# Topological Magnon Insulator in Insulating Ferromagnet

Lifa Zhang<EMAIL_ADDRESS>Department of Physics and Centre for
Computational Science and Engineering, National University of Singapore,
Singapore 117542, Republic of Singapore  Jie Ren<EMAIL_ADDRESS>Theoretical
Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Jian-Sheng Wang  Department of Physics and Centre for Computational Science
and Engineering, National University of Singapore, Singapore 117542, Republic
of Singapore  Baowen Li  Department of Physics and Centre for Computational
Science and Engineering, National University of Singapore, Singapore 117542,
Republic of Singapore  NUS Graduate School for Integrative Sciences and
Engineering, Singapore 117456, Republic of Singapore  NUS-Tongji Center for
Phononics and Thermal Energy Science and Department of Physics, Tongji
University, 200092 Shanghai, PR China

(Mar 27, 2013)

###### Abstract

In the ferromagnetic insulator with the Dzyaloshinskii-Moriya interaction, we
theoretically predict and numerically verify a topological magnon insulator,
where the charge-free magnon is topologically protected for transporting along
the edge/surface while it is insulating in the bulk. The chiral edge states
form a connected loop as a $4\pi$\- or $8\pi$-period Möbius strip in the spin-
wave vector space, showing the nontrivial topology of magnonic bands. Using
the nonequilibrium Green’s function method, we explicitly demonstrate that the
one-way chiral edge transport is indeed topologically protected from defects
or disorders. Moreover, we show that the topological edge state mainly
localizes around edges and leaks into the bulk with oscillatory decay.
Although the chiral edge magnons and energy current prefer to travel along one
edge from the hot region to the cold one, the anomalous transports are
identified in the opposite edge, which reversely flow from the cold region to
the hot one. Our findings could be validated within wide energy ranges in
various magnonic crystals, such as Lu2V2O7.

###### pacs:

85.75.-d, 75.30.Ds, 75.47.-m, 75.70.Ak

##  I Introduction

Topological insulator, as a novel state of quantum matter, is characterized by
an insulating bulk band gap and conducting gapless edge/surface states
protected by symmetries hasan10 ; qi11 . It has been theoretically predicted
and experimentally observed in a variety of systems and becomes a hot spot
because of its theoretical importance in condensed matter physics and wide
potential applications in dissipationless spin-based electronics (spintronics)
spin1 . However, due to the fact that the spin transport in topological
insulators is carried by electrons, dissipations can not be really avoided.

Magnon Hall effect, as a consequence of the Dzyaloshinskii-Moriya (DM)
interaction dzya58 ; moriya60  that plays a role of vector potential similar
to the Lorentz force, has been predicted and observed in magnetic insulators
katsura10 ; onose10 ; matsumoto11 . Compared with spin current, where the
dissipation from Joule heating is still inevitable due to the electronic
carriers, the magnon Hall effect is more promising in device applications
because of the long-range coherence of charge-free spin wave hirsch99 ;
murakami03 ; sinova04 . Magnons are collective excitations of localized spins
in a crystal lattice and can be viewed as quantized quasiparticles of spin
waves. Recently, magnon excitation serga04 ; demidov09 , localization
jorzick02  and interference podbielski06  have been experimentally realized.
The technical advancements offer the perspective of various magnonic devices,
and a new discipline – magnonics – has emerged and is growing exponentially
kruglyak06 ; neusser09 ; kruglyak10 ; serga10 ; lenk11 . The charge-free
property of magnon makes it promising to achieve dissipationless transport and
control in insulating magnets.

Figure 1:  (color online). (a) Pyrochlore crystallographic structure of the
sublattice of magnetic atoms V of Lu2V2O7. (b) Two tetrahedrons in the
pyrochlore lattice, where DM vectors on bonds 1-3, 1-2, and 2-4 are indicated
by orange arrows. (c) Schematic magnetic flux due to DM interaction in the
[111] plane of the pyrochlore lattice, i.e., a kagome lattice. The coupling of
two sites along the arrows is $(J+iD)S$ while the opposite direction
corresponds to $(J-iD)S$. (d) The quasi-one-dimensional kagome-lattice strip.
The area enclosed by the dotted line can be regarded as a center which is
connected to two semi-infinite leads in equilibrium at temperatures $T_{L}$
(left) and $T_{R}$ (right). The two big arrows schematically depict the
magnitudes and directions of energy flows along the lattice edges when
$T_{L}>T_{R}$. The width of the strip example is $W=5$, which is defined as
the number of atoms in the left column of each unit cell.

Therefore, it will be of great general interest for both theorists and
experimentalists that we find in this work a new intriguing quantum state that
magnon while insulated in the bulk, can nondissipatively transport along
edges/surfaces in the absence of backscattering from defects and disorders due
to the nontrivial topology of magnon’s band structures. We name this novel
state topological magnon insulator (TMI) and believe that due to the robust
dissipationless magnon transport, the TMI in insulating magnets could provide
widely potential applications in nondissipative magnonics and micro-
spintronics.

##  II Spin-wave Hamiltonian

The magnon Hall effect was experimentally observed in insulating ferromagnet
Lu2V2O7 onose10  with a pyrochlore lattice, in which the magnetic atom
vanadium has a corner-sharing-tetrahedra sublattice, that is, a stacking of
alternating kagome and triangular lattices along the [111] direction, as shown
in Fig. 1 (a). To study magnon transport in the ferromagnetic insulator, the
Hamiltonian can be written as dzya58 ; moriya60 ; bose94 :

$\mathcal{H}=\sum\limits_{\left\langle{mn}\right\rangle}{[-J\vec{S}_{m}\cdot\vec{S}_{n}+\vec{D}_{mn}\cdot(\vec{S}_{m}\times\vec{S}_{n})]}-g\mu_{B}\vec{H}_{0}\cdot\sum\limits_{n}{\vec{S}_{n}},$
(1)

where $\vec{S}_{n}$ is the spin angular momentum at site $n$; $-J$ denotes the
nearest-neighbor coupling; $\vec{D}_{mn}$ is the DM interaction between site
$m$ and $n$; the last term comes from the Zeeman effect under an external
field $\vec{H}_{0}$.

