ keeps this maximum value unchanged. When the Fermi energy
increases to touch the bottom of the upper band at $\epsilon_{F}$=$2$, the SHC
then suddenly switches down and rapidly decreases when the Fermi energy goes
through the upper band. Finally the SHC $\sigma_{xy}^{s}$ decreases to
disappear when the three spin-degenerate bulk bands are all fully occupied.

Topologically, the quantum spin Hall insulating state at $1/3$ or $2/3$
filling can be seen by calculating a selective \textbf{Z}$_{2}$-valued
invariant $\nu$ \cite{Kane2}, which is related to the parity eigenvalues
$\xi_{2m}(\Gamma_{i})$ of the $2m$-th occupied energy band at the four
time-reversal-invariant momenta $\Gamma_{i}$ \cite{Fu}. Very recently, Guo and
Franz \cite{Guo} have numerically calculated the eigenstate of $\mathcal{H}%
_{\Gamma_{i}}$ and they found that three $\xi$'s are positive and one is
negative. Although which of the four $\xi$'s is negative depends on the choice
of the inversion center, the product $\Pi_{i}\xi(\Gamma_{i})$=$(-1)^{\nu}$ is
independent of this choice and determines the nontrivial \textbf{Z}$_{2}$
invariant $\nu$=$1$. That confirms the kagom\'{e} lattice system to be a
quantum spin Hall TBI at $1/3$ (or $2/3$) filling.

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.8\linewidth]{Fig3.eps}
\end{center}
\caption{(Color online). (a) Sketch of the 2D kagom\'{e} lattice strip with
two edges along the $y$ direction. The red, blue, and green dots are used to
distinguish the independent sites A, B, and C, respectively. (b) Energy
spectrum of the kagom\'{e} lattice strip ($N_{y}$=$40$) with the Hamiltonian
$\mathcal{H}$=$\mathcal{H}_{0}+\mathcal{H}_{\text{SO}}$. The spin-orbit
coupling strength is set as $\lambda_{\text{SO}}$=$0.1t$. The red and blue
lines represent the edge states localized at the down and up edges of the
system, respectively. And the circle and triangle label the up and down spins,
respectively. (c) The Riemann surface of the Bloch function corresponding to
$1/3$ filling. The purple and orange curves correspond to spin-up and
spin-down channels, respectively. }%
\label{f32}%
\end{figure}

On the other hand, the topological aspect of the quantized spin Hall phase can
be distinguished by the difference between the winding numbers of the spin-up
and spin-down edge states across the holes of the complex-energy Riemann
surface, $I_{s}$=$I_{\uparrow}-I_{\downarrow}$ \cite{Wang5}. The SHC is then
given by $\sigma_{xy}^{s}$=$I_{s}\left(  e/4\pi\right)  $. Using this
topological index $I_{s}$ we have investigated the quantum spin Hall effect in
the Kane-Mele graphene model \cite{Wang5}. For the present kagom\'{e} lattice
model we can also study the quantum spin Hall effect in terms of this
topological winding index. For this purpose, let us first numerically
diagonalize the total Hamiltonian $\mathcal{H}$ using the strip geometry. For
convenience and without loss of generality, we suppose the system has two
edges in the $y$ direction while keeping infinite in the $x$ direction [see
Fig. \ref{f32}(a)]. The number of sites $A$ (or $B,C$) in the $y$ direction is
chosen to be $N_{y}$=$40$. The calculated energy spectrum is drawn in Fig.
\ref{f32}(b). From this figure one can clearly see that there are spin edge
states occurring in each energy gap. These gapless edge states in the
truncated kagom\'{e} lattice are topologically stable against random-potential
perturbation, provided that the perturbation is small compared to the bulk
gaps. We have numerically confirmed this fact. The Riemann surface of Bloch
function is plotted in Fig. \ref{f32}(c) for $1/3$ filling. According to Ref.
\cite{Wang5}, the winding number of spin-up (spin-down) edge state in the
lower gap is $I_{\uparrow}$=$-1$ ($I_{\downarrow}$=$1$), which gives $I_{s}%
$=$-2$ at $1/3$ filling. That means the SHC in this phase is quantized as
$\sigma_{xy}^{s}$=$-(e/2\pi)$. The Riemann surface of Bloch function at $2/3$
filling are the same as that at $1/3$ filling, except that the directions of
the curves corresponding to different spin channels are inverse. So the
winding number of spin-up (spin-down) edge state in the upper gap is
$I_{\uparrow}$=$1$ ($I_{\downarrow}$=$-1$), which gives $I_{s}$=$2$. The
corresponding SHC at $2/3$ filling is then quantized as $\sigma_{xy}^{s}%
$=$e/2\pi$. These conclusions are consistent with those calculated by using
the Kubo formula (\ref{kubo}) [see Fig. \ref{f2}(c)]. Note that although the
non-trivial \textbf{Z}$_{2}$ invariant $\nu$=$1$ in the bulk analysis can
confirm the quantum spin Hall TBI phase, it does not provide the information
on the sign of the SHC. In contrast, our winding-number analysis can resolve
this sign at different fillings (i.e., $\sigma_{xy}^{s}$=$\mp(e/2\pi)$ at
$1/3$ and $2/3$ filling, respectively).

