Berry Phase PhenomenaOptical Hall effect
and Ferroelectricity as quantum charge pumping
Naoto NagaosaCREST, Dept. Applied Physics, The University of Tokyo
M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004)
S. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 167602 (2004)
Berry phase M.V.Berry, Proc. R.Soc. Lond. A392, 45(1984)
)(XH Hamiltonian, ),,,,( 21 nXXXX parametersadiabatic change
)())(()( ttXHti t
1X
2XC
)0()( 0))(()/()(
T
nn
tXdtEiCi eeT
)()()()( XXEXXH nnn
)()(
)(|)()(
XBdSXAdX
XXdXiC
nC n
nC Xnn
Berry Phase
Connection of the wavefunction in the parameter spaceBerry phase curvature
eigenvalue and eigenstate for each parameter set X
Transitions between eigenstates are forbidden during the adiabatic changeProjection to the sub-space of Hilbert space constrained quantum system
Electrons with ”constraint”
Projection onto positive energy stateSpin-orbit interaction
as SU(2) gauge connection
Dirac electrons
doublydegenerate
positive energy states.
E
k
Bloch electrons
Projection onto each bandBerry phase
of Bloch wavefunction
k
E
Spin Hall Effect (S.C.Zhang’s talk) Anomalous Hall Effect (Haldane’s talk)
Berry Phase Curvature in k-space
Bloch wavefucntion )()( ruer nkikr
nk
nkknkn uuikA ||)( Berry phase connection in k-space
)()( kAikArx nknii i covariant derivative
)())()((],[ kiBkAkAiyx nznxknyk yx Curvature in k-space
y
VkB
m
k
y
Vyxi
m
kHxi
dt
tdxnz
xx
)(],[],[)(
xk yk
zk
knku|
nku|k
Anomalous Velocity andAnomalous Hall Effect
Non-commutative Q.M.
dt
tkdkB
k
k
dt
trdn
n )()(
)()(
dt
trdrB
r
rV
dt
tkd )()(
)()(
Duality between Real and Momentum Spaces
k- space curvature
r- space curvature
Z.FangSrRuO3
Degeneracy point Monopole in momentum space
Fermat’s principle and principle of least action
Path 1
Path 2
Path 3
Path 4Path 5
Every path has a specific optical path length or action.
Fermat : stationary optical path length → actual trajectoryLeast action : stationary action → actual trajectory
Start
Goal
Searching stationary value ~ Solving equations of motion
Trajectories of light and particle
dtdsrnrn
rnrds
drn
ds
d
rn
cc
ccc
c
)(indexrefractive:)(
)()(
])(larger ofdirection in the[turn OpticslGeometrica
,
potential:)(mass,:
)(
])(lower ofdirecton in the[turn motion ofequation sNewton'
c
cc
c
rVm
rVrdt
d
dt
dm
rV
What determine the equations of motion?Historically, experiments and observations
Any fundamental principles?(Fermat’s principle, principle of least action)
Geometrical phase (Berry phase)
Principle of least actionPhase factor → Equations of motion
Although light has spin, no effect of Berry phase in conventional geometrical optics.
Berry phase“Wave functions with spin obtaingeometrical phase in adiabatic motion.”