As shown in Fig. 1 (b), in a single tetrahedron, the DM vector is
perpendicular to the corresponding bond and parallel to the surface of the
surrounding cube onose10 ; elhajal05 ; kotov05 . Since the component of
$\vec{D}_{mn}$ perpendicular to $\vec{z}=\vec{H}_{0}/H_{0}$ does not
contribute to the Hamiltonian (1) up to quadratic order of the deviation of
$\vec{S}$ onose10 , we only retain $D_{mn}=\vec{D}_{mn}\cdot\vec{z}$. When
applying a magnetic field along $\vec{z}=[111]$ direction, all the projections
of the DM interaction between inter-layer sites $m$ and $n$ are zero
($D_{13}=D_{23}=D_{43}=0$); and all the ones for inner-layer sites are nonzero
and equal ($D_{12}=D_{24}=D_{41}=D$). Therefore, with the magnetic field along
$[111]$ direction, the kagome sublattice structure will play a key role for
the presence of TMI effect in Lu2V2O7. In addition, the two-dimensional kagome
lattice sheet could be obtained through doping one-quarter of the sites [e.g.,
site 3 in Figure 1 (b)] of a pyrochlore lattice by nonmagnetic atoms shores05
; olariu08 ; colman08 .

In the following we first discuss a general kagome lattice with DM
interaction; later we will incorporate actual parameters of a thin film of
Lu2V2O7 with a kagome layer of vanadium sublattice. Using the relation of
$S^{x}={1\over{2}}(S^{+}+S^{-})$ and $S^{y}={1\over{2i}}(S^{+}-S^{-})$, we can
rewrite the Hamiltonian (1) on a kagome lattice as

$\displaystyle\mathcal{H}=$ $\displaystyle-$
$\displaystyle\sum_{\langle{mn}\rangle}{\left({{J+iD}\over
2}S_{m}^{-}S_{n}^{+}+{{J-iD}\over 2}S_{m}^{+}S_{n}^{-}\right)}$ (2)
$\displaystyle-$
$\displaystyle\sum_{\langle{mn}\rangle}{JS_{m}^{z}S_{n}^{z}}-h\sum\limits_{n}{S_{n}^{z}},$

where $h=g\mu_{B}H_{0}$. Now, applying the standard Holstein-Primakoff
transformation holstein40 ; zhang08 , one can straightforwardly obtain the
quadratic spin-wave Hamiltonian:

$\mathcal{H}=\sum\limits_{mn}{b_{m}^{+}H_{mn}b_{n}}+E_{0},$ (3)

where $b^{+}$ ($b$) denotes the operator raising (lowering) the spin component
along $\vec{z}$ direction. $E_{0}=-JS^{2}\sum_{n}{M_{n}/2}-NhS$ is the ground-
state energy with $N$ the total number of sites and $M_{n}$ the number of
nearest neighbors of the site $n$. $H_{mn}=H_{nm}^{*}=(J\pm iD)S$ and
$H_{nn}=JSM_{n}+hS$. Figure 1 (c) illustrates the direction of the DM
interaction vector, that is, the coupling between two sites along the
direction of that arrow corresponds to $(J+iD)S$, while the coupling of the
opposite direction corresponds to $(J-iD)S$. Due to the different types of
loops in a unit cell of the kagome lattice, the DM interaction avoids
cancellation thus induces the Hall effect katsura10 . As a consequence, the
preserved DM interaction acts as a vector potential for the propagation of
magnons similar to the magnetic field for the propagation of electrons, which
is crucial for the manifestation of TMI effect. The magnetic field decides the
direction of spins at the ground state, and the induced Zeeman effect term
just shifts the dispersion relation. We set magnetic field $H_{0}=0^{+}$ in
the part of theoretical model calculations, and will input finite $H_{0}$ in
the part for real-material calculations. Except in Sec. VII, dimensionless
units and $S=1/2$, $J=1$ are used without loss of generality.

Figure 2:  (a), (b) and (c) The Berry curvature of the three bands at zero DM
interaction; (d), (e) and (f) The Berry curvature of the three bands at
nonzero DM interaction (D=0.2). For all the insets (a)-(f), the horizontal and
vertical axes correspond to wave vector $k_{x}$ and $k_{y}$, respectively; the
unit is $2\pi/a$, where $a$ is the lattice spacing. (g) The Chern numbers of
the three energy bands for the two-dimensional periodic kagome lattice. The
dotted, solid, and dashed lines correspond to Chern numbers of the first, the
second, and the third bands, respectively.

##  III Chern numbers of bulk states

The Eq. (3) resembles the tight-binding model, and each unit cell has three
sites. For a two-dimensional periodic kagome spin lattice, we can perform the
Fourier transformation as

$b_{\vec{R}_{l}+\vec{r}_{m}}=\frac{1}{N_{u}}\sum\limits_{\vec{k}}{e^{-i\vec{k}(\vec{R}_{l}+\vec{r}_{m})}}b_{m}(\vec{k}).$
(4)

Here, $N_{u}$ is the number of unit cells. $\vec{R}_{l}+\vec{r}_{m}$ is the
position of the $m$-th site in the $l$-th unit cell. Thus the spin-wave
Hamiltonian can be written in the momentum space.