The topological properties of the system in the insulating state are stable
even when the Rashba SOI is considered. To clearly see this fact, we
numerically calculated the SHC $\sigma_{xy}^{s}$ with the Kubo formula
(\ref{kubo}) when the total Hamiltonian includes the Rashba SOI \cite{Liu}%
\begin{equation}
\mathcal{H}_{R}=i\frac{\lambda_{R}}{\hslash}\sum_{\langle ij\rangle\sigma
_{1}\sigma_{2}}c_{i\sigma_{1}}^{\dag}(\mathbf{s}\times\mathbf{\hat{d}}%
_{ij})_{z}c_{j\sigma_{2}}, \label{Hrashba}%
\end{equation}
where $\lambda_{R}$ is the Rashba coefficient and $\mathbf{\hat{d}}_{ij}$ is a
vector along the bond the electron traverses going from site $j$ to $i$. The
calculated SHC is drawn in Fig. 4 for different values of $\lambda_{R}$.
Clearly, one can see that the amplitude of SHC in the insulating phase keeps
$e/2\pi$ unchanged when the Rashba SOI strength $0\mathtt{\leqslant}%
\lambda_{R}/t\mathtt{<}0$.$1$. When the Rashba SOI is sufficiently large (for
example $\lambda_{R}$=$0$.$2t$), the amplitude of SHC will depart little from
the quantized value. However, the topology of this insulating state keeps
unchanged, unless the bulk gaps disappear.

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=1.0\linewidth]{Fig4.eps}
\end{center}
\caption{(Color online). The SHC as a function of the Fermi energy
$\epsilon_{F}$ with different values of the Rashba coefficient. The black,
red, blue, and pink lines correspond to $\lambda_{R}$=$0$, $0.02t$, $0.1t$,
and $0.2t$, respectively. The intrinsic SOI strength is set as $\lambda
_{\text{SO}}$=$0.1t$.}%
\end{figure}

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=1.0\linewidth]{Fig5.eps}
\end{center}
\caption{(Color online). (a) Two edges of the 2D kagom\'{e} lattice are
reconnected with weaker bonds (orange dotted lines) and its corresponding
energy spectrum (b). (c) The restored bonds have a sign reversal along the
right half bonds (violet dotted lines) and its corresponding energy spectrum
(d). In these figures, the system size is set as $N_{y}$=$40$ and the
spin-orbit coupling $\lambda_{\text{SO}}$=$0.1t$. The bonds connected two
edges are set as $0.25$ times of other ones.}%
\label{f41}%
\end{figure}

In the following let us consider the spin-charge separation in the 2D
kagom\'{e} lattice. If we reconnect the two edges of the kagom\'{e} lattice
shown in Fig. \ref{f32}(a) with weaker bonds, two small gaps then reappear in
the edge spectrum [see Figs. \ref{f41}(a) and (b)]. Topological excitations
(edge solitons) are created by reversing the sign of the reconnected bonds
along the right half row [see Fig. \ref{f41}(c)]. In this manner two defects
with $\pi$ flux are introduced in the present kagom\'{e} lattice. As a result,
four degenerate in-gap spin states localized around these two defects are
formed in each bulk gap [see Figs. \ref{f41}(d)]. The corresponding energies
of these in-gap states are $\epsilon$=$-t$ and $1.7t$, respectively. Note that
in the square lattice version of the Kane-Mele model \cite{Ran}, the in-gap
modes are precisely at zero energy, while in the present kagom\'{e} lattice
model the in-gap modes are no longer at zero energy.

At 1/3 or 2/3 filling, occupation (unoccupation) of these in-gap modes leads
to an excess (deficit) of $1/2$ fermion number per spin and per defect
\cite{Zhang}. Four different types of solitons with the following quantum
numbers are obtained when these in-gap modes are occupied by different ways:
the chargeon $f_{+(1/2)\uparrow,+(1/2)\downarrow}$ (charge $-e$, $S_{z}$=$0$);
the holon $f_{-(1/2)\uparrow,-(1/2)\downarrow}$ (charge $e$, $S_{z}$=$0$); the
two spinons $f_{+(1/2)\uparrow,-(1/2)\downarrow}$ (charge $0$, $S_{z}$%
=$\frac{\hslash}{2}$) and $f_{-(1/2)\uparrow,+(1/2)\downarrow}$ (charge $0$,
$S_{z}$=$-\frac{\hslash}{2}$). Here the subscript $+$ ($-$) represents that
the in-gap mode is filled (empty) and $\uparrow$ ($\downarrow$) labels the
spin mode. For example, $f_{+(1/2)\uparrow,-(1/2)\downarrow}$ means that the
up-spin mode is filled and the down-spin mode is empty. For further
illustration, the charge-density distribution for the chargeon
$f_{+(1/2)\uparrow,+(1/2)\downarrow}$ is calculated and shown in Fig.
\ref{f51}(a), while the spin-density distribution for the spinon
$f_{+(1/2)\uparrow,-(1/2)\downarrow}$ is plotted in \ref{f51}(b). Here, we use
a $24\mathtt{\times}24$ lattice for calculation.

