Berry Phase Phenomena Optical Hall effect and Ferroelectricity as quantum charge pumping

1
Berry Phase PhenomenaOptical 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)
2
Berry phase M.V.Berry, Proc. R.Soc. Lond.
A392, 45(1984)
Hamiltonian,
parameters?adiabatic change
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

Berry Phase
Connection of the wavefunction in the parameter
space?Berry phase curvature
3
Electrons with constraint
doublydegeneratepositive energy states.
Dirac electrons
Bloch electrons

• Projection onto positive energy state
• Spin-orbit interaction
• as SU(2) gauge connection

Projection onto each band Berry phase
of Bloch wavefunction
Spin Hall Effect (S.C.Zhangs talk)
Anomalous Hall Effect (Haldanes talk)
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Berry Phase Curvature in k-space
Bloch wavefucntion
Berry phase connection in k-space
covariant derivative

Curvature in k-space
Anomalous Velocity and Anomalous Hall Effect
Non-commutative Q.M.
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Duality between Real and Momentum
Spaces
k- space curvature
r- space curvature
6
SrRuO3
Z.Fang
Degeneracy point ? Monopole in momentum space
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Fermats principle and principle of least action
Goal
Path 5
Path 4
Path 3
Path 2
Start
Path 1
Every path has a specific optical path length or
action.
Fermat stationary optical path length ? actual
trajectoryLeast action stationary action ?
actual trajectory
Searching stationary value Solving equations of
motion
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Trajectories of light and particle
What determine the equations of
motion?Historically, experiments and
observationsAny fundamental principles?(Fermat
s principle, principle of least action)
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Geometrical phase (Berry phase)
Principle of least actionPhase factor ?
Equations of motion
Berry phaseWave 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
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Effective Lagrangian of wave packet
R. Jackiw and A. Kerman,Phys. Lett. 71A, 581
(1979) A. Pattanayak and W.C. Schieve, Phys. Rev.
E 50, 3601 (1994)
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Light in weakly inhomogeneous medium
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Equations of motion of optical packet
Anomalous velocity
Neglecting polarization? Conventional
geometrical optics
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Berry Phase in Optics
Propagation of light and rotation of polarization
plane in the helical optical fiber
Chiao-Wu, Tomita-Chiao, Haldane, Berry
Spin 1 Berry phase
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Reflection and refraction at an interface
Shift perpendicular to both of incident axis and
gradient of refractive index
No polarization
Circularly polarized
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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
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Comparison with numerical simulation
V0 light speed in lower mediumV1 light speed
in upper mediumSolid and broken lines are
derived by the conservation law.?and are
obtained by numerically solving Maxwell equations.
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Photonic crystal and Berry phase
Shift in reflection and refractionSmall Berry
curvature?small shift of the order of wave length
Knowledge about electrons in solidsPeriodic
structure without a symmetry?Bloch wave with
Berry phase
Photonic crystal without a symmetry ? Bloch wave
of light with Berry phaseEnhancement of optical
Hall effect ?!
Example of 2D photonic crystal without inversion
symmetry
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Wave in periodic structure -- Bloch wave --
Bloch waveAn intermediate between traveling
wave and standing wave
Energy
Meaning of the height of periodic
structureElectron electrical potentialLight
(phase) velocity of lightFor low energy
Bloch waveLarge amplitude at low pointSmall
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/ma
in/bloch/bo_sub.html
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Dielectric function and photonic band
We shall consider wave ribbons with kz0.Note
Eigenmodes with kz0 are classified into TE or TM
mode.
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Berry curvature of optical Bloch wave
For simplicity, we consider the case in which
the spin degeneracy is resolved due to periodic
structure.
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Berry curvature in photonic crystal
Berry curvature is large at the region
whereseparation between adjacent bands is small.
c.f. Haldane-Raghu Edge mode
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Trajectory of wave packet in photonic crystal
Superimposed modulation around x 0 instead of
a boundaryNoteThe figure is the top view of 2D
photonic crystal. Periodic structure is not shown.
Large shift of several dozens of lattice constant
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classical theory of polarization
Averaged polarization at r
Charge determines pol. Ionicity is needed !!
Polarization of a unit cell 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.