Figure 3:  The dispersion relations of the periodic kagome strip lattices with
different width sizes. The insets (a), (b), (c), and (d) correspond to $W=2$,
$W=5$, $W=20$, and $W=80$, respectively.

Following the standard method to calculate the Berry phase berry84 ; xiao10 ;
zhang11 , we can obtain the Berry curvature of the $n$-th band as:

$B_{k_{x}k_{y}}^{n}=i\sum\limits_{n^{\prime}\neq
n}{\frac{{\varphi_{n}^{{\dagger}}\frac{{\partial H_{\rm{SW}}}}{{\partial
k_{x}}}\varphi_{n^{\prime}}\varphi_{n^{\prime}}^{{\dagger}}\frac{{\partial
H_{\rm{SW}}}}{{\partial k_{y}}}\varphi_{n}-(k_{x}\leftrightarrow
k_{y})}}{{(\varepsilon_{n}-\varepsilon_{n^{\prime}})^{2}}}}.$ (5)

Here $\varepsilon_{n}$ and $\varphi_{n}$ are the eigenvalue and eigenvector of
the spin-wave Hamiltonian. The associated topological Chern number is obtained
through integrating the Berry curvature over the first Brillouin zone as

$C^{n}=\frac{1}{{2\pi}}\int_{BZ}{dk_{x}dk_{y}B_{k_{x}k_{y}}^{n}}.$ (6)

If the DM interaction is zero, the Berry curvatures of the three bands are
shown in Fig. 2(a), (b) and (c): the maximum points have the opposite values;
the sum of the Berry curvatures are zero, that is, the Chern numbers are zero
at zero DM interaction as shown in Fig. 2(g). Therefore, the magnon Hall
effect and topological magnon insulator effect is absent. If the DM
interaction is nonzero, the Berry curvatures change dramatically and can not
cancel each other. As shown in Fig. 2(d), in the whole momentum space, the
Berry curvature of the first band is positive, which corresponds to the Chern
number with the value of 1 as shown in Fig. 2(g). And the Berry curvatures of
the third band shown in Fig. 2(f) also can not cancel each other and the Chern
number is $-1$. The Berry curvature of the second band also changes, but the
Chern number keeps zero. The topological magnon insulator is only possible
when the DM interaction is nonzero.

##  IV Finite size effect for the dispersion relation of quasi-1D kagome
lattice

According to the spin wave Hamiltonian Eq. (3), we can calculate the
dispersion relation ($\varepsilon$ vs $k_{x}$) of the quasi-1D periodic
lattice, as shown in the Fig. 1 (d). In this figure, the left most column has
5 sites, thus we denote the width as $W=5$, in each unit cell of the quasi-1D
kagome lattice there are $6W-1$ sites. As shown in Fig. 3, with increasing
strip width, more modes appear in the energy bands; the edge states will be
gradually fixed and independent of size hatsugai93a . If the width $W\geq 20$,
we find that the states in the bulk gap, that is, the edge states tend to be
fixed, and in the bulk energy bands there are more and more branches. From
$W=20$ to $W=80$, the edge states almost have no changes.

Figure 4:  The energy differences vs the width of the periodic kagome strip at
the anti-crossing points. The solid, dashed, and dotted lines correspond to
energy differences at anti-crossing points A, B, and C, respectively in Fig. 3
(d).

In the bulk energy gaps, we find the edge states have the trend to touch each
other at the points $A,B,C$ as shown in Fig. 3 (d). The energy difference of
the corresponding edge states at the points $A,B,C$ is shown in Fig. 4. As the
width increases, the energy difference decreases exponentially. If the system
width is finite, the states in two edges have nearly equal energy and momentum
near the anticrossing points $A,B,C$; thus they can couple together to open an
energy gap which decays exponentially with width increasing zhou08 . When the
strip width increases to infinity, the edges are separated too far to interact
with each other; thus they could have degeneracy in the dispersion relation.
As shown in Fig. 4, the gap at point $A$ decays faster than that at $B$ and
$C$, thus we can find the crossing at $A$ in the upper gap earlier than that
in the lower gap. And after $W\geq 20$, the energy differences at all the
three points $A,B,C$ are very small, therefore it is very reasonable that we
use $W=80$ in all the following numerical calculations to study the chiral
edge state transport in the quasi-1D lattice with large-enough width.

##  V Topological magnon edge modes

Figure 5:  (color online). The dispersion relation of chiral magnonic edge
modes with non-zero DM interactions. (a) and (b) are dispersion relations in
range of $k_{x}\in[0,8\pi/a]$ for $D/J=0.1$ and 0.4, respectively. (c) A
conventional cylinder strip with two boundaries, of which each has a period of
$2\pi$. (d) A Möbius strip which only has one boundary of $4\pi$ period. (e) A
looped Möbius strip which only has one boundary of $8\pi$ period.

Because of the DM interaction, two edge states within both energy gaps are
twisted so that for each state
$\varepsilon(\pi/a-k_{x})\neq\varepsilon(\pi/a+k_{x})$ and they cross at
$k_{x}=\pi/a$ in the first Brillouin zone, where $a$ is the lattice spacing.
As shown in Fig. 5(a) for the case of $D/J=0.1$, in the upper bulk band gap,
the two edge modes form a continuous state with a period of $4\pi$, which can
not be disturbed to open a gap by weak disorders so that edge modes are
topologically protected. However, when DM interaction is zero, the two edge
states are easy to be perturbed to separate and open a gap, because they do
not cross each other although they degenerate. In the lower bulk band gap, we
find that four edge states will contribute to magnon transport within the
energy gap. When $D/J\approx 0.4$ or larger, in both energy gaps there are
four edge states [see Fig. 5(b)]. In a period of $8\pi$, any two of the four
edge states have degeneracies and cross each other at different points in the
momentum space. All the four edge states form a continuous state with a period
of $8\pi$ to transport magnons along two edges of the strip.