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=1.0\linewidth]{Fig6.eps}
\end{center}
\caption{(Color online). (a) The charge density of a chargeon state and (b)
the spin density of a spinon on a $24\times24$ lattice with periodic boundary
condition. The chargon at coordinate ($6$, $6$) is $f_{+(1/2)\uparrow
,+(1/2)\downarrow}$, it has charge $-e$ and spin $S_{z}$=$0$. The spinon at
coordinate ($18$, $18$) is $f_{+(1/2)\uparrow,-(1/2)\downarrow}$, it has
charge $0$ and spin $S_{z}$=$\hslash/2$.}%
\label{f51}%
\end{figure}

The quantum statistics of these spin-charge separated solitons can
be readily seen by using anyon fusion argument \cite{Weeks,Ran},
which is based on the observation that the bound states of
fractional excitation acquire non-trivial Berry phases on adiabatic
exchange. Since the spin-up and spin-down bands in the present case
decouple, so without loss of generality, let us consider a bound
state of two identical spin-up solitons $f_{(1/2)\uparrow}$ (or
$f_{-(1/2)\uparrow}$), which carries charge $-e$ ($e$) and flux
$2\pi \mathtt{\sim}0$ thus is a fermion. According to the anyon
fusion rule, which states that the exchange phase $\Theta$\ of a
particle formed by combining $n$ identical anyons with exchange
phase $\theta$ is $\Theta$=$n^{2}\theta$, one easily obtain that the
exchange phase between two identical solitons $f_{(1/2)\uparrow}$
($f_{-(1/2)\uparrow}$) should be $1/4$ that of fermions, i.e.,
$\theta(f_{(1/2)\uparrow},f_{(1/2)\uparrow})$ (or
$\theta(f_{-(1/2)\uparrow },f_{-(1/2)\uparrow}$)=$\pm\pi/4$. Next
let us consider a bound state of an $f_{(1/2)\uparrow}$ and
$f_{-(1/2)\uparrow}$ soliton, which carries charge $0$ and should be
a boson. Then the exchange phase between solitons
$f_{(1/2)\uparrow}$ and $f_{-(1/2)\uparrow}$ is given by $\theta
(f_{(1/2)\uparrow},f_{-(1/2)\uparrow})$=$\mp\pi/4$. Since the
spin-down band is the Hermitian conjugate of the spin-up band one
obtain $\theta
(f_{\alpha_{1}\downarrow},f_{\alpha_{2}\downarrow})$=$-\theta(f_{\alpha
_{1}\uparrow},f_{\alpha_{2}\uparrow})$. Substituting these results
into the formula calculating the exchange phase between spin-charge
separated solitons
$\theta(f_{\alpha_{1}\uparrow\beta_{1}\downarrow},f_{\alpha_{2}\uparrow
\beta_{2}\downarrow})$=$\theta(f_{\alpha_{1}\uparrow},f_{\alpha_{2}\uparrow}%
)$+$\theta(f_{\beta_{1}\downarrow},f_{\beta_{2}\downarrow})$, one immediately
concludes that the spinons, holon, and chargeon are all bosons. However, each
spinon has nontrivial mutual exchange phase $\pi$ with the chargeon and holon.

Before ending this paper, we would like to point out that the ways creating
the spin-charge separated solitons in the kagom\'{e} lattice are not the only
one. Recently Guo and Franz \cite{Guo}\ suggested one possible realization of
fractional charges $\pm e/3$ or $\pm2e/3$ by trimerizing the kagom\'{e}
lattice, like the 1D way proposed by Su and Schrieffer \cite{Su}. Based on
this possible realization, if we introduce the intrinsic SOI in the considered
Hamiltonian, then the spin-charge separated excitations are expected to appear
with fractional charges $\pm e/3$ and spins $\pm\hslash/2$ and $\pm\hslash/6$.
We leave this and the other interesting related problems to future study.

