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Topological chiral magnonic edge mode in a magnonic crystal

Ryuichi Shindou, Ryo Matsumoto, and Shuichi Murakami

Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, Japan

(Dated: April 17, 2012)

Topological phases have been explored in various fields in physics such as spintron-

ics, photonics, liquid helium, correlated electron system and cold-atomic system. This

leads to the recent foundation of emerging materials such as topological band insula-

tors, topological photonic crystals and topological superconductors/superfluid. Here,

we propose topological magnonic crystal which provides protected chiral edge modes

for magnetostatic spin waves. Based on a generic linearized Landau-Lifshitz equation,

we show that a magnonic crystal with the dipolar interaction usually acquires spin

wave band with non-zero Chern integer. We argue that such magnonic systems are

accompanied by the same integer numbers of chiral magnonic edge modes, along which

the spin wave can propagate in a unidirectional manner without being backward scat-

tered. Being a robust ‘one-way spin-wave guide’, the proposed chiral magnonic edge

mode makes it possible the magnonic Fabry-Perot type interferometer, which can be

utilized as the spin-wave logic gate.

Topological phases in condensed matters have been at-

tracting much attention because of their fascinating phys-

ical properties. Discoveries of topological band insula-

tors1–7open up emerging research paradigm on spin-orbit

interaction physics. Relativistic spin-orbit interaction in

the topological band insulator endows its Bloch electron

bands with non-trivial global structures, which lead to

novel surface metallic states5and topological magneto-

electric effect.8Superconductor analogues of topological

insulators have exotic edge modes9or bound states,10

which are composed only of ‘real-valued’ fermion field

dubbed as Majorana fermion.

bound states are experimentally confirmed,11,12foster-

ing much prospect of the realization of quantum comput-

ers.13Photonics analogue of topological phase with chiral

edge modes are proposed theoretically14,15and are subse-

quently designed in actual photonics crystals.16Unidirec-

tional propagations of electromagnetic wave along these

chiral edge modes were experimentally observed, which

provides these metamaterials with unique photonic func-

tionality.

Some aspects of these

Here, we theoretically propose a magnonics analogue

of topological phases, which has topologically-protected

chiral edge mode for spin waves.

lective propagation of precessional motions of magnetic

moments in magnets. Depending on its wavelength, spin

waves are classified into two categories. One is exchange

spin-wave with the shorter wavelength, whose motion is

driven by the quantum-mechanical exchange interactions

(‘exchange-dominated’ region). The other is magneto-

static spin wave with the longer wavelength,17,18whose

propagation is caused by the long-range dipolar interac-

tion (‘dipolar’ region). Magnonics research investigates

how these spin waves propagate in the sub-micrometer

length scale and sub-nanosecond time scale.19–25Like in

other solid-state technologies such as photonics, phonon-

ics and plasmonics, the main application direction is to

explore ability of the spin wave to carry and process

information. Especially, the propagation of spin waves

Spin wave is a col-

in periodically modulated magnetic materials dubbed as

magnonic crystals21–23,26,27are of one of its central con-

cern. Owing to the periodic structuring, the spin wave

spectrum in magnonic crystal acquires allowed frequency

bands of spin wave states and forbidden-frequency bands

(band gap).

We introduce a topological Chern number in these

spin-wave bands with a band gap. Based on a simple

magnonic crystal with wide varieties of material parame-

ters, we predict that the first Chern number for the low-

est spin-wave band usually takes non-zero integer values,

when the wavelength is in the dipolar region. The rele-

vant length scale turns out to be from sub-µm to hun-

dreds of µm, well within the range of nanosize fabrica-

tion. The nonzero Chern integer in magnonic crystals

results in the same integer numbers of topological chiral

edge modes, along which the spin wave can propagate in a

unidirectional way. Thanks to the topological protection,

this propagation is generically free from any static back-

ward scatterings. The unidirectional motion can be ex-

perimentally measurable especially in yttrium iron garnet

(YIG), where the coherence length of magnons is on the

order of centimeters.20We argue that these edge modes

can be easily channelized, twisted, split and manipulated,

which enables to construct novel magnonic devices such

as spin-wave logic gate.

Chern number in boson systems

To introduce topological Chern number for the magne-

tostatic spin wave, let us consider a general quadratic

Bogoliubov-de Gennes (BdG) Hamiltonian for boson

fields in the two-dimensional momentum space (k-space);

H =1

2

?

k

?

β†

kβ−k

?· Hk·

?

βk

β†

−k

?

. (1)

with β†

bosonic bands, and the 2N by 2N Hermite matrix Hk

is composed of the N by N normal parts (particle-hole

k≡ [β1,k,··· ,βN,k]. N denotes the number of

arXiv:1204.3349v1 [cond-mat.mes-hall] 16 Apr 2012

Page 2

2

FIG. 1: Magnonic crystal with chiral edge modes. Pe-

riodic array of holes is introduced into YIG, forming a square

lattice. Iron (Fe) is filled inside every hole.

channel) and N by N anomalous parts (particle-particle

channel).