We can understand the topology of the edge states as follows. With zero DM
interaction, the edge states are similar to the two boundaries of the
conventional cylindric strip as shown in Fig. 5 (c), both of which have a
period of $2\pi$ in Brillouin zone and transport separately along two edges.
Due to the nonzero DM interaction, two edge states are twisted so that they
cross each other and go into the other energy band after $2\pi$ in momentum
space, thus form a closed loop with a period of $4\pi$. These edge states are
similar to the one-sided Möbius strip with only one boundary, as shown in Fig.
5 (d), where a line drawn starting from a point at the boundary will meet back
at the “other side” after a circle of $2\pi$, then go back to the original
point after a whole period of $4\pi$. The two edge states form one closed loop
winding the bulk energy gap between the two bands, which are thus
topologically protected from distortions. At larger DM interaction, four edge
states contribute to the transport in the bulk gap, cross each other, and
connect to form a closed $8\pi$-period loop which can be interpreted as the
only one boundary of a looped Möbius strip as shown in Fig. 5 (e). This looped
Möbius strip also has only one boundary winding around the strip surface,
thereby having the same topology as that of the conventional Möbius strip
shown in Fig. 5 (d).

The topological chiral edge state is related to the band topology
characterized by Chern numbers of the bulk states hatsugai93a ; qi06 ;
scarola07 ; yao09 ; zhang11 , as shown in Fig. 2(b). Since there are three
sites in each unit cell, the two-dimensional infinite kagome lattice with
Hamiltonian (3) has three bands. When the DM interaction is absent, all the
Chern numbers of three bands are zero so that there is no TMI effect.
Accordingly, the winding numbers of edge states are zero as well, thus they
are not topologically protected. With nonzero DM interaction, the Chern
numbers of the lowest and highest energy bands become $\pm 1$ that indicate
the nontrivial topology; the one of the middle energy band is still zero.
According to the relation between the Chern number and the winding number
hatsugai93a , the winding numbers of edge states in both bulk gaps have the
same value of 1 or $-1$, which is consistent with the only one closed loop
winding the bulk gap regardless of the period of $4\pi$, $8\pi$ or others.

Figure 6:  (color online). The local energy current and density of state for
edge magnon transport at equilibrium. (a) The uniform kagome lattice. (b) The
lattice with a defect at the upmost site of the left fifth column. The red
arrows, the blue dots, and the small black dots correspond to the local energy
current, the local density of magnon, and atom sites, respectively. The color
of the arrows and dots indicate the magnitude of the local current and density
of states, respectively . Parameters are $\varepsilon=1.5$, $T_{L}=T_{R}=1.0$,
$D/J=0.1$, and $a=1$.

##  VI Topological magnon transport: the NEGF method

To intuitively illustrate the topological magnon transport carried by chiral
edge states, we choose some unit cells of the kagome lattice strip as a center
region and set the rest as two semi-infinite leads in equilibrium at
temperatures $T_{L}$ and $T_{R}$, respectively [see Fig. 1 (d)]. We then apply
the nonequilibrium Green’s function method haug96  to calculate the local
density of magnons and the local energy current density of magnons. For the
nonequilibrium magnon transport in such system, the Hamiltonian can be written
as follows

$\mathcal{H}=\sum\limits{\mathcal{H}_{\alpha}}+\Bigl{(}\sum\limits_{lm}({b_{l}^{L+}H_{lm}^{LC}b_{m}^{C}+b_{m}^{C+}H_{ml}^{CR}b_{l}^{R}})+{\rm
h.c.}\Bigr{)},$ (7)

where
$\mathcal{H}_{\alpha}=\sum\limits_{lm}{b_{l}^{\alpha+}H_{lm}^{\alpha}b_{m}^{\alpha}},\alpha=L,C,R$,
here ‘$L,C,R$’ denote the left lead, the center part and the right lead,
respectively. The Hamiltonian matrix of the full system is

$H=\left({\begin{array}[]{*{20}c}{H_{L}}&{H_{LC}}&0\\\
{H_{CL}}&{H_{C}}&{H_{CR}}\\\ 0&{H_{RC}}&{H_{R}}\\\ \end{array}}\right).$ (8)

The retarded Green’s function is defined as

$G^{r}(t,t^{\prime})=-i\theta(t-t^{\prime})\langle[b(t),b^{+}(t^{\prime})]\rangle,$
(9)

where we set $\hbar=1$ for notational simplicity. In nonequilibrium steady
states, the Green’s function is time-translationally invariant so that it
depends only on the difference in time. Thus, the Fourier transform of
$G^{r}(t-t^{\prime})=G^{r}(t,t^{\prime})$ is obtained as

$G^{r}[\varepsilon]=\int_{-\infty}^{+\infty}G^{r}(t)e^{i\varepsilon t}dt.$
(10)

Without interaction, the free Green’s functions for three parts in equilibrium
can be written as:

$\begin{array}[]{l}((\varepsilon+i\eta)-H_{\alpha})g_{\alpha}^{r}[\varepsilon]=I,\quad\alpha=L,C,R,\\\
g_{\alpha}^{a}[\varepsilon]=g_{\alpha}^{r}[\varepsilon]^{\dagger}.\end{array}$
(11)

And there is an additional equation relating $g^{r}$ and $g^{<}$:

$g_{\alpha}^{<}[\varepsilon]=f_{\alpha}(\varepsilon)[g_{\alpha}^{a}[\varepsilon]-g_{\alpha}^{r}[\varepsilon]],$
(12)

where $f_{\alpha}(\varepsilon)=\langle
b^{+}b\rangle=[e^{\varepsilon/T_{\alpha}}-1]^{-1}$ is the Bose-Einstein
distribution function at the ${\alpha}$ part with temperature $T_{a}$; we have
set $k_{B}=1$.

