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Berry Phase Phenomena
Optical Hall effect
and
Ferroelectricity as quantum charge pumping
Naoto Nagaosa
CREST, 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)
X  ( X1, X 2 , , , X n )
i t (t )  H ( X (t )) (t )
H ( X )n ( X )  En ( X )n ( X )
H (X )
Hamiltonian,
parametersadiabatic change
X2
C
eigenvalue and eigenstate for each parameter set X
Transitions between eigenstates are forbidden
during the adiabatic change
Projection to the sub-space of Hilbert space
constrained quantum system
X1
T
 (T )  e

i n ( C ) ( i /  ) 0 dtEn ( X ( t ))
e
 (0)
 n (C )  i  dX   n ( X ) |  X n ( X ) 
C
  dX  An ( X )    dS  Bn ( X )
Berry Phase
C
Connection of the wavefunction in the parameter spaceBerry phase curvature
Electrons with ”constraint”
E
E
doubly
degenerate
positive
energy states.
k
k
Dirac electrons
Projection onto positive energy state
Spin-orbit interaction
as SU(2) gauge connection
Spin Hall Effect (S.C.Zhang’s talk)
Bloch electrons
Projection onto each band
Berry phase
of Bloch wavefunction
Anomalous Hall Effect (Haldane’s talk)
Berry Phase Curvature in k-space
 nk (r )  eikr unk (r )
Bloch wavefucntion
An (k )  i  unk | k | unk 
Berry phase connection in k-space
xi  ri  An (k )  i ki  An (k )
covariant derivative
[ x, y]  i( k x Any (k )   k y Anx (k ))  iBnz (k )
Curvature in k-space
k
dx(t )
V k x
V
 i[ x, H ]  x  i[ x, y ]
  Bnz (k )
dt
m
y m
y
Anomalous Velocity and
Anomalous Hall Effect
kz
 | unk  k 
k  | unk 
kx
ky
Non-commutative Q.M.
Duality between
Real and Momentum Spaces



d r (t )  n ( k )   d k (t )

 Bn (k ) 

dt
dt
kspace
curvature
k



d k (t ) V ( r )   d r (t )

 B( r ) 

dt
dt
rspace
curvature
r
Degeneracy point
 Monopole in momentum space
SrRuO3
Z.Fang
Fermat’s principle and principle of least action
Goal
Path 5
Path 4
Path 3
Path 2
Path 1
Every path has a specific optical path length or action.
Fermat : stationary optical path length → actual trajectory
Least action : stationary action → actual trajectory
Searching stationary value ~ Solving equations of motion
Start
Trajectories of light and particle

Geometrica l Optics [turn in the direction of larger n(rc )]
d   d   
n(rc ) rc   n(rc )

ds 
ds 


n(rc ) : refractive index , n(rc )ds  dt
Newton' s equation of motion
 
d d 
m  rc   V (rc )
dt  dt 

m : mass, V (rc ) : potential

[turn in the directon of lower V (rc )]
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 action
Phase factor → Equations of motion
Berry phase
“Wave functions with spin obtain
geometrical phase in adiabatic motion.”
Although light has spin,
no effect of Berry phase in conventional
geometrical optics.
Topological effects (wave optics)
in trajectory of light (geometrical optics)
→ wave packet
Effective Lagrangian of wave packet
d
d
L   i  H   variaton  i
 H 
dt
dt


W : wave packet centered at position rc and momentum kc


d
Leff  W i  H W  variation  EOM of rc and kc
dt

R : position operator


Condition rc  W R W  Leff
R. Jackiw and A. Kerman,Phys. Lett. 71A, 581 (1979)
A. Pattanayak and W.C. Schieve, Phys. Rev. E 50, 3601 (1994)
Light in weakly inhomogeneous medium
 
 
   (r ) 2   (r ) 2  


H   dr 
E (r ) 
H (r ),  (r ) and  (r ) : slowly varying
2
 2




    (r )  2   (r )  2  
RH   dr r 
E (r ) 
H (r )
2
 2


  
dk


W 
w
(
k

k
,
r
)
z
a
c c  c k 0 ,
3
(2 )


2
zc  1
ak : creation operator of circularly polarized photon

W RH W

Condition for the center of gravity  rc 
W H W
Leff
  



 kc  rc  kc  ( zc |  k | zc )  i ( zc | zc )  v(rc )kc
c
 

 zc  

1
| zc )   , v(rc ) 