This bosonic BdG Hamiltonian is diagonalized in terms

of a para-unitary matrix Tkinstead of a unitary matrix,28

?Ek

with [γ†

k,β−k], and Ekis a diagonal ma-

trix. From the commutation relation for the γ-field, the

orthogonality and completeness of the new basis are de-

rived as

T†

kHkTk=

E−k

?

, (2)

k,γ−k]T†

k= [β†

T†

kσ3Tk= σ3, Tkσ3T†

k= σ3

(3)

respectively, where the diagonal matrix σ3 takes ±1 in

the particle/hole space, i.e. [σ3]jm= δjmσj with σj =

+1 for j = 1,··· ,N and σj= −1 for j = N +1,··· ,2N.

In terms of the para-unitary matrix, the field strength

(Berry’s curvature) Bjand the gauge connection (gauge

field) (Aj,x,Aj,y) are introduced in the momentum space

for j-th magnonic band as;

Bj≡ ∂kxAj,y− ∂kyAj,x,

Aj,ν≡ iTr[Γjσ3T†

(4)

kσ3(∂kνTk)], (5)

respectively with j = 1,··· ,2N. Γjis a 2N by 2N diag-

onal matrix taking +1 for the j-th diagonal component

and zero otherwise. The Chern number (the first Chern

number) associated with the j-th band is given by the

momentum integral of the respective Berry’s curvature

over the two-dimensional first Brillouin zone (BZ),

?

This quantity can be shown to be an integer Cj =

n (see Supplementary Information for details), which

corresponds to a number of topological chiral edge

modes. Note that, due to the commutation relation

of the boson fields, the Chern integer defined in equa-

tions (4,5,6) generally takes a form different from those

Cj=

1

2π

BZ

d2kBj. (6)

Chern integers previously defined in electronic29and pho-

tonic14,15,30systems. In the following, we show that a

two-dimensional magnonic crystal (MC) with the dipolar

interaction usually supports spin-wave bands with non-

zero Chern integers.

2-d magnonic crystals and dipolar interaction

The MC considered is a ferromagnetic system with its

magnetization and exchange interaction modulated pe-

riodically in the 2-dimensional (x-y) direction. For sim-

plicity, we assume that the system is translationally sym-

metric along the z-direction, whereas the subsequent re-

sults are expected to be similar when the thickness of the

system in the z-direction becomes finite. The system is

composed of two kinds of ferromagnets; iron and YIG.

The unit cell of the MC is an ax× ay rectangle, inside

which iron is embedded into a circular region, while the

remaining region is filled with YIG (Fig. 1). The uniform

magnetic field H0is applied along the z direction, such

that the static ferromagnetic moment Msin both regions

is fully polarized in the z direction. Propagation of the

transverse moments (mx,my) is described by a linearized

Landau-Lifshitz equation26,27,31,32;

1

|γ|µ0

dm±

dt

= ± 2iMs

?∇ · Q∇?m±∓ 2im±

∓ iH0m±± ih±Ms

?∇ · Q∇?Ms

(7)

with ∇ ≡ (∂x,∂y), m±= mx± imyand h±= hx± ihy.

(hx,hy) stands for the transverse component of long-

ranged magnetic dipolar field h, which is related to

the ferromagnetic moment m ≡ (mx,my,Ms) via the

Maxwell equation, i.e. ∇×h = c−1∂tezand ∇·(h+m) =

0. The former two terms in the right hand side of equa-

tion (7) comes from short-ranged exchange interaction,

where Q denotes the square of the exchange interac-

tion length. Ms and Q take the values of iron inside

the circular region, while taking the values of YIG oth-

erwise. The filling fraction of the circular region with

respect to the total area of the unit cell is parameter-

ized by f. We restrict ourselves to the wavelength much

longer than atomic lattice constants of YIG and iron,

so that the use of this Landau-Lifshitz equation is jus-

tified. Since the velocities of relevant spin-wave modes

are much smaller than the speed of light, we can further

employ the magneto-static approximation, replacing the

Maxwell equations by ∇ × h = 0 and ∇ · (h + m) = 0;

hν= −∂νΨ, ∆Ψ = ∂xmx+ ∂ymy,

with ν = x,y. This in combination with equation (7)

gives a closed equation of motion (EOM) for the trans-

verse moments. In terms of normalized transverse fields;

(8)

β(r) ≡

m+(r)

?2Ms(r), β†(r) ≡

m−(r)

?2Ms(r). (9)

with a proper commutator [β(r),β†(r?)] = δ(r −r?), the

coupled EOM reduces to an equivalent generalized eigen-

value problem with a Hermitian matrix Hk(see Methods

Page 3

3

for details);

id

dt

?