In summary, we have theoretically studied the quantum spin Hall effect and the
spin-charge separation in the 2D kagom\'{e} lattice. By using the topological
winding numbers of the spin-edge states on the complex-energy Riemann surface,
we have obtained that the SHC is quantized as $\pm(e/2\pi)$ when the system is
in the insulating phase, which is consistent with the \textbf{Z}$_{2}$
topological invariant analysis and the numerical linear-response calculation.
Furthermore, we have constructed the spin-charge separated solitons in
kagom\'{e} lattice by connecting the system's boundaries in twist. The quantum
statistics of these solitons has also been discussed.

This work was supported by NSFC under Grants No. 10604010, No.
10904005, and No. 60776063, and by the National Basic Research
Program of China (973 Program) under Grant No. 2009CB929103.

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\end{document}

0909.2465.extracted_bib
\begin{thebibliography}{99}                                                                                               %


\bibitem {Thouless}D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den
Nijs, Phys. Rev. Lett. \textbf{49}, 405 (1982).

\bibitem {Wenbook}X.-G. Wen, \textit{Quantum Field Theory of Many-Body
Systems} (Oxford University Press, New York, 2004).

\bibitem {Halperin}B. I. Halperin, Phys. Rev. B \textbf{25}, 2185 (1982).

\bibitem {Hatsugai}Y. Hatsugai, Phys. Rev. Lett. \textbf{71}, 3697 (1993).

\bibitem {Haldane}F. D. M. Haldane, Phys. Rev. Lett. \textbf{61}, 2015 (1988).

\bibitem {Ohgushi}K. Ohgushi, S. Marakami, and N. Nagaosa, Phys. Rev. B
\textbf{62}, R6065 (2000).

\bibitem {Wang2}Z. Wang and P. Zhang, Phys. Rev. B \textbf{76}, 064406 (2007);
Phys. Rev. B \textbf{77}, 125119 (2008).

\bibitem {Shindou}R. Shindou and N. Nagaosa, Phys. Rev. Lett. \textbf{87},
116801 (2001).

\bibitem {Wang3}Z. Wang, P. Zhang, and J. Shi, Phys. Rev. B \textbf{76},
094406 (2007).

\bibitem {Kane1}C. L. Kane and E. J. Mele, Phys. Rev. Lett. \textbf{95},
226801 (2005).

\bibitem {Kane2}C. L. Kane and E. J. Mele, Phys. Rev. Lett. \textbf{95},
146802 (2005).

\bibitem {Bernevig}B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science
\textbf{314}, 1757 (2006).

\bibitem {Koenig}M. Koenig, S. Wiedmann, Christoph Br\"{u}ne, A. Roth, H.
Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science \textbf{318}, 766 (2007).

\bibitem {Ran}Y. Ran, A. Vishwanath, and D.-H. Lee, Phys. Rev. Lett.
\textbf{101}, 086801 (2008).

\bibitem {Qi}X.-L. Qi and S.-C. Zhang, Phys. Rev. Lett. \textbf{101}, 086802 (2008).

\bibitem {Lee}D.-H. Lee, G.-M. Zhang, and T. Xiang, Phys. Rev. Lett.
\textbf{99}, 196805 (2007).

\bibitem {Hou}C.-Y. Hou, C. Chamon, and C. Mudry, Phys. Rev. Lett.
\textbf{98}, 186809 (2007).

\bibitem {Shitade}A. Shitade, H. Katsura, J. Kune\v{s}, X.-L. Qi, S.-C. Zhang,
and N. Nagaosa, Phys. Rev. Lett. \textbf{102}, 256403 (2009).

\bibitem {Zhang}P. Zhang, Z. Wang, N. Hao, and W. Zhang (unpublished).

\bibitem {Wang5}Z. Wang, N. Hao, and P. Zhang, arXiv: 0906.5118 (2009), to
appear in Phys. Rev. B.

\bibitem {Mie}A. Mielke, J. Phys. A \textbf{24}, L73 (1991); \textbf{24}, 3311
(1991); \textbf{25}, 4335 (1992).

\bibitem {Guo}H.-M. Guo and M. Franz, arXiv: 0905.3385 (2009), to appear in
Phys. Rev. B.

\bibitem {Sinova}J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth,
and A. H. MacDonald, Phys. Rev. Lett. \textbf{92}, 126603 (2004).

\bibitem {Fu}L. Fu and C. L. Kane, Phys. Rev. B \textbf{76}, 045302 (2007).

\bibitem {Liu}G. Liu, P. Zhang, Z. Wang, and S.-S. Li, Phys. Rev. B 79, 035323 (2009).

\bibitem {Weeks}C. Weaks, G. Rosenberg, B. Seradjeh and M. Franz, Nature Phys.
\textbf{3}, 796 (2007).

\bibitem {Su}W. P. Su and J. R. Schrieffer, Phys. Rev. Lett. \textbf{46}, 738 (1981).
\end{thebibliography}