topology in the momentum space and emergent phenomena...

Topology in the momentum space and emergent phenomena:

from electrons waves in solids

Shuichi Murakami Department of Physics , Tokyo Institute of Technology TIES , Tokyo Institute of Technology CREST, JST

IAS Winter School & Workshop on Advanced Concepts in Wave Physics:

Topology and Parity-Time Symmetries

HKUST, Jan. 11, 2016

http://www.titech.ac.jp/home-j.html

Contents:

§1 Berry curvature

§2 integer quantum Hall effect

and Chern number

§3 Various Hall effects by Berry curvature

§4 topological magnonic crystals

§5 topological plasmonic crystals

§1 Berry curvature

Eigenvalue equation dependent on parameter

Berry curvature

ˆk k k k

H u r E u r

k

, ( ) .nk nkn z

x y

u uB k i c c

k k

Phenomena due to Berry curvature of band structure

• Hall effect Quantum Hall effect

chiral edge modes

• Spin Hall effect (of electrons) Topological insulators

helical edge/surface modes

•Spin Hall effect of light one-way waveguide in photonic crystal

• Magnon thermal Hall effect topological

magnonic crystal

Electrons

Photons

Gapless Gapped

Electrons

Magnons

Plasmons • plasmon Hall effect ? topological

plasmonic crystal ?

topological phase

edge/surface modes Hall effect

Berry phase by an adiabatic change of a system

M.V.Berry, Proc. R. Soc. Lond. A392, 45 (1984)

: dependent on time-dependent parameters

: Closed path

Initially, the state is one of the eigenstate

: n-th eigenstate

Adiabatic change (T: very large) : the state remains the n-th eigenstate,

Berry phase :

C

provided there is no degeneracy .

)()()()( RRERRH nnn

),,()(

YXHRH

R

)()0( TRR

)0()(: RTRTt

)0(:0 Rt

),(),()( tYtXtR

)0( t .))0(()0( Rt n

))(()( tRt n

))(()( 0))((

)( tReet n

dttREti

t

ni

dttRdt

dtRit n

t

n ))(())(()(0

1nE

nE

1nE

tT

Berry phase for the loop C (t=0t=T)

: real

Properties: 1. It is determined by the shape of the loop C (does not depend on the time evolution )

2. Gauge invariant Invariant under gauge transformation

dttRdt

dtRit n

t

n ))(())(()(0

( ( )) ( ( )( ) )n R nC

i R t R tC dR )()0( TRR

R

C

( )R t

)()()(

Ri

nnneRR

For example, if

: Berry connection

: Berry curvature

where we used

Rewritten as a surface integral

Sn

Cn

SdB

RdAC

)(

),,( ZYXR

))(())(()( tRtRiRA nRnn

( ) ( )n nB R A R

nm mn

nkmmjn

ijk

nm

nkmmnjijk

nknjijkin

EE

HHi

i

iB

2)(

)(RBn

R

)()0( TRR

S

mn

nkm

nkm

nknnnknknknnn

EE

H

EEHHEH

Berry curvature in R space

: “vector potential in R space”

: “magnetic field in R-space”

antimonopole

monopole

Berry curvature

Berry connection

Monopole density

: gauge transformation

Similar to magnetic field !

: integer

Quantization of monopole charge

( )ni

i

A R i nR nRR

( ) ( )n nRB R A R

( ) ( )n nkR B R

( )( ) ( )

( ) ( )

( ) ( )

( ) ( )

i R

n n

n n R

n n

n n

u R u R e

A R A R

B R B R

R R

( ) ( )n l l

l

R q R R lq

Waves in spatially periodic systems: -- Berry curvature in k-space – (e.g.) electrons in a crystal

Schrodinger eq. in crystals Wavefunction labeled by wavevector

Bloch theorem

: periodic function

Eigenvalue eq. for

: periodic potential

From the k-dependent eigenvalue eq.

one can define the Berry curvature

Bloch theorem

2

2

pV r r E r

m

V r V r a

a

ik r

k kr e u r

k

k

u r

k

u r

2

2 k k k

i kV r u r E u r

m

ˆk k k k

H u r E u r

ˆk k k k

H u r E u r

( ) i . .nk nkn

x y

u uB k c c

k k

Berry curvature in k space

( : band index)

: periodic part of the Bloch wf.

