Transcript スライド 1 - University of British Columbia
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) H
(
X
) Hamiltonian,
i
H
(
X t
(
t
)
n
) (
X X
)
H
(
X E n
(
t
( ))
X
) (
t n
( )
X
(
X
1 , )
X
2 , , ,
X n
) eigenvalue and eigenstate for each parameter set X parameters adiabatic change
X
2
C
Transitions between eigenstates are forbidden during the adiabatic change Projection to the sub-space of Hilbert space constrained quantum system
X
1 (
T
)
e i
n
(
C
)
e
(
i
/ ) 0
T dtE n
(
X
(
t
)) ( 0 )
n
(
C
)
C dX i
C dX
n
(
X
) |
X
n
(
X
)
A n
(
X
)
dS
B n
(
X
)
Berry Phase
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
)
e ikr u nk
(
r
) Bloch wavefucntion
A n
(
k
)
i
u nk
|
k x i
r i
A n
(
k
)
i
k i
|
u nk
Berry phase connection in k-space
A n
(
k
) covariant derivative [
x
,
y
]
i
(
k x A ny
(
k
)
k y A nx
(
k
))
iB nz
(
k
) Curvature in k-space
dx
(
t
)
i
[
x
,
H
]
dt k m x
i
[
x
,
y
]
V
y k z
k x m
B nz
(
k
)
V
y
Anomalous Velocity and Anomalous Hall Effect
Non-commutative Q.M.
k
|
u nk
|
u
k nk
k x k y
Duality between Real and Momentum Spaces
d
r
(
t
)
dt
(
k n
k
)
B n
(
k
)
d
k- space curvature
k
(
t
)
dt d
k
(
t
)
dt
V
(
r r
)
B
(
r
)
d r
r- space curvature
dt
(
t
)
Degeneracy point Monopole in momentum space SrRuO3 Z.Fang
Fermat’s principle and principle of least action
Goal Path 5 Path 4 Path 2 Path 3 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 Start
Searching stationary value ~ Solving equations of motion
Trajectories of light and particle
Geometrica l Optics [turn in the direction of larger
n
(
r c
) ]
ds
n d
(
r c
)
n
( :
r
c
)
d ds r
c
refractive
n
index (
r c
) ，
n
(
r c
)
ds
dt
[turn in the directon of lower
V
(
r c
) ] Newton' s equation of motion
m m
:
d dt
d dt
mass,
r c V
(
r c
V
) : (
r c
)
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 geometrical phase
obtain 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
L
d i dt
H
variaton
i d dt
H
W L
eff : wave packet centered
W i d dt
H W
at position variation
r c
and momentum EOM of
r c
and
k
c k
c
R
: position Condition
r c
operator
W
R W
L
eff 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
H
R H
d r
d r
r
(
r
) 2 2
E
(
r
) 2 (
r
)
E
2 (
r
) (
r
) 2
H
2 (
r
) 2 (
r
) ,
H
2 (
r
) (
r
) and (
r
) : slowly varying
W
d k
( 2 ) 3
w
(
k
k
c
,
r c
)
z c
a
k
a
k
: creation operator of circularly Condition for the center of gravity 0 ,
z c
2 1 polarized
r
c
W
photon
R H W W H W L
eff
k
c
r
c
k
c
(
z c
|
k
c
|
z c
)
z z c c
,
v
(
r c
) |
z c
)
i
(
z c
|
z
c
)
v
(
r c
)
k c
(
r c
1 ) (
r c
) ,
k
i e
k
k
e
k
Equations of motion of optical packet
r
v c
( : position
r c
) : light ,
k
c
: speed momentum
Anomalous velocity
|
z
k
k
c
) : : : state Berry Berry of polarizati connection curvature on
k
e
k
:
i e
k
polarizati on
k
e
k
vector
k
k
k
i
k
k
k
k
3 3 |
r
c k
c z
c
)
v
( [
r c
)
v k c
(
i k
c k
c
k
r c
)]
k c k
c c
|
z c
(
z c
) |
k c
|
z c
) 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
|
z c out
t
[
e
i
z in
,
e i
z in
]
dk
[
k
]
S
dS k
[
k
]
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
j z
r
c
k
c
(
z c
| 3 |
z c
)
k
c k c
z
const.
j z I
j z T
,
j z I
j z R I
: incident,
T
: transmitte d,
R
: reflected
y c A
(
z c A
| 3 |
z c A
) cos
A k c
sin (
z c I I A
T
,
R
| 3 |
z c I
) cos
I
Comparison with numerical simulation
V
0 : light speed in lower medium
V
1 : 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 --
Bloch wave
An intermediate between traveling wave and standing wave
Meaning of the height of periodic structure
Electron : electrical potential Light : (phase) velocity of light
For low energy Bloch wave
Large amplitude at low point Small amplitude at high point Strength of periodic structure
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 k
z
=0.