Topological effects (wave optics)in trajectory of light (geometrical optics)→ wave packet
Effective Lagrangian of wave packet
Hdt
diH
dt
diL variaton
cc
cc
krWHdt
diWL
krW
andofEOMvariation
momentumandpositionatcenteredpacketwave:
eff
R. Jackiw and A. Kerman,Phys. Lett. 71A, 581 (1979)
A. Pattanayak and W.C. Schieve, Phys. Rev. E 50, 3601 (1994)
effCondition
operatorposition:
LWRWr
R
c
WHW
WRWr
a
zazrkkwkd
W
Hc
k
ckccc
gravity ofcentertheforCondition
photonpolarizedcircularlyofoperatorcreation:
1,0),()2(
2
3
)(2
)()(
2
)(
varyingslowly:)(and)(,)(2
)()(
2
)(
22
22
rHr
rEr
rrdR
rrrHr
rEr
rdH
H
Light in weakly inhomogeneous medium
kkkk
cc
cc
cc
ccccckcccc
eeirr
rvz
zz
krvzzizzkrkLc
,)()(
1)(,)|
)()|()||(eff
Equations of motion of optical packet
)|)|
)]([
)||()(
ckcc
ccc
ckccc
ccc
zkiz
krvk
zzkk
krvr
c
c
Anomalous velocity
curvatureBerry:
connectionBerry:
onpolarizatiofstate:)|
speedlight:)(
momentum:,position:
k
k
c
c
cc
z
rv
kr
33
vectoronpolarizati:
k
ki
e
eei
kkkkk
k
kkkk
Neglecting polarization→ Conventional geometrical optics
Berry Phase in Optics
Propagation of light and rotation of polarization plane in the helical optical fiber
Chiao-Wu, Tomita-Chiao, Haldane, Berry
],[| iniinitoutc zezez
S
kkk dSdk ][][ Spin 1 Berry phase
Reflection and refraction at an interface
No polarization Circularly polarized
Shift perpendicular to both of incident axis and gradient of refractive index
Conservation law of angular momentum
Conservation of total angular momentum as a photon
EOM are derived under the condition of weak inhomogeneity.Application to the case with a sharp interface?
RTA
k
zzzzy
Ic
IIc
IcA
Ac
AcA
c
,
sin
cos)||(cos)||( 33
reflected:d,transmitte:incident,:
,
const.)||( 3
RTI
jjjj
k
kzzkrj
Rz
Iz
Tz
Iz
zc
cccccz
Comparison with numerical simulation
V0: light speed in lower mediumV1: light speed in upper medium
Solid and broken lines are derived by the conservation law.●and ■ are obtained by numerically solving Maxwell equations.
Photonic crystal and Berry phase
Knowledge about electrons in solidsPeriodic structure without a symmetry→Bloch wave with Berry phase
Example of 2D photonic crystal without inversion symmetry
Photonic crystal without a symmetry → Bloch wave of light with Berry phase
Enhancement of optical Hall effect ?!
Shift in reflection and refractionSmall Berry curvature→small shift of the order of wave length
Wave in periodic structure -- Bloch wave --
Wave packet of Bloch wave (right Fig.)Red line = periodic structure + constant incline
http://ppprs1.phy.tu-dresden.de/~rosam/kurzzeit/main/bloch/bo_sub.html
Strength of periodic structure
Energy
Meaning of the height of periodic structureElectron : electrical potentialLight : (phase) velocity of light
For low energy Bloch waveLarge amplitude at low pointSmall amplitude at high point
Bloch waveAn intermediate between traveling wave and standing wave
Dielectric function and photonic band
We shall consider wave ribbons with kz=0.Note: Eigenmodes with kz=0 are classified into TE or TM mode.
fieldmagnetic:,fieldelectric:
bandthoffunctionsBloch:
2
1
1)( of case in theenergy bandth :
,
HE
nu
uuuui
rnE
HE
kn
knkkn
H
knk
H
kn
E
knk
E
knkn
kn c
Berry curvature of optical Bloch wave
cc kncknccc ErkrkL
)(eff
)(
)(
)(
1,modulationmoderate:)(
2
r
r
rr
For simplicity, we consider the case in which the spin degeneracy is resolved due to periodic structure.
Berry curvature in photonic crystal
Berry curvature is large at the region whereseparation between adjacent bands is small.
c.f. Haldane-Raghu Edge mode
Trajectory of wave packet in photonic crystal
)(
)(
)(
1
modulation edsuperimpos :)(
,)]([
,)(
2
rε
x
rε
x
Exk
kExr
c
ccc
kncc
kcknkcc
Large shift of several dozens of lattice constant
Superimposed modulation around x = 0 instead of a boundaryNote:The figure is the top view of 2D photonic crystal. Periodic structure is not shown.
classical theory of polarization
d
dRdRdR rrrurrRp ')'(')'(')( dd
polarization due to displacements of rigid ions
Ionic polarization+
• It is not well-defined in general. It depends on the choice of a unit cell.