: “magnetic field in k-space”

antimonopole

monopole

Berry curvature

Monopole density

( ) i . .nk nkn

x y

u uB k c c

k k

xki

knknexux

)()(

knu

n

( ) ( )n nkk B k

§２

integer quantum Hall effect

and Chern number

: Hall conductivity

: current

: electric field

Expressed in terms of

Berry curvature

crystal x xy yj E

( )nB k

xyyE

xj

Kubo formula for Hall conductivity (noninteracting electron, T=0) Thouless et al. PRL49,405(1982), Kohmoto, Ann.Phys.(NY) 160,343 (1985)

2D system + electric field along y:

Perturbation theory

Current density

We then use

Hall conductivity

band index

Bloch wavevector

(0, )E E

( )

eEy

E E

2 2( )

( )1 1( ) ( ) . .

x

x x

evj f E j f E c c

L L

y

E E

eE

,y

i iv y yy H E E

2

22( )

( ) . .x y

xy

v vief E c c

L E E

( , ), ( , )n k m k , :m n

:k

: Berry curvature

Hall conductivity

band index

Bloch wavevector

Hall conductivity

Brillouin zone

: Fermi distribution

2

22( )

( ) . .x y

xy

v vief E c c

L E E

( , ), ( , )n k m k , :m n

:k1

i

i

Hv

k

2

22( ),

( ) . .x y

xy nkm nk n

nk mk

H Hnk mk mk nk

k kief E c c

L E E

2 2

,2( ) ( )

(2 )xy n znkBZ

n

e d kf E B k

h

, ( ) .nk nkn z

x y

u uB k i c c

k k

( )nk

f E

yk

xk

Quantization of Hall conductivity for insulators Thouless et al. PRL49,405(1982), Kohmoto, Ann.Phys.(NY) 160,343 (1985)

: Chern number of the n-th band : always an integer

Integer quantum Hall effect

for gapped system (insulator) :

no states on the Fermi energy

2 2

,2( ) ( )

(2 )xy n znkBZ

n

e d kf E B k

h

, ( ) .nk nkn z

x y

u uB k i c c

k k

FE

E

k

band filled

2

n

nxy Chh

e

kdkBCh znn

2

BZ )()( )(2

1

2

integerxy

e

h

From Stokes theorem…

Chern number is always an integer

Phase of the Bloch wavefunction is important

BZ

(Proof):

This is not correct when cannot be chosen as a single function over the BZ. ( ),nk

u x yr

BZ periodicity

Divide BZ into two patches

( ) , ( ) ( )n n nnk nk kA k i u u B k A k

k

¶= = Ñ ´

¶s r r

r r rr rrr

2

,BZ( )

2n n z

d kCh B k

p= ò

r

( ),nk

u x yr

2

,BZ BZ( ) ( ) 0

2 2n n z n

d k dkCh B k A k

p p¶= = × =ò ò

rr rr

Ñ

( )) ( )( 'i k

n nkkeu k u k

( )1 1

2 2

1 ( )integ

(

r2

)

e

nC C

C

nnCh dk dk

kdk

k

k kA Ap p

q

p

= × - ×

¶= × =

¢

¶

ò ò

ò

rrr

r

r

rr

r r

Ñ Ñ

Ñ0k

Chern number & topological chiral modes

Band gap Chern number for n-th band = integer

topological chiral edge modes

Berry curvature

bulk mode: Chern number= Ch1

Ch1 topological edge modes



(Ch1+Ch2) topological edge modes

( )2

Ch2

n nBZ

d kk

p= Wò

r

bands below

Ch #(clockwise chiraledgestates in the gap at )n

n E

N EÎ

= ºå

k

Relation between the number of edge states and Chern number:

Laughlin gedanken experiment:

Number of electron carried from left end to right end = Ch

Increase the flux charge transport along x

This charge transport is

between the edge states

on the left and right ends.

gapless edge modes exist.