For the quadratic Hamiltonian, the magnon transport is ballistic. The lesser
Green’s function can be solved as

$G^{<}[\varepsilon]=G^{r}[\varepsilon]\Sigma^{<}[\varepsilon]G^{a}[\varepsilon]$
(13)

where $G^{a}=(G^{r})^{\dagger}$ and the self energy

$\Sigma^{r,<}[\varepsilon]=H_{CL}g_{L}^{r,<}[\varepsilon]H_{LC}+H_{CR}g_{R}^{r,<}[\varepsilon]H_{RC}.$
(14)

The retarded Green’s function has the same form as for the electron case

$G^{r}[\varepsilon]=\Bigl{[}\varepsilon+i\eta-
H_{C}-\Sigma^{r}[\varepsilon]]^{-1}.$ (15)

The local density of magnon is given by zhang07

$\rho_{n}={{i\hbar G_{nn}^{<}(\varepsilon)}\over{\pi a}}.$ (16)

The local energy current is given by jing09

$\mathfrak{j}_{mn}(\varepsilon)={\varepsilon\over{2\pi}}{\mathop{\rm
Re}\nolimits}[G_{mn}^{<}[\varepsilon]H_{nm}-G_{nm}^{<}[\varepsilon]H_{mn}].$
(17)

At the interface between the left lead and the center part, it reads

$\mathfrak{j}_{mn}(\varepsilon)={\varepsilon\over{2\pi}}{\mathop{\rm
Re}\nolimits}[G_{mn}^{CL,<}[\varepsilon]H_{nm}^{LC}-G_{nm}^{LC,<}[\varepsilon]H_{mn}^{CL}].$
(18)

Taking the trace to sum over all the local current in the interface, and
integrating over all the energy, we then get the Landauer-like formula as

$\displaystyle J$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\sum\limits{j_{mn}}}d\varepsilon$ (19)
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\varepsilon{\varepsilon\over{2\pi}}{\rm{Tr}}\bigl{\\{}{\mathop{\rm
Re}\nolimits}\bigl{(}G^{CL,<}[\varepsilon]H^{LC}-G^{LC,<}[\varepsilon]H^{CL}\bigr{)}\bigr{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{{2\pi}}\int_{0}^{\infty}{\varepsilon\;[f_{L}(\varepsilon)-f_{R}(\varepsilon)]T[\varepsilon]}d\varepsilon$

where the transmission is

$T[\varepsilon]={\rm{Tr}}\bigl{\\{}G^{r}[\varepsilon]\Gamma_{L}[\varepsilon]G^{a}[\varepsilon]\Gamma_{R}[\varepsilon]\bigr{\\}}.$
(20)

with the $\Gamma_{\alpha}$ functions given by
$\Gamma_{\alpha}=i(\Sigma_{\alpha}^{r}-\Sigma_{\alpha}^{a}).$

Figure 7:  (a) Local energy current vs the coordinate along the $y$ direction.
The solid and dashed lines correspond to the local currents in two different
columns in one unit cell. (b) Local density of states of magnons vs the
coordinate along the $y$ direction. The solid, dashed, and dotted lines
correspond to the local density of states in three different columns in one
unit cell. The energy of magnon is $\varepsilon=1.5$ in the lower bulk band
gap.

Based on the formula Eq. (17), we can calculate the equilibrium or
nonequilibrium magnon transport in the lattice. Fig. 6 shows the edge state
magnon transport in the bulk gap at a fixed magnon energy $\varepsilon=1.5$ in
the thermal equilibrium. The forward (left-to-right) thermal current carried
by magnons travels along one edge, and the backward (right-to-left) current
with the same magnitude transports along the other one, as shown in Fig. 6
(a). Near both edges the local magnon currents form many chiral vortices due
to the nonzero DM interaction. Moreover, both the current and the magnon
density of states are symmetrically localized at two edges. We plot the local
current and the local density of states for the edge mode with
$\varepsilon=1.5$ in Fig. 7. We find both the local current and the local
density of states decay exponentially from the edge to the center with some
oscillations. The oscillations come from the vortex of the energy current in
the kagome lattice. Thus the magnon with energy $\varepsilon=1.5$ in the lower
bulk band gap indeed localizes at the two edges of the quasi-1D lattice. The
magnon with other energies in the bulk gaps also has the similar picture.

When a defect is present at one edge, the current take a detour around it and
transports ahead without backscattering, as illustrated in Fig. 6 (b).
Although the defect dramatically affects the local density of magnons and
destroys the local current vortex, the global currents along two edges keep
intact compared to those of the uniform lattice, and their summation keeps
zero since the net transport vanishes at equilibrium.

Figure 8:  (color online). The local energy current and density of state for
edge magnon transport at nonequilibrium. The red arrows, the blue dots, and
the small black dots correspond to the local energy current, the local density
of magnon, and atom sites, respectively. The color of the arrows and dots
indicate the magnitude of the local current and density of states,
respectively. The uniform kagome lattice are with (a) $T_{L}=1.2,T_{R}=0.8$,
and (b) $T_{L}=0.8,T_{R}=1.2$. The lattice with a defect at the upmost site of
the left fifth column are with (c) $T_{L}=1.2,T_{R}=0.8$, and (d)
$T_{L}=0.8,T_{R}=1.2$. Other parameters are $\varepsilon=1.5$, $D/J=0.1$,
$a=1$, $W=80$.