βk

β†

−k

?

= σ3Hk

?

βk

β†

−k

?

, (10)

where β†

of β†(r) with respect to the reciprocal vector G; β†

[··· ,β†

EOM is diagonalized in terms of a para-unitary matrix

Tkintroduced in equation (2) as

?

kdenotes a vector composed of the Fourier series

k≡

k(G),···] and β(r) =?

k,Gβk(G)e−i(k+G)r. The

id

dt

γk

γ†

−k

?

=

?Ek

−E−k

??

γk

γ†

−k

?

.

[Ek]j gives the dispersion of the j-th magnonic band

(j = 1,···), while [Tk]a,j (a = ··· ,G,···) stands for

the periodic part of the j-th magnonic Bloch wavefunc-

tion. Correspondingly, the Chern integer is defined for

each magnonic band via equations (6), (4) and (5).

The topological Chern integer thus calculated always

reduces to zero, whenever the system considered is ei-

ther time-reversal symmetric; H∗

symmetric with the mirror plane being perpendicular to

the xy plane, e.g. H(kx,ky)= H(kx,−ky). Owing to the

dipolar interaction, however, the anomalous part of Hk

acquires a complex-valued phase factor which depends

on the momentum in the chiral way (see Methods for de-

tails). As a result, these two symmetries are generically

absent in the 2-d MCs and the bosonic Chern integer can

take a non-zero integer value.

This situation is quite analogous to what the rela-

tivistic spin-orbit interaction does in ferromagnetic met-

als33,34and topological band insulators.1,2,4,5Moreover,

contrary to the relativistic spin-orbit interaction, the

strength of the dipolar interaction is an experimentally

tunable parameter in the MCs.21Namely, the dimen-

sional analysis on equation (7) indicates that, when the

characteristic length scale of the MC (the unit cell size

λ ≡√axay) becomes larger than the typical exchange

length√Q, the dipolar interaction prevails over the ex-

change interaction.

−k= Hk, or mirror-

Chiral magnonic bands in MC

We found that the Chern integer of the lowest magnonic

band, C1, is always quantized to be 2 for the longer

λ, while the integer reduces to zero for the shorter λ

(Fig. 2a). The respective quantization is protected by

a finite direct band gap between the lowest band and

the second lowest band (Fig. 2b). In the intermediate

regime of λ, these two bands get closer to each other.

With the four-fold rotational symmetry (r ≡ ay/ax= 1),

the gap closes at the two X-points at a critical value

of λ (∼ 0.28µm), where the two bands form gapless

Dirac spectra (Fig. 2c). Without the four-fold symmetry

(r ?= 1), the band touching at one of the two X points

and that of the other occur at different values of λ. These

band-touchings are P1(π,0,λc,1) and P2(0,π,λc,2) in the

FIG. 2:

magnonic band and band dispersions of several lowest

magnonic bands. a, Chern-integer phase diagram for the

lowest magnonic band as a function of the linear dimension

of the unit cell size (λ ≡√axay) and the aspect ratio of the

unit cell (r ≡ ay/ax) with f = π×10−2. The phases are char-

acterized by the Chern integer of the lowest magnonic band,

C1. b, Band dispersions of the lowest three magnonic bands

with r = 1, λ = 0.35µm and f = π ×10−2. A magnonic band

gap appears between the first and the second lowest band,

whose size is on the order of 10−3THz. c, Band dispersions

of the lowest and second lowest magnonic band at the critical

point which intervenes the phase II (C1 = 2) and the phase

IV (C1 = 0). Dirac cones with same chiralities are formed at

the two inequivalent X points.

Chern-integer phase diagram for the lowest

3-dimensional parameter space subtended by two crystal

momenta kxand kyand the unit cell size λ (Fig. 3).

As in the fermionic case,35,36a band-touching point in

the 3-d parameter space generally plays a role of the dual

magnetic monopole (charge). The dual magnetic field

considered is a vector field defined in the parameter space

(kx,ky,λ), generalized from equation (4) as the ‘rotation’

of three-component gauge field Aj≡ (Aj,x,Aj,y,Aj,λ);

Bj≡ ∇ × Aj, (11)

with ∇ ≡ (∂kx,∂ky,∂λ). Here the third component of

the gauge field Aj,λis newly introduced in the same way

Page 4

4

FIG. 3:

monopoles in the 3-dimensional parameter space sub-

tended by the crystal momenta (kx,ky) and the lin-

ear dimension of the unit cell size (λ ≡√axay) for

r ≡ ay/ax > 1. Small green spheres denote the magnetic

monopoles, which emit the dual magnetic field (blue arrows).