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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
P is path dependent in general !!
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Ferroelectricity in Hydrogen Bonded
Supermolecular Chain
S.Horiuchi et al 2004
Polarization is huge compared with the
classical estimate
Neutral and covalent
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Ferroelectricity in Phz-H2ca
S. Horiuchi _at_ CERC et al.
With F. Ishii _at_ERATO-SSS
First-principles calculation
Isolated molecule ? 0.1 µC/cm2 (too small !)
Hydrogen bond ( covalency)
Polarization as a Berry phase
Bulk
Isolated molecule
Large polarization with covalency
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Geometrical meaning of polarization in 1D
two-band model
dP Solid angle of the ribon
Generalized Born charge
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Strings as trajectories of band-crossing points
flux density

• only along strings (trajectories of
  band-crossing points) with k in
  -p/a,p/a
• d-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

C-p/a,p/a
B
C
Band-crossing point
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Biot-Savart law, asymptotic behavior charge
pumping
Transverse part of the polarization current
A Biot-Savart law
L strings
string
Asymptotic behavior (leading order in 1/Eg)
Strength 1/Eg Direction same as a magnetic
field created by an electric current
Eg
Quantum charge pumping due to cyclic change of Q
around a string
ne
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Specific models

• Simplest physically relevant models

Different choices of f and g
Geometrically different structures of strings
B and polarization current A
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Quantum Charge Pumping in Insulator
or Pressure
Electron(charge)flow
Large polarization even in the neutral molecules
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Dimerized charge-ordered systems
TTF-CA (TMTTF)2PF6 (DI-DCNQI)2Ag
TTF-CA polarization perpendicular to
displacement of molecules. D2 triggers the
ferroelectricity.
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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
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Motivation of this study
Goal dissipationless functionality
of electrons in solidsKey concept topological
effects of wave phenomena of electrons
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
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
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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/u
goki1.html
Polarization and spin
Linear S 0
Right circular S 1
Left circular S -1
http//www.physics.gla.ac.uk/Optics/projects/singl
ePhotonOAM/
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Action and quantum mechanics
Quantum mechanicsWave-particle
dualityEverything is described by a wave
function.Action in classical mechanics phase
factor of wave functionSearching a trajectory
of classical particle Solving a wave function
approximately
Similar relation holds between geometrical and
wave optics.
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Wave and geometrical optics, Quantum and
classical mechanics
Wave optics ? Eikonal ? Fermats 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.
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Hall effect of 2DES in periodic potential
M.-C. Chang and Q. Niu, Phys. Rev. B 53, 7010
(1996)
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Optical path length and action
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)
Particle in inhomogeneous potentialAction Sum
of (kinetic energy potential) x (infinitesimal
time) along a trajectory
PointOptical path length and action can be
defined for any trajectories,regardless of
whether realistic or unrealistic.
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Why is it interpreted as the optical Hall effect ?
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.)
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)
QHE, AHE, spin HE optical HENOTE spin is not
indispensable in QHE
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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.
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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

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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

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Internal Angular momentum of light
Spin angular momentum
Linear S0
Right circular S1
Left circular S-1
Orbital angular momentum
L0
L1
L2
L3
http//www.physics.gla.ac.uk/Optics/projects/singl
ePhotonOAM/
The above OAM is interpreted as internal angular
momentum when optical packets are
considered.More generally, Berry phase ?
internal rotation ?
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Rotation of optical packet
Non-zero Berry curvature Rotation
Periodic structure without inversion? rotating
wave packet
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Molecular orbitals(extended Huckel)
Transfer integral t is estimated by t
ES, E10eV( S overlap integral)
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Transfer integrals along the stacking
direction(b-axis)
-2.2 (x10-3)
Phz stack
LUMO
1.5
-1.4
HOMO
-5.2
H2ca stack
LUMO
2.7
-4.9
5.5
HOMO
-1.6
48
Polarization is huge compared with the
classical estimate
neutral
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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
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Simple example (electron in periodic potential)
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Magnetic field by circuit
energy perturbation due to atomic
displacement
(i)
(ii)
Case (ii) can not explain the obs. value