As shown in Fig. 8 (a) and (b), when two leads are held at different
temperatures, the magnons and energy current prefer to flow along one edge
from the hot lead to the cold one. The transport around the other edge however
shows an interesting anomalous behavior that the magnons and energy reversely
flow from the cold lead to the hot one. Nevertheless, we note that this does
not violate the second law of thermodynamics because the forward (hot-to-cold)
energy current transported along one edge is larger than the backward one
(cold-to-hot) along the other edge so that the total transport is still from
the hot part to the cold one. If we only swap two temperatures
($T_{L}\leftrightarrow T_{R}$), the transport will prefer the other edge but
with the directions of local edge currents unchanged. If we merely reverse the
DM interaction ($D\rightarrow-D$), the transport will change to prefer the
other edge with the local currents reversed but with the total average current
unchanged. If we swap both the temperatures and the DM interaction, the local
currents will just reverse their directions but keep the same magnitudes,
which is a consequence of the time-reversal invariance.

It is worthy to notice that for the chiral edge state, although both the
current and the magnon density of states mainly localize around two edges,
they leak into the bulk with oscillatory decay. The oscillatory motion results
from the quantum interference due to the edge boundaries, which is similar to
the properties of the localized edge phonon modes jiang09  and electron
transport in graphene qiao10 . This phenomenon indicates that even for the
topological chiral edge state, the transport within the bulk of a topological
insulator can not be really avoided. A topological insulator is not a perfect
“insulator”, not only referring to the edges/surfaces, but also for the bulk.

When a defect is present around one edge, the one-way edge current in the TMI
is able to take a detour around it and transport ahead without backscattering,
see Fig. 8 (c) and (d). Although the defect dramatically affects the local
density of magnons and the vortex pattern of local currents, the global
currents along two edges keep intact compared to those of the uniform lattice.
This means that the chiral edge state in the bulk gap is indeed topologically
protected from the lattice defect or weak disorders.

##  VII  Thin film of Lu2V2O7

Figure 9:  Two tetrahedrons in the pyrochlore lattice of the atom vanadium of
the ferromagnet Lu2V2O7. On a single tetrahedron, all the DM vectors on bonds
1-2, 2-4, 4-1, 1-3, 2-3, and 4-3 are shown by the arrows.

In the insulating ferromagnet Lu2V2O7, the orbitals of the $d$ electrons are
ordered to point to the center of mass of the vanadium tetrahedron and a
virtual hopping process stabilizes the ferromagnetic order of the vanadium
spin in this orbital-ordered state onose10 ; ichikawa05 . The vanadium
sublattice in Lu2V2O7 has a pyrochlore structure composed of corner-sharing
tetrahedra, that is, a stacking of alternating kagome and triangular lattices
along the [111] direction. Considering the strong constraint of the crystal
symmetry and using Moriya’s rules moriya60 , possible DM (Dzyaloshinskii-
Moriya) interactions on a single tetrahedron can be determined as onose10
$\vec{D}_{12}={D_{0}\over{\sqrt{2}}}(-\hat{e}_{y}-\hat{e}_{z}),\,\vec{D}_{24}={D_{0}\over{\sqrt{2}}}(-\hat{e}_{x}-\hat{e}_{y}),\,\vec{D}_{41}={D_{0}\over{\sqrt{2}}}(-\hat{e}_{x}-\hat{e}_{z}),\,\vec{D}_{13}={D_{0}\over{\sqrt{2}}}(-\hat{e}_{x}+\hat{e}_{y}),\,\vec{D}_{23}={D_{0}\over{\sqrt{2}}}(+\hat{e}_{x}-\hat{e}_{z}),\,\vec{D}_{43}={D_{0}\over{\sqrt{2}}}(-\hat{e}_{y}+\hat{e}_{z})$.
Here $D_{0}$ denotes the strength of the DM interaction; the number 1,2,3, and
4 denote the site in a single tetrahedron. If we apply a magnetic field
$\vec{H}_{0}$, then all the spin angular momentum in the direction along
$\vec{l}=\vec{H}_{0}/H_{0}$, with $H_{0}$ the magnitude of $\vec{H}_{0}$. We
know the component of the DM vector perpendicular to $\vec{l}$ does not
contribute to the spin-wave Hamiltonian, thus we only retain the projections
of the DM interaction along $\vec{l}$ direction, i.e.,
$D_{mn}^{l}=\vec{D}_{mn}\cdot\vec{l}$. If we apply a magnetic field along
$\vec{l}=[111]$ direction, then $D_{13}^{l}=D_{23}^{l}=D_{43}^{l}=0$ and
$D_{12}^{l}=D_{24}^{l}=D_{41}^{l}=\frac{-\sqrt{2}}{\sqrt{3}}D_{0}$ explanD .
Therefore, if the magnetic field is applied along $[111]$ direction, the
magnon Hall effect and topological magnon insulator effect only come from the
noncancellation of different types of DM interaction loops in the unit cell of
the kagome lattice. The effective DM interactions between inter-layer sites
have no contributions. According to the experimental observation in Ref.
onose10 , the DM interaction is obtained $D_{0}/J=0.32$, thus we use
$D={-\sqrt{2}\over{\sqrt{3}}}D_{0}={-\sqrt{2}\over{\sqrt{3}}}\cdot
0.32J=-0.26J$ for the DM interaction of the thin film of Lu2V2O7 with kagome
layer. Since in Ref. onose10 , we have $JS=8D_{s}/a_{0}^{2}$ with
$a_{0}=9.94{\rm\AA}$ the spacing between unit cells of the pyrochlore
structure and $D_{s}=21{\rm meV}{\rm\AA}^{2}$ the spin stiffness constant,
then we get the coupling $J=3.4$ meV and $a=\frac{\sqrt{2}}{2}a_{0}=7.03$ Å.
Based on these parameters, we calculate the dispersion relations for the
quasi-1D kagome lattice, as shown in Fig. 10 (b).