We only show the first quadrant of the Brillouin zone. Three

phases found in Fig. 2a with r > 1 are separated by the

λ = λc,2 plane and the λ = λc,1 plane. A magnetic monopole

located on each plane means that the ‘phase transition’ is

accompanied by the change of the Chern integer.

Schematic configuration of dual magnetic

as previously done in (5);

Aj,λ≡ iTr[Γjσ3T†

kσ3(∂λTk)]. (12)

j specifies either one of the two magnonic bands which

form the band-touching. At the band-touching point, the

dual magnetic field for the respective bands has a dual

magnetic charge, whose strength is quantized to be 2π

times integer (see Supplementary Information).

We found that dual magnetic charges for the lowest

band at the band touching points at P1and P2are both

+2π (Fig. 3); ∇ · B1 = 2πδ(λ − λc,1)δ(kx− π)δ(ky) +

2πδ(λ−λc,2)δ(kx)δ(ky−π), where λc,1< λc,2for ay> ax

(r > 1) and λc,1> λc,2for ay< ax(r < 1). The Gauss

theorem suggests that, when the unit cell size λ is varied

across one of these band-touching points (λc,1 or λc,2),

the Chern integer for the lowest magnonic band changes

by unity, e.g.

C1|λ>λc,1− C1|λc,1>λ=

1

2π

?

S1

dn · B1= 1,

where S1is a small sphere enclosing the magnetic charge

at P1in the 3-d parameter space. This leads to the phase

diagram in Fig. 2a, which describes the Chern integer of

the lowest magnonic band as a function of the unit cell

size λ and the aspect ratio r. There are four phases,

divided by the curves λ = λc,1and λ = λc,2.

A dual magnetic charge is a quantized object, so that

it can neither disappear by itself nor change gradually.

Upon any small change of material parameters, two mag-

netic charges can only move around in the 3-d parameter

space. Moreover, in the presence of the spatial inver-

sion symmetry Hk = H−k, their locations are always

restricted at the X points. As a result, the global struc-

ture of the phase diagram depicted in Fig. 2a widely

holds true for other combinations of material parame-

ters. In fact, for r = 1 and f = π × 10−2, we found

λc= 0.370µm for iron (circular region) and YIG (host),

and λc= 0.372µm for cobalt (circular region) and YIG

(host). When varying the filling fraction for iron (cir-

cular region) and YIG (host) with r = 1, we found

λc = 0.274µm for f = 4π × 10−2, and λc = 0.348µm

for f = 9π × 10−2. From these observations, we expect

that, even if materials in the circular region is replaced

simply by the vacuum, the MC systems with larger unit

cell size still belongs to topological non-trivial phases,

having nonzero Chern integers.

Chiral magnonic edge mode in MC

The chiral phases with non-zero Chern integers have chi-

ral magnonic edge modes, which are localized at the

boundary with the phase with zero Chern integer (phase

IV) or the vacuum. These chiral modes carry spin wave

in a unidirectional manner, whose dispersions go across

the band gap between the lowest and the second lowest

band. As an illustrative example, we consider a bound-

ary (y axis) between the MC in phase III and MC in the

phase IV, whose Chern integers for the lowest magnonic

bands differ by unity (Fig. 4a). The existence of a chiral

magnonic edge mode between these two is shown from a 2

by 2 Dirac Hamiltonian derived near their phase bound-

ary (λ = λc,1). The Hamiltonian generally takes a fol-

lowing form (see Supplementary Information),

Heff= ω0τ0+ κ(x)τ3− ia∂xτ1− ib∂yτ2.

τj denotes the Pauli matrices subtended by the two-

fold degenerate eigenstates with their eigen-frequency ω0,

which are formed by the lowest magnonic band and sec-

ond lowest one at P1. a and b are constant material pa-

rameters which are positive-valued. The difference of the

Chern integers (C1) for the two phases is represented as a

change of the sign of the Dirac mass term κ(x); we hence

suppose that positive κ(x) for x > 0 is for the phase III

and and negative κ(x) for x < 0 is for the phase IV (see

Fig. 4a); limx→±∞κ(x) = ±κ∞. The Hamiltonian has a

following eigenstate;37,38

?xκ(x?)dx??1

(13)

ψk(r) ∝ eikye−1

a

i

?

.

which is localized at the boundary (x = 0). In terms

of the surface crystal momentum k along the y axis, the

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5

FIG. 4: Chiral magnonic edge modes. a, geometry of

the system. bc, dispersions for Dirac Hamiltonian with the

P¨ oschl-Teller potential15κ(x) = κ∞tanh(x/d) for κ∞d =

0.9a (b) and κ∞d = 2.9a (c).

(‘bulk’) have a gap, and one chiral edge state runs across the

gap. In c there are some nonchiral edge states, whereas in b

there is not. Nonetheless, the number of chiral edge modes is

one, which is determined solely from the difference between

the Chern integers for the two phases.