Note: Eigenmodes with
k z
=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.
(
r
) : moderate modulation , 1 (
r
) 2 (
r
(
r
) )
E n k
c
:
n k
n
u E
,
n k k
H
n
th band energy in the :
i
1
k
2
u E
n k
n k
Bloch
k
u E
n k
functions of case of
u n
th
H
n k
band (
k
u H
n k r
) 1
E
: electric field ,
H
: magnetic field
L
eff
k c
r
c
k c
n
k c
(
r c
)
E n
k 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.
r
c k
c
(
x c
[ )
k
c
(
x c E n k
c
)]
E n k
c
,
k
c
(
x
) : superimpos
k
c
, ed modulation
ε
( 1
r
)
ε
2 ( (
r
x
) )
Large shift of several dozens of lattice constant
classical theory of polarization
Averaged polarization at
r P
(
r
)
f
(
r
R
)
p
(
R
)
R
Charge determines pol.
Ionicity is needed !! Polarization of a unit cell
R p
(
R
)
d
d
r
'
R
d
(
r
' )
u R
d
polarization due to displacements of rigid ions +
d
r
'
R
d
(
r
' )
r
' 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
dP i
(
Q
)
e
A i
l
(
d Q
2 ) 3
d
3
k
n l
(
k
)
l
d
l
r
l
d
l
l
r
l
A i
2
e
l
( 2 ) 3
d
3
k
n l
(
k
) Im
l
k i
l
Q
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
P cl e
* 0
e
*
u
.
01
e P obs
30
P cl
(
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/cm 2 (too small !) Hydrogen bond ( covalency)
P
( )
Polarization as a Berry phase
(2 2
e
) 3
dk
dk
dk
occ
n
1
u
( )
kn
k
u
( )
kn
0.7
0.6
0.5
Bulk
0.4
0.3
0.2
0.1
Isolated molecule
0 0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
Large polarization with covalency Asymm etry in Bond length O-H (ang.)
Geometrical meaning of polarization
H
(
k
,
Q
)
in 1D two-band model
0
h
1 ( 0 (
k
(
k
,
k
,
Q
,
Q Q
) ) ) with Pauli
h
(
h ih
3
k
2 ( ( ,
k Q k
, , matrices )
Q Q
) )
h
1 0 ( (
k k
, ,
Q
)
Q
)
ih
2
h
3 ( (
k k
, ,
Q Q
) )
dP
: Solid angle of the ribon
dP A
(
Q
)
A
(
Q
)
e
dk
4
h
ˆ
d Q
h k
ˆ
h Q
ˆ Generalized Born charge
Strings as trajectories of band-crossing points
flux density
:
B
(
Q B
)
Q
3 4
A dk
2 (
Q
)
d h
ˆ
dQ
d h
ˆ
dk
d h
ˆ
dQ
1.