• It is not a bulk polarization.
R
RpRrrP )()()( f
Polarization of a unit cell R
Averaged polarization at r Charge determines pol.Ionicity is needed !!
quantum theory of polarization
Covalent ferroelectric: polarization without ionicity
“r” is ill-defined for extended Bloch wavefunction
l
lil
li
il
llllllli
QkndeA
QdA
ddndeQdP
Im)()2(2
)()2()(
33
33
kk
rrkk
P is given by the amount of the charge transfer due to the displacement of the atoms
Integral of the polarization current along the path C determines P
P is path dependent in general !!
Ferroelectricity in Hydrogen Bonded Supermolecular Chain
S.Horiuchi et al 2004
uePcl* )/(30 *eePP clobs
ee 01.0* Neutral and covalent
Polarization is “huge” compared with the classical estimate
Ferroelectricity in Phz-H2ca
S. Horiuchi @ CERC et al.
Hydrogen bond
( covalency)
P()
2e
(2 )3 dk dk dk u()
kn k
u()kn
n1
occ
Polarization as a Berry phase
First-principles calculationIsolated molecule → 0.1 μC/cm2 (too small !)
Large polarization with covalency
With F. Ishii @ERATO-SSS
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13
Ps(
μC
/cm
2)
Asymmetry in Bond length O-H (ang.)
Isolated molecule
Bulk
Geometrical meaning of polarization in 1D two-band model
Q
h
k
hh
dkeQA
QdQAdPˆˆ
ˆ4
)(
)(
dP : Solid angle of the ribon
matrices Pauli with ),(),(),(),(
),(),(),(),(
),(),(),(
3021
2130
0
QkhQkQkihQkh
QkihQkhQkhQk
QkhQkQkH
Generalized Born charge
Strings as trajectories of band-crossing points
1. only along strings (trajectories of band-crossing points) with k in [aa
-function singularity along strings (monopoles in k space)
2. Divergence-free
3. Total flux of the string is quantized to be an integer
(Pontryagin index, or wrapping number): [c.f. Thouless]
)()( QAQB Q
0)(
QB
0),(
Qkh
0)( QB
flux density:
S QC
nQBSdQAQd )()(
dQ
hd
dk
hd
dQ
hddkB
ˆˆˆ
24
3
C×[/a,/a]
B C
Q
Band-crossing point
Biot-Savart law, asymptotic behavior & charge pumping
L
t
QQQdQA
3|'|4
)'(')(
Transverse part of the polarization current A
Biot-Savart law:L : strings
Asymptotic behavior (leading order in 1/Eg)string
Eg
)(QA
Strength ~ 1/Eg
Direction: same as a magnetic field created by an electric current
Quantum charge pumping due to cyclic change of Q around a string
S QC
nQBSdQAQd )()(
ne
Specific modelsSimplest physically relevant models
QkgkfQkh
cQkhkcHk
kk
)()(),(
),()( ',',',0,
Different choices of f and g
Geometrically differentstructures of strings Band polarization current A
xE
zE
E
Quantum Charge Pumping in Insulator
Electron(charge)flow
Large polarization even in the neutral molecules
orPressure
Dimerized charge-ordered systems
TTF-CA(TMTTF)2PF6
(DI-DCNQI)2Ag
TTF-CA: polarization perpendicular to displacement of molecules. triggers the ferroelectricity.
Conclusions
・ Generalized equation of motion for geometrical optics taking into account the Berry phase assoiciated with the polarization ・ Optical Hall Effect and its enhancement in photonic crystal
・ Covalent (quantum) ferroelectricity is due to Berry phase and associated dissipationless current
・ Geometrical view for P in the parameter space - non-locality and Biot-Savart law
・ Possible charge pumping and D.C. current in insulator Ferroelectricity is analogous to the quantum Hall effect
Motivation of this study
Goal : dissipationless functionality of electrons in solidsKey concept : topological effects of wave phenomena of electrons
What is corresponding phenomena in optics?