The bulk magnonic bands

corresponding eigen-frequency is given by E = ω0+ bk.

This connects the lowest magnonic band lying at E ≤

ω0−κ∞and the second lowest band at E ≥ ω0+κ∞(see

Fig. 4b,c). The mode is chiral, since it propagates along

the boundary in a unidirectional way, vk≡ ∂kE = b. The

argument so far can be generalized into other situations.

The generalization suggests that the phase with C1= 1

at r > 1 (Phase III) or r < 1 (Phase I) has a chiral edge

mode at its boundary with vacuum, whose dispersion

crosses the direct band gap at the (π,0) or (0,π)-point

respectively, while the phase with C1= 2 (Phase II) has

both at its boundary with vacuum.

The chiral magnonic edge modes can be easily twisted

or split by the change of the size (λ) and shape (r) of

the unit cell, which we demonstrate in Fig. 5a,b.

Fig. 5a, the MC in the phase II (r = 1) is connected

with the other MC in the phase III (r > 1), whose Chern

integer for the lowest band differ by unity. The bound-

ary between these two MC systems supports the chiral

In

magnonic edge mode which runs across the direct band

gap at (0,π) point. The existence of such an edge mode

can be shown from the other 2 by 2 Dirac Hamiltonian

derived near λ = λc,2. This means that the two chiral

magnonic edge modes propagating along the boundary

of the the MC in the phase II (r = 1) are spatially di-

vided into two, where one mode goes along the boundary

of another MC in the phase III (r > 1), while the other

goes along the boundary between these two MCs. This

configuration realizes a ‘spin-wave current splitter’, an

alternative to those proposed in other geometries.40

Discussion

By calculating a newly-introduced bosonic Chern inte-

ger for spin wave bands, we argue that two-dimensional

normally-magnetized magnonic crystal acquires chiral

edge modes for magnetostatic wave in the dipolar regime.

Each mode is localized at the boundary of the sys-

tem, carrying magnetic energies in a unidirectional way.

Thanks to the topological protection, spin-wave propa-

gations along these chiral edge modes are generally ro-

bust against imperfections of the lattice periodicity and

boundary roughness; they are free from any types of

elastic backward scatterings with moderate strength.39

This robustness makes it possible a magnonic analogue

of Fabry-Perot interferometer as discussed below.

The interferometer is made up of a coupled of chiral

magnonic edge modes encompassing a single topological

MC (see Fig. 5c). Parts of the MC are spatially con-

stricted by the hole inside, so as to play a role of the

‘point contact’ between these edge modes.41A unidirec-

tional spin-wave propagation is induced in a chiral mode

via an antenna attached to the boundary. The wave is

divided into two chiral edge modes at a point contact.

Two chiral propagations merge into a single chiral propa-

gation at the other point contact. Depending on a phase

difference between these two, the superposed wave ex-

hibits either a destructive interference or a constructive

one, which is detected as an electric signal from the other

antenna. Note that a local application of magnetic field

changes the velocity of either one of the two chiral edge

modes locally as vk= b → vk= b + ∆b(r). This mod-

ification results in a phase shift of the wave, which thus

changes the interference pattern observed in the output

signal. With the use of this local magnetic field as an

external control,19,42the interferometer can serve as a

solid-state based magnonic logic gate.

Though chiral edge modes are robust against static

perturbations, magnetic energies excited in the edge

mode decay into either phonon states or other magnon

states in the bulk via inelastic scatterings. The associated

decay time or coherence length depends on specific ma-

terials, and spin wave propagation along the chiral edge

mode survives only within this coherence length. Due to

the absence of conduction electrons, however, spin waves

in magnetic insulators are generally known to have long

coherence lengths. For example, the coherence length in

YIG in the magnetostatic regime becomes on the order of

Page 6

6

FIG. 5: Examples of magnonic circuit made by chiral

magnonic edge modes. a, Schematic picture of a ‘spin-

current splitter’. Two-fold degenerate chiral magnonic edge

modes are spatially separated from each other at the interface

region between two MCs with different shapes of the unit cell;

one with a square unit cell (r = 1), while the other with a

rectangular unit cell (r > 1). b, Combination of several MCs

with different sizes and shapes of the unit cell channelizes

chiral magnonic edge modes at our disposal. Three MCs are

used, whose material parameters are taken as those of the

phases I, II and III. c, Magnonic analogue of the Fabry-Perot

interferometer. An electric current in an input antenna in-

duces a chiral magnetostatic wave in the edge mode (yellow

arrows). The input wave is split into two at the point con-

tact (PC1). After travelling along different trajectories, these

two waves make an interference at the other point contact

(PC2). The output wave is measured as an electric signal via

the other antenna (output). The phase of the chiral magne-

tostatic wave in the respective trajectory can be controlled

by the local magnetic field applied near the edge modes (PS1

and PS2).

centimeter.20Thus, all the characteristic spin-wave prop-

agations depicted in Fig. 5a,b,c are experimentally real-

izable especially in sizable MC systems made out of these

magnetic insulators. Measuring these propagations in the

space- and time-resolved manner is also experimentally

possible in terms of Brillouin light scatterings,43which is

by itself remarkable and which surely leads to the devel-

opment of innovative spintronic device in future.