Gradual change of vector pot. = change of wavenumber

Edge states at the right end

0

2

0 0Ch Chx xy

y

y

y y y

e ej

TL h TL TLE

TL

f ffs

-= = = -Þ=

ChQ eÞ = - ×

0

0at 0

at

t

t Tf

F = =

F = =jr

Er

y

y

AL

F= -

Brillouin zone Total “flux” inside the Brilouin zone

k

(Example) Topological number :

Invariant under continuous change of the system

Number of chiral edge modes

2

( ) ( )BZ

1( )

2n n zCh B k d k

band filled

2

n

nxy Chh

e

E

FE

2 2 2

filled band

21 3xy n

n

e e eCh

h h h

2

filled band

xy n

n

eCh

hs

Î

= å

yk

xk

2D Brillouin zone torus

Attach

Brillouin zone Total “flux” inside the Brilouin zone

Total flux through torus surface

= number of magnetic monopoles inside

the torus

Topological number :

Invariant under continuous change of the system

Quantization of Hall conductivity

＝quantization of monopole charge

2

( ) ( )BZ

1( )

2n n zCh B k d k

band filled

2

n

nxy Chh

e

yk

xk

2

2 2 ( )2S

d rg r

p- = Wò

r 2

( )2

j jBZ

d kC k

p= Wò

r

They cannot be continuously

deformed to each other

Topology & topological number

Classified by genus g (number of holes)

genus= topological number

(unchanged by

continuous deformation)

g =0 g =1

Classified by Chern number

Chern number=

topological number

(unchanged by

continuous deformation)

A magnonic band

(separated by a gap)

Berry curvature

Number of chiral edge modes genus

Gauss curvature

jC

Phys. Rev. Lett. 59, 1776 - 1779 (1987)

R. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui

A. C. Gossard, J. H. English

chiral edge state


electron

2D electron gas in a magnetic field

Ch=-1

Ch=-1

Ch=-1

Number of edge states

= -2


= -1


= 0

Electronic states form Landau levels with spacing cwh

§3 Various Hall effects

by Berry curvature



chiral edge modes





magnonic crystal

Electrons

Photons

Gapless Gapped

Electrons

Magnons


plasmonic crystal ?

topological phase


transverse velocity

Hall effect

Semiclassical theory Adams, Blount; Sundaram,Niu, …

( : band index)


: Berry curvature

Boltzmann

transport

Motion of a wavepacket under slowly varying background

For electrons:

: force onto the electron

( )1( )n

n

E kx k k

k

k eE

k

( ) n nn

u uk i

k k

knu

xki

knknexux

)()(

n

obtains a phase during

propagation

Wavepacket motion and Berry phase (Wavepacket)

Wavepacket center shifts by

Semiclassical eq. of motion (Adams, Blount; Sundaram,Niu)

Example 1:

spin Hall effect of electrons

semiclassical eq. of motion for

wavepackets

( : band index)


: Berry curvature

- SM, Nagaosa, Zhang, Science (2003)

- Sinova et al., Phys. Rev. Lett. (2004)

Adams, Blount; Sundaram,Niu, …

Intrinsic spin Hall effect in metals& semiconductors

Spin-orbit coupling Berry curvature depends on spin

Force // electric field

( )1( )n

n

E kx k k

k

k eE

( ) n nn

u uk i

k k

knu

xki

knknexux

)()( n

Example 2:

