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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
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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
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§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
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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
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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
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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
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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
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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
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Waves in spatially periodic systems: -- Berry curvature in k-space – (e.g.) electrons in a crystal
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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
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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
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§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
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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
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: 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
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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
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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
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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
bulk mode: Chern number= Ch2
bulk mode: Chern number= Ch3
(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
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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= -
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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
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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
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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
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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
Integer quantum Hall effect
electron
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2D electron gas in a magnetic field
Ch=-1
Ch=-1
Ch=-1
Number of edge states
= -2
Number of edge states
= -1
Number of edge states
= 0
Electronic states form Landau levels with spacing cwh
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§3 Various Hall effects
by Berry curvature
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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
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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
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transverse velocity
Hall effect
Semiclassical theory Adams, Blount; Sundaram,Niu, …
( : band index)
: periodic part of the Bloch wf.
: 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
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obtains a phase during
propagation
Wavepacket motion and Berry phase (Wavepacket)
Wavepacket center shifts by
Semiclassical eq. of motion (Adams, Blount; Sundaram,Niu)
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Example 1:
spin Hall effect of electrons
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semiclassical eq. of motion for
wavepackets
( : band index)
: periodic part of the Bloch wf.
: 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
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Example 2:
Hall effect of light
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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
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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
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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
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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
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Example 3:
Magnon thermal Hall effect
• Matsumoto, Murakami, Phys. Rev. Lett. 106, 197202 (2011).
• Matsumoto, Murakami, Phys. Rev. B 84, 184406 (2011)
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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
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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
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Magnetic dipole interaction
• Dominant in long length scale (microns) • Similar to spin-orbit int. Berry curvature • Long-ranged nontrivial, controlled by shape
Magnetic domains
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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
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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
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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)
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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 ?? (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
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§4 Topological plasmonic crystals
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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
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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
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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??
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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
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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//
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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)
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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
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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
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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
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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
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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)
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Simulation (multiphysics, COMSOL)
Surface corrugation forming a triangular lattice
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Result of the simulation
Dirac
point
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Result of the simulation
Dirac
point
Linear splitting
velocity~2*108 m/s
~plasmon velocity for
flat surface
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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
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§6 Topological magnonic crystals
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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)
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Chern number & topological chiral modes
Band gap Chern number for n-th band = integer
topological chiral edge modes
• Analogous to chiral edge states of quantum Hall effect.
• N>0 cw, N<0: ccw mode
Berry curvature
bulk mode: Chern number= Ch1
Ch1 topological edge modes
bulk mode: Chern number= Ch2
bulk mode: Chern number= Ch3
(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
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Integer quantum Hall effect
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
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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.
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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
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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
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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
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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
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“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
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• 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
topological chiral edge modes
Topologically nontrivial
= chiral edge modes
+1 chiral mode -1 chiral mode
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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)
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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
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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