Methods

Derivation of equation (10) and material parameters

To introduce the Chern integer for the magnonic bands, we

reduced the Landau-Lifshitz equation (7) into an equivalent

generalized eigenvalue problem with bosonic quadratic Hamil-

tonian as in equation (10). To do this, notice first that the

commutator between the transverse moments depends on the

spatial coordinate, i.e. [ˆ mx(r), ˆ my(r?)] = Ms(r)δ(r −r?). To

remove this spatial dependence, we normalize the fields as;

β(r) ≡

m+(r)

?2Ms(r), β†(r) ≡

m−(r)

?2Ms(r).(14)

which leads to a proper commutator of the Holstein-Primakov

(HP) boson; [ˆβ(r),ˆβ†(r?)] = δ(r − r?). In terms of the HP

boson fields, the Landau-Lifshitz equation is also properly

symmetrized as,

dt=4iα?∇ · Q∇?αβ − 4iβ?∇ · Q∇?α2

− iH0β − iα∂+Ψ,

∆Ψ =∂+

?Ms(r)/2 and |γ|µ0 being omitted for clar-

are spatially modulated with a lattice periodicity; α(r) =

?

m(r) and Ψ(r) take a form;

?

Ψ(r) =Ψk(G)e−i(k+G)·r,

dβ

(15)

?αβ†?+ ∂−

?αβ?,∂± ≡ ∂x± i∂y, (16)

with α(r) ≡

ity.The static magnetization and exchange interaction

Gα(G)eiG·rand Q(r) =?

GQ(G)eiG·rwith the recipro-

cal vectors G. Thus, it follows from the Bloch theorem that

β(r) =

k

?

?

G

βk(G)e−i(k+G)·r,

?

k

G

where the k-summation is taken over the first Brillouin zone.

In terms of these Fourier modes, an equivalent generalized

eigenvalue problem with a quadratic Hamiltonian for the HP

field is derived as

?

ˆ H =1

2

k

β†

β−k≡ [··· ,β−k(−G),···β−k(G),···].

σ3 in the right hand side takes ±1 in the particle/hole

space, which comes from the commutation relation of bosons,

[βk,β†

tween equations (15,16) and equation (17) dictates that Hk

thus introduced is given by a following Hermitian matrix

(H†

k= Hk);

?α · α + Bk+ H01

id

dt

βk

β†

−k

?

=??

βk

β†

−k

?

?

,ˆ H?= σ3Hk

β†

kβ−k

?

βk

β†

−k

βk

β†

−k

?

, (17)

??Hk

k(−G),···],

??

,

k≡ [··· ,β†

k(G),···β†

k] = 1 and [β†

k,βk] = −1.The comparison be-

Hk≡

α · Ik· α

α · α + Bk+ H01α · I∗

k· α

?

(18)

Page 7

7

iron

1.711

21.24

cobalt

1.401

28.90

YIG

0.140

183.57

Ms [µA/m]

√Q [˚ A]

TABLE I: Saturated magnetization Msand exchange interac-

tion length√Q of iron, cobalt and yttrium iron garnet (YIG).

with

[α]G,G? ≡ α(G − G?), [Ik]G,G? ≡ δG,G?e−2iθk(G),

[Bk]G,G? ≡ 4

G1,G2

× (k + G1) · (k + G2)

− 4

G1,G2

× (G − G?) · (G − G?− G1),

and

?

α(G − G1)Q(G1− G2)α(G2− G?)

?

Q(G1)α(G2)α(G − G?− G1− G2)

(19)

eiθk(G)≡(k + G)x+ i(k + G)y

|k + G|

.(20)

After taking the summation over G1and G2in equation (19),

one can further decompose [Bk]G,G? into three parts,

?

+ 4i

[Bk]G,G? =4

µ=x,y

?Qα2?

?

?

G−G?(k + G)µ(k + G?)µ

?Qα(∂µα)?

?Q(∂µα)(∂µα)?

µ=x,y

G−G?(G − G?)µ

+ 4

µ=x,y

G−G?

(21)

with

?Qα2?

G≡1

G≡1

G≡1

S

?

?

?

Q(r)α2(r)e−iG·rd2r,

?Qα(∂µα)?

?Q(∂µα)(∂µα)?