Hall effect of light

Spin Hall effect of light

Onoda, SM, Nagaosa, Phys. Rev. Lett. (2004)

Onoda, SM, Nagaosa, Phys. Rev. E (2006)

Semiclassical eq. of motion

: slowly varying

: Berry curvature

: gauge field

gradient of refractive index

spin Hall effect of light”

Shift of a trajectory of light beam

: polarization

In the vacuum

Left circular pol.

right

Geometrical optics

“Fermat’s principle”

spin-orbit coupling of light

(transverse only)

Theory: Fedorov (1955) Experiment: Imbert, PRD (1972) Hosten, Kwiat, Science (2008)

Imbert shift

zkkiz

rvkk

zkzkkrvr

)(

)(

)(ˆ)(

1

1)(

3k

kk

Berry phase

Anomalous velocity = = transverse shift

For refracted light,

Anomalous velocity =

Left circular pol.

Right circular pol.

Imbert shift of light beam = Berry curvature of photon

Theory: Fedorov (1955) Experiment: Imbert, PRD (1972) Hosten, Kwiat, Science (2008)

Imbert shift

Onoda, SM, Nagaosa, Phys. Rev. Lett. (2004) Onoda, SM, Nagaosa, Phys. Rev. E (2006)

Semiclassical eq. of motion

1

1)(

3k

kk

)||()( zzkk

krvr

k

)(rvkk

Hosten, Kwiat, Science 319,787 (2008)

Magnitude of the shift

Width of the beam is much larger not easy to observe.

Experiments on imbert shift : Shift of light beam in reflection/refraction

28 total reflections shift is enhanced

Imbert, Phys. Rev. D5, 787 (1972)

total reflection refraction

Good agreement with theory

right circular pol.

k and S antiparallel

left circular pol.

k and S parallel

z z

k k

S

S

Spin Hall effect of light

strong spin orbit coupling in photon nonzero Berry curvature

k and S are either

parallel or antiparallel. Left circular

Right circular

1

1)(

3k

kk

E

E

Example 3:

Magnon thermal Hall effect

• Matsumoto, Murakami, Phys. Rev. Lett. 106, 197202 (2011).

• Matsumoto, Murakami, Phys. Rev. B 84, 184406 (2011)

V(r)： confinement

potential

From electrons to magnons (spin waves)

electrons magnons

• Exchange magnons (quantum-mechanical)

e.g. Lu2V2O7

(Katsura et al. (2011), Onose et al.(2011))

• Magnetostatic spin waves (classical)

e.g. YIG (yttrium iron garnet)

R. Matsumoto, S. Murakami,

Phys. Rev. Lett. 106, 197202 (2011).

: Berry curvature

– electron: : charge=-e, fermion

– Spin wave (magnon): charge=0, boson

wave = represented by Bloch wavefunction

( ) n nn

u uk i

k k

( )1( )n

n

E kx k k

k

k e E x B

( )1( )n

n

E kx k k

k

k V

Magnon Thermal Hall effect in ferromagnet

Q xy yxj T

Heat current

Magnon (spin wave) = low-energy excitations in magnetts

Berry curvature

dispersion

Matsumoto, Murakami, Phys. Rev. Lett. 106, 197202 (2011).

Matsumoto, Murakami, Phys. Rev. B 84, 184406 (2011)

Thermal Hall conductivity

Dipolar interaction spin-orbit coupling

Magnetic dipole interaction

• Dominant in long length scale (microns) • Similar to spin-orbit int. Berry curvature • Long-ranged nontrivial, controlled by shape

Magnetic domains

k k~μm-1 k~nm-1

GHz~THz ~GHz

Spin wave dispersion for thin-film ferromagnet MagnetoStatic Forward Volume Wave

Magnetostatic mode

dipolar interaction

Classical, long-ranged

Magnon

Exchange

quantum mechnical

short ranged

“spin-orbit coupling” nonzero Berry curvature

2E k

w

Generalized eigenvalue eq.