Figure 10:  (color online). (a) The current density vs energy of magnon for
uniform and edge-defect kagome lattices with the parameters of Lu2V2O7. The
solid and dotted lines correspond to the energy current in the bulk band gaps
for uniform and edge-defect lattices, respectively. (b) The dispersion
relation of the kagome lattice with the parameters of Lu2V2O7. $J=3.4$ meV,
$D/J=-0.26$, $H_{0}=1$ T, $T_{L}=21$ K, and $T_{R}=19$ K.

As shown in Fig. 10 (a), the energy current of magnon is not affected by
defect or disorder in the range of $[4.45,5.98]\cup[8.79,10.31]$ meV. These
energy intervals coincide with the bulk gaps in the magnon spectrum where the
topological magnonic edge states can be identified [see Fig. 10 (b)]. Although
certain distortions of edge states will occur as results of the defect or
disorder, the total energy current carried by edge magnons does not change in
the whole bulk energy gap. This indicates that the defect or disorder does not
open a gap in the magnon spectrum so that the topology of the chiral magnon
edge state is robust. According to the energy ranges, the topological magnon
states have frequencies within $[1.08,1.45]\cup[2.13,2.49]$ THz.

Applying different external magnetic fields will not change the main
properties of magnons, but shift the corresponding dispersion relations, so
that the frequency of topological edge magnons can be tuned flexibly with a
wide range. Also, when the inter-layer exchange couplings are considered, they
only play the role of effective on-site potentials, which just shift the whole
bands and leave the main band structural properties unchanged. In addition,
the two-dimensional kagome lattice sheet could be obtained by doping with
nonmagnetic atoms as we mentioned before so that the inter-layer exchange
couplings are ignorable. Therefore, we expect that one can observe the TMI for
the thin film of Lu2V2O7 in a wide energy range of magnons. Our findings about
the TMI could also be applied for other magnetic crystals, including even
antiferromagnetic materials where the existence of magnons is possible.

##  VIII Possible Experiments

To realize magnonic devices as well as the predicted TMI, the excitation and
detection of magnons is the major challenge. Recent years have witnessed a
fast development in experimental techniques such as ferromagnetic resonance
podbielski06 , pulse-inductive microwave magnetometer silva99 , time-resolved
scanning Kerr microscopy barman03 , optical pump-probe techniques kimel07 , as
well as Brillouin light scattering (BLS) perzlmaier05  which takes a special
role since it allows the direct measure of dispersions and band structures. We
could observe the topological edge modes by using these techniques to measure
the magnon dispersion relation and could also verify the TMI by detecting the
magnon transport in the bulk band gap where magnons could be selectively
excited by non-thermal optical pulses kimel04 ; bigot09 ; zhangy12 ; kamp11 ;
nish12  or induced by external spin-polarized current kiselev03 , so that we
can avoid the thermal transport from bulk states but extract the one only from
edge channels. We hope our theoretical predictions about TMI could open a new
window into the application of nondissipative magnon transport, especially for
the novel magnonic device design, which could also shed light on the
information technology based on magnonics, and micro-spintronics.

_Note added:_ Recently, we have learned of a submission note  thanks to its
authors, studying the similar topological chiral magnonic edge mode.
Differently, their results are based on a linearized Landau-Lifshitz equation
and account for the dipolar interaction instead of the DM one studied here.

###### Acknowledgements.

L. Z. and B. L. are supported by the grant R-144-000-285-646 from Ministry of
Education of Republic of Singapore. J. R. acknowledges the support of the U.S.
Department of Energy through the LANL/LDRD Program. J.-S. W. acknowledges
support from a URC research grant R-144-000-257-112 of NUS.