S

Q(r)α(r)(∂µα(r))e−iG·rd2r,

S

Q(r)(∂µα(r))(∂µα(r))e−iG·rd2r,

where the 2-d integrals in the right hand side are taken over

the unit cell and S denotes the area of the cell.

Notice that, owing to complex-valued phase factors in the

anomalous part, Ik ?= I∗

and the mirror symmetries within the 2-d plane are absent

in equation (18). Without periodic modulations, α = α1

and Q = Q1, these phase factors can be erased by a proper

gauge transformation, β†

k

that both symmetries are recovered. In the presence of the

periodic modulations, α ?= α1 and Q ?= Q1, however, this is

not the case and these two symmetries are generally gone.

k, both the time-reversal symmetry

k→ β†

?I∗

kand β−k→ β−k

√Ik, so

The specific MC system considered in this paper is com-

posed of two kinds of ferromagnets;26,27,32the respective

(square root of) magnetization and (square of) exchange in-

teraction length are specified by (αj,Qj) (j = 1,2). The unit

cell of the MC is square shaped, inside which one of these two

ferromagnets (α1,Q1) are embedded within a circular region,

while the remaining region is filled with the other (α2,Q2). If

α(r) has a discontinuity at the boundary between these two

regions, the last term in equation (21), [Q(∂µα)(∂µα)], gener-

ally diverges, since having the second derivative with respect

to a spatial coordinate. Physically, such infrared divergences

are removed by a smooth variation of the magnetization at

the boundary. For simplicity, we interpolate α(r) as a linear

function of the radial coordinate measured from the center of

the circular region;

α(r) =

α1

α1−

α2

(|r| < R0)

|r|−R0

R1−R0(α1− α2) (R0 < |r| < R1)

(R1 < |r|)

.(22)

A discontinuity in Q(r) is also removed by the same linear

interpolation. The results in Fig. 2 were obtained from the

numerics with (r0,r1) ≡ (R0/λ,R1/λ) = (0.10,0.125), while

(Ms,0,Ms,1) = (1.752,0.194) [µA/m] and (√Q0,√Q1) =

(33,130) [˚ A].27As for the material parameters of iron (Fe),

cobalt (Co), and YIG, we used the values from the literature32

(Table I). To obtain the para-unitary matrix Tk which diago-

nalizes Hk, we employed the method based on the Cholesky

decomposition.28

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Acknowledgement

We would like to thank Masayuki Hashisaka for discus-

sions. RS also thanks to Tsutomu Momoi for informing

him of Ref. 28. This work is partly supported in part by

Grant-in-Aids from the Ministry of Education, Culture,

Sports, Science and Technology of Japan (No. 21000004

and 22540327) and by Grant for Basic Scientific Re-

search Projects from the Sumitomo Foundation.

also acknowledge financial support from the Global Cen-

ter of Excellence Program by MEXT, Japan, throught

the “Nanoscience and Quantum Physics” Project of the

Tokyo Institute of Technology.

We

Additional Information

The authors declare no competing financial interests.

Author contribution

All authors contributed equally to the manuscript.

Supplementary Information for ‘Topological chi-

ral magnonic edge mode in a magnonic crystal’

Quantization of the Chern number

Page 9

9

The Chern number associated with the j-th magnonic

band is constructed from the projection operator1

?

where Pjdenotes the projection operator filtering out the

j-th band at the momentum k. The integral is over the

first Brillouin zone (BZ) in the two-dimensional k space.

Equation (3) suggests that the projection operator takes

a form,

Cj≡i?µν

2π

BZ

dkTr?(1 − Pj)?∂kµPj

??∂kνPj

??, (S23)

Pj≡ TkΓjσ3T†

kσ3, (S24)

where Γj is a diagonal matrix taking +1 for the j-

th diagonal component and zero otherwise. With this

definition, equation (S24) allows that?

to a surface integral of the rotation of the gauge field

Aj≡ (Aj,x,Aj,y) over the BZ, i.e. equations (4-6).

To see its quantization,2,3notice first that such a sur-

face integral reduces to zero, provided that the gauge

field is defined uniquely and smoothly over the whole

BZ. When any of |[Tk]a,j| has a zero on the BZ, however,

the gauge field cannot be determined uniquely over the

whole BZ. In this case, we need to decompose the BZ into

two overlapped regions3(H1and H2with H1∪H2= BZ

and H1∩ H2= ∂H1= −∂H2≡ Γ), in such a way that

one of [Tk]a,j(say [Tk]1,j) does not have any zero within

one region (H1), while another (say [Tk]2,j) has no zero

inside the other (H2). In the former region, we then take

the gauge, ?a|u(1)

ways real positive, while take another gauge (|u(2)

the other (H2), making ?2|u(2)

tive. Provided that the j-th magnonic band considered

is isolated from the others (Ek,j ?= Ek,m?=j for any k),

these two wavefunctions are always related to each other

by a certain U(1) transformation,

jPj = 1 and

PjPm = δjmPj. Cj defined in equation (S23) reduces

k,j? ≡ [Tk]a,j, such that ?1|u(1)

k,j? is al-

k,j?) in

k,j? to be always real posi-

|u(2)

k,j? = |u(1)

k,j?eiθk. (S25)

Now that the gauge of |u(a)

are uniquely defined in H1and H2respectively, A(a)

i[σ3]jj?u(a)

inside each of these two regions. The Stokes theorem is

applied separately, so that equation (6) is calculated as,

?