 ,  k
 (rc )  (rc )
 zc  
 
     
 iek  k e k
Equations of motion of optical packet


rc : position , kc : momentum

v(rc ) : light speed
| zc ) : state of polarizati on

 k : Berry connection

 k : Berry curvature
 

 


 k    iek  k e k

ek : polarizati on vector






k





 k   k   k  i k   k  3  3
k
Neglecting polarization
→ Conventional geometrical optics
Anomalous velocity



 kc 
rc  v(rc )  kc  ( zc |  k | zc )
c
kc

 
kc  [v(rc )]kc
 
| zc )  ikc   k | zc )
c
Berry Phase in Optics
Propagation of light and rotation of polarization plane in the helical optical fiber
Chiao-Wu, Tomita-Chiao, Haldane, Berry
| zcout t [e i zin , ei zin ]
   dk  [ k ]    dS k  [ k ] 
S
Spin 1 Berry phase
Reflection and refraction at an interface
Shift perpendicular to both of
incident axis and gradient of
refractive index
No polarization
Circularly polarized
Conservation law of angular momentum
EOM are derived under the condition of weak inhomogeneity.
Application to the case with a sharp interface?
Conservation of total angular momentum as a photon



kc 
j z  rc  kc  ( zc |  3 | zc )   const.
kc  z

j zI  j zT ,
j zI  j zR
I : incident, T : transmitte d, R : reflected
( zcA |  3 | zcA ) cos  A  ( zcI |  3 | zcI ) cos  I
y 
kc sin  I
A
c
AT ,R
Comparison with numerical simulation
V0: light speed in lower medium
V1: 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
Shift in reflection and refraction
Small Berry curvature
→small shift of the order of wave length
Knowledge about electrons in solids
Periodic structure without a symmetry
→Bloch wave with Berry phase
Photonic crystal without a symmetry
→ Bloch wave of light with Berry phase
Enhancement of optical Hall effect ?!
Example of 2D photonic crystal without
inversion symmetry
Wave in periodic structure -- Bloch wave --
Meaning of the height of periodic
structure
Electron : electrical potential
Light
: (phase) velocity of light
Energy
Bloch wave
An intermediate between
traveling wave and standing wave
Strength of periodic structure
For low energy Bloch wave
Large amplitude at low point
Small amplitude at high point
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
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.
Berry curvature of optical Bloch wave
For simplicity, we consider the case in which
the spin degeneracy is resolved due to periodic structure.


1
 2 (r )
 (r ) : moderate modulation ,   
 (r )
 (r )

Enk : nth band energy in the case of  (r )  1
c


 E
 H


1
E
H
 nk  i unk   k unk  unk   k unk
2





 nk   k   nk
 E, H
unk : Bloch functions of nth band
E : electric field , H : magnetic field
   

Leff  kc  rc  kc   nk   (rc ) Enk
c
c

Berry curvature in photonic crystal
Berry curvature is large at the region where
separation between adjacent bands is small.
c.f. Haldane-Raghu
Edge mode
Trajectory of wave packet in photonic crystal
Superimposed modulation around x = 0
instead of a boundary
Note:
The figure is the top view of 2D photonic
crystal. Periodic structure is not shown.
 


rc   ( xc ) k Enk  kc   k ,
c
c
c


kc  [ ( xc )]Enk ,
c
 ( x) : superimpos ed modulation
1
 2 ( x)
  
ε (r )
ε (r )
Large shift of several dozens of
lattice constant
classical theory of polarization
Averaged polarization at r
P(r)   f (r  R)p(R)
Charge determines pol.
Ionicity is needed !!
R
Polarization of a unit cell R
p( R ) 
  dr' 
d
R d


(r ' ) u R  d   dr '  R  d (r ' )r '
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.
quantum theory of polarization
Covalent ferroelectric: polarization without ionicity
“r” is ill-defined for extended Bloch wavefunction
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

i
dP (Q)  e (2 ) 3  d 3k nl (k ) l  dl r l  dl  l r l 
i l 
 A  dQ
l l
i
3
3
A  2e (2 )  d k nl (k ) Im
i

k
Q
l
P is path dependent in general !!
Ferroelectricity in Hydrogen Bonded
Supermolecular Chain
S.Horiuchi et al 2004
Polarization is “huge” compared with
the classical estimate
Pcl  e u
*
e*  0.01e
Pobs  30 Pcl (e / e* )
Neutral and covalent
Ferroelectricity in Phz-H2ca
S. Horiuchi @ CERC et al.
With F. Ishii @ERATO-SSS
First-principles calculation
Isolated molecule → 0.1 μC/cm2 (too small !)
Hydrogen bond
( covalency)
P
()