## References

  * (1) M. Z. Hasan, and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * (2) X.-L. Qi, and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * (3) I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
  * (4) I. Dzyaloshinskii, J. Phys. Chem. Solids 4 241 (1958).
  * (5) T. Moriya, Phys. Rev. 120, 91 (1960).
  * (6) H. Katsura, N. Nagaosa and P. A. Lee, Phys. Rev. Lett. 104, 066403 (2010).
  * (7) Y. Onose, T. Ideue, H. Katsura , Y. Shiomi , N. Nagaosa, and Y. Tokura, Science 329, 297 (2010).
  * (8) R. Matsumoto and S. Murakami, Phys. Rev. Lett. 106, 197202 (2011).
  * (9) J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).
  * (10) S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003).
  * (11) J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004).
  * (12) A. A. Serga, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rev. Lett. 92, 117203 (2004)
  * (13) V. E. Demidov, M. P. Kostylev, K. Rott, P. Krzysteczko, G. Reiss, and S. O. Demokritov, Appl. Phys. Lett. 95, 112509 (2009)
  * (14) J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul, C. Fermon, K. Y. Guslienko, A. N. Slavin, D. V. Berkov, and N. L. Gorn, Phys. Rev. Lett. 88, 047204 (2002).
  * (15) J. Podbielski, F. Giesen, and D. Grundler, Phys. Rev. Lett. 96, 167207 (2006).
  * (16) V. V. Kruglyak, and J. Hicken, Magn. Magn. Mater. 306, 191 (2006).
  * (17) S. Neusser, and D. Grundler, Adv. Mater. 21, 2927 (2009).
  * (18) V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43, 264001 (2010).
  * (19) A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 264002 (2010).
  * (20) B. Lenk, H. Ulrichs, F. Garbs, M. Münzenberg, Physics Reports 507, 107 (2011).
  * (21) I. Bose, and U. Bhaumik, J. Phys.: Condens. Matter 6, 10617 (1994).
  * (22) M. Elhajal, B. Canals, R. Sunyer, and C. Lacroix, Phys. Rev. B 71, 094420 (2005).
  * (23) V. N. Kotov, M. Elhajal, M. E. Zhitomirsky, and F. Mila, Phys. Rev. B 72, 014421 (2005).
  * (24) M. P. Shores, E. A. Nytko, B. M. Bartlett, and D. G. Nocera, J. Am. Chem. Soc. 127, 13462 (2005)
  * (25) A. Olariu, P. Mendels, F. Bert, F. Duc, J. C. Trombe, M. A. de Vries, and A. Harrison, Phys. Rev. Lett. 100, 087202 (2008).
  * (26) R. H. Colman, C. Ritter, and A. S. Wills, Chem. Mater. 20, 6897 (2008).
  * (27) T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).
  * (28) L. Zhang, J.-S. Wang, and B. Li, Phys. Rev. B 78, 144416 (2008).
  * (29) M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
  * (30) D. Xiao , M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
  * (31) Y. Hatsugai, Phys. Rev. B 48, 11851 (1993); Phys. Rev. Lett. 71, 3697 (1993).
  * (32) B. Zhou, H.-Z. Lu, R.-L. Chu, S.-Q. Shen, and Q. Niu, Phys. Rev. Lett. 101, 246807 (2008).
  * (33) X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, Phys. Rev. B 74, 045125 (2006).
  * (34) V. W. Scarola, and S. Das Sarma, Phys. Rev. Lett. 98, 210403 (2007)
  * (35) W. Yao, S. A. Yang, and Q. Niu, Phys. Rev. Lett. 102, 096801 (2009).
  * (36) L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys. Rev. Lett. 105, 225901 (2010); J. Phys.: Condens. Matter 23, 305402 (2011).
  * (37) H. Haug, and A. P. Jauho, _Quantum Kinetics in Transport and Optics of Semiconductors_ (Springer, Berlin, 1996)
  * (38) A. P. Jauho, N. S. Wingreen, and Y. Meir Phys. Rev. B 50, 5528 (1994).
  * (39) J.-W. Jiang, J. Chen, J.-S. Wang, and B. Li, Phys. Rev. B 80, 052301 (2009)
  * (40) Z. Qiao , S. A. Yang, W. Feng, W. -K. Tse, J. Ding, Y. Yao, J. Wang, and Q. Niu, Phys. Rev. B 82, 161414(R) (2010).
  * (41) W. Zhang, T. S. Fisher, and N. Mingo, Numer. Heat Transfer, Part B 51, 333 (2007).
  * (42) L. Jing, et al., J. Appl. Phys. 106, 054312 (2009).
  * (43) H. Ichikawa, et al., J. Phys. Soc. Jpn. 74, 1020 (2005).
  * (44) T. Moriya, Phys. Rev. 120, 91 (1960).
  * (45) The projection of DM interaction along $\vec{l}=\vec{H}_{0}/H_{0}$ can be written as $D_{12}^{l}={D_{0}\over{\sqrt{2}}}{{-H_{y}-H_{z}}\over H_{0}},\,D_{24}^{l}={D_{0}\over{\sqrt{2}}}{{-H_{x}-H_{y}}\over H_{0}},\,D_{41}^{l}={D_{0}\over{\sqrt{2}}}{{-H_{x}-H_{z}}\over H_{0}},\,D_{13}^{l}={D_{0}\over{\sqrt{2}}}{{-H_{x}+H_{y}}\over H_{0}},\,D_{23}^{l}={D_{0}\over{\sqrt{2}}}{{+H_{x}-H_{z}}\over H_{0}},\,D_{43}^{l}={D_{0}\over{\sqrt{2}}}{{-H_{y}+H_{z}}\over H_{0}}$. If the magnetic field is in the direction of $[111]$, then we have $H_{x}=H_{y}=H_{z}=H_{0}/\sqrt{3}$.
  * (46) T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, J. Appl. Phys. 85, 7849 (1999).
  * (47) A. Barman, V. V. Kruglyak, R. J. Hicken, A. Kundrotaite, and M. Rahman, Appl. Phys. Lett. 82, 3065 (2003).
  * (48) A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V. Pisarev, and T. Rasing, J. Phys.: Condens. Matter 19, 043201 (2007).
  * (49) K. Perzlmaier, M. Buess, C. H. Back, V. E. Demidov, B. Hillebrands, and S. O. Demokritov, Phys. Rev. Lett. 94, 057202 (2005).
  * (50) A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and T. Rasing, Nature 429, 850 (2004).
  * (51) J.-Y. Bigot, M. Vomir, and E. Beaurepaire, Nature Phys. 5, 515 (2009).
  * (52) Y. Zhang, T.-H. Chuang, Kh. Zakeri, and J. Kirschner, Phys Rev Lett. 109, 087203 (2012).
  * (53) T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. Mährlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, Nat. Photonics 5, 31 (2011).
  * (54) J. Nishitani, T. Nagashima, and M. Hangyo, Phys. Rev. B 85, 174439 (2012)
  * (55) S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature 425, 380 (2003).
  * (56) R. Shindou, R. Matsumoto, and S. Murakami, arXiv:1204.3349, (2012).