Example of our studyTopological interpretation of quantization in quantum Hall effect
↓Intrinsic anomalous Hall effect and spin Hall effect
due to the geometrical phase of wave function
Geometrical optics : simple and useful for designing optical devices
Wave optics : complicated but capable of describing specific phenomena for wave
Topological effects of wave phenomena
Photonic crystals as media with eccentric refractive indices
→ Extended geometrical optics
Polarization and Angular momentum
Linear S = 0 Right circular S = +1 Left circular S = -1
http://www.physics.gla.ac.uk/Optics/projects/singlePhotonOAM/
Polarization and spin
Rotation and angular momentum
Rotation of center of gravity Rotation around center of gravity
http://www.expocenter.or.jp/shiori/
ugoki/ugoki1/ugoki1.html
Action and quantum mechanics
particleclassicaloftrajectoryactualactionstationary:
)(inand connects which trajectoryth theof funtional a
path)(trajectoryththeforaction:
),(),(),(
integralPath
st
00
000000st321
S
ttrrn
nS
rterdrteeerdrt
n
iSiSiSiS
Quantum mechanics“Wave-particle duality”“Everything is described by a wave function.”“Action in classical mechanics ~ phase factor of wave function”
Searching a trajectory of classical particle~ Solving a wave function approximately
Similar relation holds between geometrical and wave optics.
“Wave and geometrical optics”, “Quantum and classical mechanics”
Wave optics → Eikonal → Fermat’s principle → Geometrical optics
Quantum mechanics → Path integral → Principle of least action → Classical mechanics
Optical path, Action ~ Phase factor
Roughly speaking,Trajectory is determined by the phase factor of a wave function.
Hall effect of 2DES in periodic potential
0n with Hamiltonia:
)(
aroundOAM:
2][
][
0
0
BEH
uHEumL
rL
LBm
eEBE
BreEek
kBEr
cccccc
c
ccc
ccc
knkknknkkn
ckn
knknkn
cc
kncknkc
functionBloch:kn
knkkn
knkknkn
u
uui
constantlattice:
fieldmagnetic:
fieldelectric:
20
a
Beq
p
aB
B
E
z
M.-C. Chang and Q. Niu, Phys. Rev. B 53, 7010 (1996)
Optical path length and action
Particle in inhomogeneous potentialAction = Sum of (kinetic energy – potential) x (infinitesimal time) along a trajectory
Light in media with inhomogeneous refractive indexOptical path length= Sum of (refractive index x infinitesimal length) along a trajectory= Time from start to goalLight speed = 1/(refractive index)Time for infinitesimal length = (infinitesimal length) / (light speed)
Point
Optical path length and action can be defined for any trajectories,regardless of whether realistic or unrealistic.
Why is it interpreted as the optical Hall effect ?
Hall effect of electronsClassical HE : Lorentz forceQHE : anomalous velocity (Berry phase effect)Intrinsic AHE : anomalous velocity (Berry phase effect)Intrinsic spin HE : anomalous velocity (Berry phase effect)[Spin HE by Murakami, Nagaosa, Zhang, Science 301, 1378 (2003)]
Transverse shift of light in reflection and refraction at an interfaceThe shift is originated by the anomalous velocity.(Light will turn in the case of moderate gradient of refractive index.)
QHE, AHE, spin HE ~ optical HENOTE: spin is not indispensable in QHE
Earlier Studies
1. Suggestion of lateral shift in total reflection (energy flux of evanescent light) F. I. Fedorov, Dokl. Akad. Nauk SSSR 105, 465 (1955)
2. Theory of total and partial reflection (stationary phase)H. Schilling, Ann. Physik (Leipzig) 16, 122 (1965)
3. Theory and experiment of total reflection (energy flux of evanescent light )C. Imbert, Phys. Rev. D 5, 787 (1972)
4. Different opinionsD. G. Boulware, Phys. Rev. D 7, 2375 (1973)N. Ashby and S. C. Miller Jr., Phys. Rev. D 7, 2383 (1973)V. G. Fedoseev, Opt. Spektrosk. 58, 491 (1985)
Ref. 1 and 3 explain the transverse shift in analogy with Goos-Hanchen effect (due to evanescent light). However, Ref.2 says that the transverse shift can be observed in partial reflection.