• Landau-Lifshitz (LL) equation

• Maxwell equation

• Boundary conditions

B. A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013 (1986)

Magnetostatic modes

in ferromagnetic films (YIG)

Berry curvature

(Only the dipolar coupling is considered.)

/2

/2

ˆˆ ˆ( ) ( ) ( ) ( ) ( , ) ( )L

z H ML

H z z H z z dz G z z zm m m m m

0 0

1 0, , : thickness of the film, , m(z)

0 1

x y

H M z

x y

m imH M L

m imw g w g s

æ öæ ö + ÷÷ çç ÷= = = ÷ = çç ÷÷ çç ÷ç ÷-ç-è ø è ø

, ,( ) Im

n n

n zk k

k km m

k

H

Berry curvature

for magnetostatic forward volume wave mode

: Band structure for magnetostatic

forward volume-wave mode : Berry curvature

Berry curvature is zero for backward volume wave and surface wave

• R. Matsumoto, S. Murakami, PRL 106,197202 (2011), PRB84, 184406 (2011)



chiral edge modes




• Magnon thermal Hall effect ?? (magnonic crystal)

Fermions

Bosons

Gapless Gapped Topological edge/surface

modes in gapped systems Exert a force transverse motion

Electric field

Electric field

Spatial gradient of refractive index

Temperature gradient

§4 Topological plasmonic crystals

Optical one-way waveguide in photonic crystal: -- Topological photonic crystal --

Periodic array of ferrite rods in air (Periodicity =4cm) One-way propagation

(105 times difference) (A B)/(BA) =105

Chiral mode !

A B

Obstracle (metal)

Ch=0 Ch=1 1 Chiral edge mode

Theory: Haldane, Raghu, PRL100, 013904 (2008) Experiment: Wang, Chong, Joannopoulos, Soljačić, Nature 461, 772 (2009)

Photonic gap

Photonic gap

Surface plasmon polariton(SPP)

Electromagnetic modes propagating

on a metal surface with negative dielectric constant

Dielectric constant

It becomes negative for ω < ωp

1次元プラズモニック結晶のエルミートな2バンドモデル

Intensity

Metal with negative dielectric constant

metal

vacuum

ωp : plasma frequency

metal

Plasmonic

Band gap

wavenumber 0

fre

qu

en

cy

• Hermitian formulation is required

• We need to formulate Berry curvature and Chern number

“ quantum Hall effect for plasmons by introducing spatial periodicity

Non-hermitian • Formulation of perturbation theory is not easy.

• Eigenfrequency may become complex

Previous work

Topological plasmonic crystals??

Towards reformulation of SPP equations…:

For calculation of Berry curvature:

we need to calculate wavefunctions analytically

we focus on small surface corrugation & apply perturbation theory

Surface

corrugation

metal metal

Surface profile

Even for small s(x), the change of the dielectric constant at the point P is large

perturbation theory cannot be applied in the usual way.

We introduce a coordinate transformation

Hermitian eigenvalue equation for 1D plasmonic crystals Maxwell eq. for flat surface

Eigenvectors for unperturbed problem

Include surface corrugation as a perturbation： coordinate transformation

（For simplicity the corrugation is set to be sinusoidal）

metal

vacuum

metal

vacuum

Boundary condition for electromagnetic wave Hermitian eigenvalue problem for SPP

Kitamura, Murakami, PRB 88, 045406 (2013)

a: Wavenumber in vacuum

b: Wavenumber in metal

a: Dielectric const. in vacuum

b: Dielectric const. in metal

• Boundary condition for E//

• Boundary condition for H//

Hermiticity

0 K -K

Fre

quency

Wavenumber

・In the metal

Automatically satisfied by ・In vacuum

Satisfied when

Outside of the light cone: Hermitian

Inside of the light cone: non-Hermitian

（radiation of the SPP into vacuum）

Eigenvalue eq. for SPP

Wavenumber

Fre

qu

en

cy

0 2K K

Zone Boundary

（Amplitudes for two waves)