B
(
Q
) with
k
0 only along strings (trajectories of band-crossing points) in [ /
a
,/
a
2. Divergence-free
B
(
Q
)
h
(
k
, 0
Q
) 0
k
space) 3. Total flux of the string is quantized to be an integer
Q
(Pontryagin index, or wrapping number): [c.f. Thouless]
C d
Q
A
(
Q
)
S d S
Q
B
(
Q
)
n
C × [ /
a
, /
a
]
B
C
Band-crossing point
Biot-Savart law, asymptotic behavior & charge pumping
Transverse part of the polarization current
A
Biot-Savart law:
A t
(
Q
)
L
d Q
' 4 | (
Q
Q
Q Q
' | 3 ' )
L
: strings string Asymptotic behavior (leading order in 1/E g )
A
(
Q
) Strength ~ 1/E g Direction: same as a magnetic field created by an electric current E g
C d Q
A
(
Q
)
S d S Q
B
(
Q
)
n
Q
around a string
ne
Specific models
Simplest physically relevant models
h
(
k
,
H
Q
)
f
k
(
c k k
, ) 0
g
(
k
( )
k
, )
Q
'
h
(
k
,
Q
) , '
c k
, ' Different choices of
f
and
g
Geometrically different structures of strings
B
and polarization current
A
Quantum Charge Pumping in Insulator
E
or
z
Pressure
E
Electron(charge)flow
E x
Large polarization even in the neutral molecules
Dimerized charge-ordered systems
TTF-CA (TMTTF) 2 PF 6 (DI-DCNQI) 2 Ag TTF-CA: polarization perpendicular to 2 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 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
Polarization and spin
Linear S = 0 Right circular S = +1 http://www.expocenter.or.jp/shiori/ ugoki/ugoki1/ugoki1.html
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 (
t
,
r
integral )
d
r
0
e iS
1
e iS
2
e iS
3 (
t
0 ,
r
0 )
d
r
0
e iS
st (
t
0 ,
r
0 )
S n
: action for a funtional the
n
th of the
n
trajectory th ( path) trajectory which connects
r
0
S
st : stationary action actual trajectory of classical and
r
in particle (
t
t
0 ) 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
B
: electric : magnetic field field
B
0
a
2
p
e z q
B a
: lattice constant
r
c k
c
k
c
e E E n k
c
[
B
]
e r
c
B k
c
n
k c n k
i u n k
n k
k
n k
k
u n k
u n k
: Bloch function
E n k
c
L
L n k
c n k
c H
0 [
B
]
E n k
c
e
B
L n k
c
: : OAM
m
around
k c
Hamiltonia 2
m
r c u n k
c
( n with
E n k
c
E
H
B
0 ) 0
k
c u n k
c
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
• 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 :
P E
d r
E
(
r
)
H
(
r
) Rotation
J
E L E
r
c d
r
of
r
P E
, eneryg
S
E
E
(
r
) current
H
(
r
)
d r
(
r
:
r c
)
L E
E S
E
(
r
)
H
(
r
) Momentum :
P
d r
D
(
r
)
B
(
r
) Angular
J
d r
D
(
r
)
r
momentum (
r
)
D
E
( (
r
r
) ),
B
: (
B
(
r
r
) ) (
r
)
H
(
r
)
W W
: wave
S E W
packet centered is very similar at
r
c
to Berry curvature
Non-zero Berry curvature ~ Rotation Periodic structure without inversion → rotating wave packet
Molecular orbitals(extended Huckel ） Phz LUMO N N 3.1eV
吸収端 1.7eV
1.2 eV H2ca Cl O H O 2.88eV
H O Cl O LUMO 4t ~ 0.12 eV 4t ~ 0.2 eV HOMO N N N ~1 eV (B 2g ) O HOMO H O Cl (B 1g ) O Cl (A g ) H O Transfer integral
t is estimated by
ｔ
E
~10eV （
S: overlap integral
）
= ES
, N PM3
Transfer integrals along the stacking direction
（
b-axis
）
-2.2 (x10 -3 ) Phz
stack
LUMO -1.4
1.5
HOMO LUMO -4.9
-5.2
2.7
5.5
HOMO -1.6
H 2 ca
stack
Polarization is “huge” compared with the classical estimate
P e
*
cl
e
*
u
0 .
01
e P obs
neutral
30
P cl
(
e
/
e
* )
Wave packet
Image of wave ： Image of particle ： we cannot distinguish where it is.
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)
H
d r
(
r
)
R
V
( (
r
r
) ) : :
d r
r
periodic potential (
r
)
r
2
m
(
r
) potential for weak
V
(
r
) electric
e
(
r
) field (
r
) perturabat ion
W
d k
( 2 ) 3
d k
( 2 ) 3
w
(
k
k
c
,
r c
)
c
n k w
(
k
k
c
,
r c
) 2 1 , 0 ,
d k
( 2 ) 3
c
n k
: creation operator of
k
w
(
k
k
c
,
r c
) 2
k
c n
th band
L
eff
k
c
r
c
E n k
c
e
(
r
c
)
E n k
c
: energy of
n
th band
r
c k
c
k
c E e
n k
c
r c
(
r c
)
e E
“Magnetic field” by circuit
Q
energy perturbation due to atomic displacement (i)
E G
4
t P
ea
4
t
(
Q
/
E G
) (ii)
E G
4
t P
4
t
ea
(
t
2
Q
/
E G
3 )
P obs
.
3
eV
ea t
/ 20 0 .
1
eV
Case (ii) can not explain the obs. value