Polarization as a Berry
phase
occ
2e

(2 ) 3
( ) 
()
dk
dk
dk
u
u
kn

    
k kn
n1
0.7
0.6
Bulk
2
Ps( μC/cm )
0.5

0.4
0.3
0.2
Isolated
molecule
0.1
0
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
Asymm etry in Bond length O-H (ang.)
Large polarization with covalency
Geometrical meaning of polarization
in 1D two-band model



 
H (k , Q)   0 (k , Q)  h (k , Q)  




  0 (k , Q)  h3 (k , Q) h1 (k , Q)  ih2 (k , Q) 



 
 
 h1 (k , Q)  ih2 (k , Q)  0 (k , Q)  h3 (k , Q) 
with Pauli matrices 
dP : Solid angle of the ribon
 

dP  A(Q)  dQ

dk ˆ  hˆ
hˆ 
A (Q)  e 
h 

4
 k Q 
Generalized Born charge
Strings as trajectories of band-crossing points
 
 

flux density: B(Q)  Q  A(Q)
3 dk
dhˆ  dhˆ
dhˆ 
B     



4 2
dQ  dk dQ 

 
1. B (Q )  0 only along strings (trajectories of band-crossing points)



with k in [/a,/a
h (k , Q)  0
-function singularity along strings (monopoles in k space)
  
2. Divergence-free   B(Q)  0
3. Total flux of the string is quantized to be an integer
(Pontryagin index, or wrapping number): [c.f. Thouless]

C

Q
B
C
  
  
dQ  A(Q)   dSQ  B(Q)  n
C×[/a,/a]
S
Band-crossing point
Biot-Savart law, asymptotic behavior
& charge pumping
Transverse part of the polarization current A

 
t 
dQ'  (Q  Q' )
Biot-Savart law:
 
A (Q)   
L 4 | Q  Q' |3
Asymptotic behavior (leading order in 1/Eg)
 
Strength ~ 1/Eg
A(Q )
Direction: same as a magnetic field
created by an electric current
L : strings
string
Eg
Quantum charge pumping due to cyclic change of Q around a string
  
  
C dQ  A(Q)  SdSQ  B(Q)  n
ne
Specific models
Simplest physically relevant models

 
H   ck ,  0 (k )  , '  h (k , Q)    , ' ck , '
k

h (k , Q)  f (k )  g (k )Q

Different choices of f and g
Geometrically different
structures of strings B
and polarization current A

Quantum Charge Pumping in Insulator
Ez
or
Pressure

E
Electron(charge)flow
Ex
Large polarization even in the neutral molecules
Dimerized charge-ordered systems
TTF-CA
(TMTTF)2PF6
(DI-DCNQI)2Ag
TTF-CA: polarization perpendicular to
displacement of molecules.
2 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 solids
Key concept : topological effects of wave phenomena of electrons
Example of our study
Topological interpretation of quantization in quantum Hall effect
↓
Intrinsic anomalous Hall effect and spin Hall effect
due to the geometrical phase of wave function
What is corresponding phenomena in optics?
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
Rotation and angular momentum
Rotation of center of gravity
Rotation around center of gravity
http://www.expocenter.or.jp/shiori/
ugoki/ugoki1/ugoki1.html
Polarization and spin
Linear S = 0
Right circular S = +1
Left circular S = -1
http://www.physics.gla.ac.uk/Optics/projects/singlePhotonOAM/
Action and quantum mechanics
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
Path integral

 iS1 iS2

 iSst

iS3
 (t , r )   dr0 e  e  e    (t0 , r0 )   dr0 e  (t0 , r0 )


S n : action for the n th trajectory (path)


 a funtional of the n th trajectory which connects r0 and r in (t  t0 )
Sst : stationary action  actual trajectory of classical particle
Similar relation holds between geometrical and wave optics.
“Wave and geometrical optics”,
“Quantum and classical mechanics”
Wave optics → Eikonal
→ Fermat’s principle → Geometrical optics
Optical path, Action
~ Phase factor
Quantum mechanics → Path integral
→ Principle of least action → Classical mechanics
Roughly speaking,
Trajectory is determined by the phase factor of a wave function.
Hall effect of 2DES in periodic potential