Hermitian when a, b are pure imaginary

Kitamura, Murakami, PRB 88, 045406 (2013)

At Brillouin zone boundary

• eigenfrequency

Band gap at the zone boundary

K ：wavenumber at zone boundary

0 K -K

Fre

qu

en

cy

Wavenumber

（Schematic） Surface corrugation

g : size of corrugation

• eigenfunction

+ + + - - - + + + - - -

+ + + - - - + + +

- - - + + +

E

Two-dimensional plasmonic crystal

Hexagonal Brillouin zone

Corrugation forming triangular lattice Dirac cones at the Brillouin zone corners.

K’

K’

K’

Solution for flat surface

Boundary condition at K point

Surface

corrugation

Reciprocal lattice vector

Dirac cones at K and K’ gap opening by time-reversal symm. breaking

K

k

Solutions at the zone corners (K,K points) Solutions at flat surface hybridization due to surface corrugation:

: splits into doublet & singlet at K & K’

Singlet

Doublet

k

K K’

K’

K’

K

K

Group velocity

Dirac cone

( : SPP velocity for flat surface 2

v»

v

K

k k

Group velocity

( : velocity of SPP at flat surface)

• Dirac cone at K and K’

because of zero gap, time-reversal symmetry breaking will open a gap

topological plasmonic crystal

lowest band has a nonzero topological number (Chern number)=+1 or -1

Dirac cones at K and K’ gap opening by time-reversal symm. breaking

2

v»

v

Breaking time-reversal symmetry

gap at zone boundaries(K,K’)

Magnetic field off-diagonal components for dielectric tensor

Gap opens at the Dirac cones in K, K’

Topological plasmonic crystals

The l owest band has a nontrivial topological

number (Chern number =+1)

Simulation （multiphysics, COMSOL)

Surface corrugation forming a triangular lattice

Result of the simulation

Dirac

point

Result of the simulation

Dirac

point

Linear splitting

velocity～2*108 m/s

～plasmon velocity for

flat surface



chiral edge modes





magnonic crystal

Electrons

Photons

Gapless Gapped

Electrons

Magnons


plasmonic crystal ?

topological phase


§6 Topological magnonic crystals

Topological chiral modes

in magnonic crystals

• R. Shindou, R. Matsumoto, J. Ohe, S. Murakami, Phys. Rev. B 87,174427

(2013)

• R. Shindou, J. Ohe, R. Matsumoto, S. Murakami, E. Saitoh,Phys. Rev. B87,

174402 (2013)

Chern number & topological chiral modes

Band gap Chern number for n-th band = integer


• Analogous to chiral edge states of quantum Hall effect.

• N>0 cw, N<0: ccw mode

Berry curvature


Ch1 topological edge modes



(Ch1+Ch2) topological edge modes

( )2

Ch2

n nBZ

d kk

p= Wò

r

bands below

Ch #(clockwise chiraledgestates in the gap at )n

n E

N EÎ

= ºå

k

( ) n nn

u uk i

k k


Ch=-1

Ch=1

Ch=1 2 chiral

edge states

1 chiral

edge mode

0 edge mode

B

Topological photonic crystals Theory: Haldane, Raghu, PRL100, 013904 (2008) Experiment: Wang et al., Nature 461, 772 (2009)

Ch=0

Ch=1

1 Chiral

edge mode

Photonic gap

Photonic gap

Landau-Lifshitz equation

Maxwell equation (magnetostatic approx.)