=

2π

Γ

k,j? and the gauge of |u(2)

k,j?

j,ν≡

k,j|σ3∂kν|u(a)

k,j? (a = 1,2) are smooth functions

Cj=

1

2π

1

H1

dk∇k× A(1)

dk ·?A(1)

j

+

1

2π

?=

?

1

2π

H2

dk∇k× A(2)

?

j

?

j

− A(2)

j

Γ

dk · ∇kθk,

(S26)

with ∇k ≡ (∂kx,∂ky). Two regions share a boundary

(Γ), which forms a closed loop. θkintroduced in equa-

tion (S25) generally have a 2πn phase winding (n = Z),

when k winds along this loop. Accordingly, the right

hand side is an integer, i.e.

numerical calculations of Cj, we employed an algorithm

based on the ‘manifestly gauge-invariant’ description of

the Chern integer.4

Cj = Z.In the actual

Magnetic Monopole and Dirac Hamiltonian

When two bosonic (magnonic) bands form a band-

touching point in the 3-dimensional p-parameter space

with p ≡ (kx,ky,λ), the dual magnetic fields associated

with these two bands via equations (5,11,12) generally

have a quantized source of their divergence at the point

(p = pc). To see this, notice first that, away from the

band-touching point, p ?= pc, the gauge invariant dual

magnetic fields for these two bands are uniquely deter-

mined, which are therefore divergence-free by their def-

inition. At p = pc, however, the projection to each of

these two bands cannot be defined, which endows the

respective dual magnetic field with some singular struc-

ture. To study this structure, one can use the degenerate

perturbation theory in a generalized eigenvalue problem.

The eigenvalue problem takes a form

?

HpTp= σ3Tp

Ep

−Ep

?

,

with p ≡ (kx,ky,λ) and p ≡ (−kx,−ky,λ). The diago-

nal matrix σ3takes +1 for the particle space while takes

−1 in the hole space. Epis a diagonal matrix, whose ele-

ments give bosonic (magnonic) bands and are physically

required to be all positive definite. Hp is a Hermitian

matrix obtained from a quadratic bosonic Hamiltonian

considered. We decompose this into the zero-th order

part and the perturbation part;

Hp= H0+ (Hp− H0) ≡ H0+ Vp.

with H0≡ Hp=pc. Suppose that H0has two-fold degen-

erate eigenstates |uj? (j = 1,2) with its eigen-frequency

ω0(> 0);

(S27)

H0|uj? = σ3|uj?ω0,

where the states are normalized as ?uj|σ3|um? = δjm.

On introducing the perturbation Vp, the degeneracy is

split into two frequency levels. The eigenstate for the

respective eigen-frequency is determined on the zero-th

order of p − pcas;

Tp= T0Up+ O(|p − pc|),

where T0diagonalizes H0with T†

and a unitary matrix Updiagonalizes a 2 by 2 Hamilto-

nian Veffformed by the two-fold degenerate eigenstates;

?

?u2|Vp|u1? ?u2|Vp|u2?

Accordingly, near p = pc, the dual magnetic field de-

fined in equations (5,11,12) is given only by this unitary

matrix;

(S28)

0σ3T0= T0σ3T†

0= σ3

Veff≡

?u1|Vp|u1? ?u1|Vp|u2?

?

.(S29)

Bj= ∇ × Aj, Aj= iTr[ΓjU†

p∇Up],

Page 10

10

with ∇ ≡ (∂kx,∂ky,∂λ), while equation (S29) reduces

to some 2 by 2 Dirac-type Hamiltonian. From this, we

can prove the quantization of the dual magnetic charge

at the band-touching point exactly in the same way as in

the fermionic systems,5,6where the sign and the strength

of the magnetic charge is determined only by the 2 by 2

effective Dirac-type Hamiltonian. With a proper gauge

transformation and scale transformation, the effective

Hamiltonians at the band-touching points Pj (j = 1,2)

take a form;

Heff= ω0τ0+ (λ − λc,j)τ3+ apxτ1+ bpyτ2,

with a > 0, b > 0, (px,py) ≡ (kx− π,ky) for j = 1 and

(px,py) ≡ (kx,ky− π) for j = 2. From this, equation

(13) is derived by the replacement of pµ→ −i∂µ.

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