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

QQ

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