Linearized EOM

exchange field (quantum mechanical short-range)

Dipolar field (classical, long range)

External field

• Saturation magnetization Ms • exchange interaction length Q

2D Magnonic Crystal : periodically modulated magnetic materials

YIG (host) Iron (substitute)

ax

ay

H// z modulated

bosonic Bogoliubov – de Gennes eq.

chiral magnonic band in magnonic crystal

`exchange’ regime

`dipolar’ regime

2nd Lowest band

Lowest magnon band

λ=0.35um, r=1

dipolar interaction non-trivial Chern integer (like spin-orbit interaction)

x y

y

x

a a

ar

a

: unit cell size : aspect ratio of unit cell

YIG (host) Iron (substitute)

ax

ay

H// z

Phase diagram

Magnonic gap between

1st and 2nd bands

C1 (Chern number)

for 1st band

=Number of

topological chiral

modes within the gap

2

1 1( )2BZ

d kC k

p= Wò

r

bulk

bulk f=4.2GHz

bulk

f=4.5GHz

bulk

f=4.4GHz

edge

Simulation (by Dr. Ohe) DC magnetic field : out-of-plane AC magnetic field : in-plane

Magnonic crystals with ferromagnetic dot array R. Shindou, J. Ohe, R. Matsumoto, S. Murakami, E. Saitoh, PRB (2013)

dot (=thin magnetic disc) cluster: forming “atomic orbitals”

convenient for (1) understanding how the topological phases appear

(2) designing topological phases

decorated square lattice

Each island is assumed to behave as monodomain

H//z

Equilibrium spin configuration

2.4, 2 1.2, 1.70, 1.0x y se e r V M= = = D = =r r

Magnetostatic energy

Magnonic crystals: decorated square lattice

Hext

Hext

Hext < Hc=1.71

Hext > Hc

Collinear // Hext

Tilted along Hext

“atomic orbitals” within a single cluster

Spin wave excitations: “atomic orbitals”

relative phase for precessions

H<Hc: noncollinear H>Hc:

collinear // z

nJ=1 and nJ=3 degenerate at H=0

nJ=0 softens

at H=Hc

nJ=2 is lowest at H=0: favorable for dipolar int.

nJ=0

(s-orbital)

nJ=+1

(px+ipy-orbital)

nJ=3

(px-ipy-orbital) nJ=2

(dx2-y2-orbital)

Equilibrium configuration

Energy levels of atomic orbitals

• Spin-wave bands and Chern numbers

H=0

H=0.47Hc

H=0.82Hc

H=1.01Hc

H=1.1Hc

H=1.4Hc

Red: Ch=-1

Blue: Ch=+1

Time-reversal

symmetry

Topologically

trivial

•Small H<<Hc

•Large H>>Hc

Dipolar interaction is

weak

Nontrivial phases (i.e. nonzero Chern number )

in the intermediate magnetic field strength


Topologically nontrivial

= chiral edge modes

+1 chiral mode -1 chiral mode

H=0.47Hc

H=0.76Hc

H=0.82Hc

Magnonic crystals: edge states and Chern numbers (1)

Red: Ch=-1

Blue: Ch=+1

+1 chiral mode

-1 chiral mode

-1 chiral mode

+1 chiral mode

Edge states

bulk Strip geometry

(bulk+edge)

Magnonic crystals: tight-binding model with atomic orbitals

Gap closes at M

(example) :

H=0.47Hc H=0.82Hc

gap between 3rd and 4th bands

retain only nJ=0 and nJ=1 orbitals

tight binding model

complex phase for hopping

px+ipy orbitals

nJ=0

(s-orbital)

nJ=+1

(px+ipy-orbital)

+i = Model for quantum anomalous Hall effect

(e.g. Bernevig et al., Science 314, 1757 (2006))

Gap closing + topological transition

Change of Chern number at gap closing event

parameter

Ci

Cj

C’i

C’j

( Ci +CI =C’i +C’I: sum is conserved)

DCi = -DCj = ±1

: Dirac cone at gap closing

volume mode (=bulk)

gap closes

magnonic

(volume-mode)

bands

topology in the momentum space and emergent phenomena...

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