Summary
• Topological effects in wave phenomena of electrons→ What are the corresponding phenomena of light?• Equations of motion of optical packet with internal rotation• Deflection of light due to anomalous velocity• QHE, Intrinsic AHE, Intrinsic spin HE ~ Optical HE• Photonic crystal without inversion symmetry→ Optical Bloch wave with Berry curvature (internal rotation)• Enhancement and control of optical HE in photonic crystals
Future prospects and challenges
• Tunable photonic crystal → optical switch?
• Transverse shift in multilayer film → precise measurement
• Optical Hall effect of packet with internal OAM (Sasada)
• Localization in photonic band with Berry phase
• Surface mode of photonic crystal and Berry curvature
• Magnetic photonic crystal → Chiral edge state of light (Haldane)• Effect of absorption (relation with Rikken-van Tiggelen effect)• Quasi-photonic crystal (rotational symmetry) → rotation → Berry
phase? (Sawada et al.)• Phononic crystal → sonic Hall effect
Internal Angular momentum of light
Linear S=0 Right circular S=1 Left circular S=-1
http://www.physics.gla.ac.uk/Optics/projects/singlePhotonOAM/
Spin angular momentum
Orbital angular momentum
L=0 L=1 L=2 L=3
The above OAM is interpreted as internal angular momentum when optical packets are considered.More generally, Berry phase → internal rotation ?
Rotation of optical packet
)()()(),()()(
)()(
:momentumAngular
)()(:Momentum
rHrrBrErrD
rBrDrrdJ
rBrDrdP
)()()(,
)()(
:currentenerygofRotation
)()(:currentEnergy
rHrErrrdSPrL
SLrHrErrdJ
rHrErdP
cEEcE
EEE
E
curvatureBerrytosimilarveryis
atcenteredpacketwave:
WSW
rW
E
c
Non-zero Berry curvature ~ Rotation
Periodic structure without inversion→ rotating wave packet
N
N
PM3
N
N
HOMO
N
N
HOMO
LUMO
2.88eV
LUMO
吸収端1.7eV
4t ~ 0.2 eV
4t ~ 0.12 eV
1.2 eV
~1 eV
Phz
3.1eV
H2ca
(B2g)
(B1g)
(Ag)
O
O
Cl
O Cl
O
H
H
O
O
Cl
O Cl
O
H
H
Molecular orbitals(extended Huckel )
Transfer integral t is estimated by t = ES,E~10eV ( S: overlap integral )
PhzPhz stack stack
HH22caca stack stack
LUMO
HOMO
LUMO
HOMO
Transfer integrals along the stacking directionTransfer integrals along the stacking direction (( b-axisb-axis))
-4.9 5.5
1.5-1.4
-5.2
-2.2 (x10-3)
2.7
-1.6
Polarization is “huge” compared with the classical estimate
uePcl* )/(30 *eePP clobs
ee 01.0* neutral
Wave packet
Wave packet (Green) in potential (Red)
http://mamacass.ucsd.edu/people/pblanco/physics2d/lectures.html
Image of wave : we cannot distinguish where it is.Image of particle : we can distinguish where it is.
Wave packet : well-defined position of center + broadening.
Simple example (electron in periodic potential)
ccccc
knkncc
krkkwkkd
rkkwkd
nccrkkwkd
W
2
3
2
3
3
),()2(
,1),()2(
bandthofoperatorcreation:,0),()2(
ionperturabatfieldelectricweakforpotential:)(
potentialperiodic:)(
)()(
)()()(2
)(2
r
rV
rrrrdR
rrerVm
rrdH r
Eerek
Er
crc
knkc
c
cc
)(bandthofenergy:
)(eff
nE
reErkL
c
c
kn
ckncc
“Magnetic field” by circuit
ttEG 44 )/( GEQeaP
ttEG 44
)/( 32GEQteaP
eV3 eVt 1.0
(i)
(ii)
20/. eaPobs
Case (ii) can not explain the obs. value
energy perturbation due to atomic displacement
Q
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