E : electric field

B : magnetic field
 0 p 

B  2 ez  B
a q
a : lattice constant


 nk  i unk  k unk



 nk   k   nk
unk : Bloch function
       
rc   k Enk [B]  kc   nk
c
c
c

 


kc  eE  erc  B

 
e
Enk [B]  Enk 

B  Lnk
c
c
c
2m



Lnk : OAM around rc
c



Lnk  m  k unk  ( Enk  H 0 )  k unk
c
c
c
c
c
c


H 0 : Hamiltonia n with E  B  0

M.-C. Chang and Q. Niu, Phys. Rev. B 53, 7010 (1996)

Optical path length and action
Light in media with inhomogeneous refractive index
Optical path length
= Sum of (refractive index x infinitesimal length) along a trajectory
= Time from start to goal
Light speed = 1/(refractive index)
Time for infinitesimal length = (infinitesimal length) / (light speed)
Particle in inhomogeneous potential
Action
= Sum of (kinetic energy – potential) x (infinitesimal time) along a trajectory
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 ?
Transverse shift of light in reflection and refraction at an interface
The shift is originated by the anomalous velocity.
(Light will turn in the case of moderate gradient of refractive index.)
Hall effect of electrons
Classical HE
: Lorentz force
QHE
: 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)]
QHE, AHE, spin HE ~ optical HE
NOTE: 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 opinions
D. 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
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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
Spin angular momentum
Linear S=0
Right circular S=1
Left circular S=-1
Orbital angular momentum
L=0
L=1
L=2
L=3
http://www.physics.gla.ac.uk/Optics/projects/singlePhotonOAM/
The above OAM is interpreted as internal angular momentum when
optical packets are considered.
More generally, Berry phase → internal rotation ?
Rotation of optical packet

   
Energy current : PE   dr E (r )  H (r )

    
Momentum : P   dr D(r )  B(r )
Rotation of eneryg current :



    
J E   dr r  E (r )  H (r )  LE  S E

   
  
  
LE  rc  PE , S E   dr (r  rc )  E (r )  H (r )
Angular momentum :

    
J   dr r  D(r )  B(r )
 
    
  
D(r )   (r ) E (r ), B(r )   (r ) H (r )






W : wave packet centered at rc


W S E W is very similar to Berry curvature 
Non-zero Berry curvature ~ Rotation
Periodic structure without inversion
→ rotating wave packet

H2ca
Molecular orbitals(extended Huckel)
O
H
Cl
O
Phz
LUMO
O
N
1.2 eV
Cl
H
O
LUMO
3.1eV
N
4t ~ 0.12 eV
吸収端
1.7eV
2.88eV
HOMO
4t ~ 0.2 eV
~1 eV
N
HOMO
 (B2g)
N
O
Cl
H
O
 (B1g)
 (Ag)
O
H
N
N
Cl
O
Transfer integral t is estimated by t = ES,
E~10eV( S: overlap integral)
Transfer integrals along the stacking direction(b-axis)
-2.2 (x10-3)
LUMO
-1.4
Phz stack
1.5
HOMO
-5.2
LUMO
H2ca stack
2.7
-4.9
5.5
HOMO
-1.6
Polarization is “huge” compared with
the classical estimate
Pcl  e*u Pobs  30 Pcl (e / e* )
e*  0.01e
neutral
Wave packet
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.
Wave packet (Green) in potential (Red)
http://mamacass.ucsd.edu/people/pblanco/physics2d/lectures.html
Simple example (electron in periodic potential)
     2r

 
H   dr  (r ) 
 V (r )  e (r ) (r )
 2m





R   dr r   (r ) (r )

V (r ) : periodic potential

 (r ) : potential for weak electric field  perturabat ion

   
dk

 0 , c  : creation operator of nth band
W 
w
(
k

k
,
r
)
c
c c
nk
nk
(2 )3




 2
dk
dk     2 
 (2 )3 w(k  kc , rc )  1,  (2 )3 k w(k  kc , rc )  kc
 

 kc  rc  Enk  e (rc )
Leff
c
Enk : energy of nth band
c
   
rc   k Enk
c
c




kc  e rc  (rc )  eE
“Magnetic field” by circuit
Q energy perturbation due to atomic
displacement
(i) EG    4t  4t
P  ea(Q / EG )
(ii) EG    4t  4t
P  ea(t 2Q / EG3 )
  3eV t  0.1eV
Pobs.  ea / 20
Case (ii) can not explain the obs. value
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