Berry Phase PhenomenaOptical Hall effect and Ferroelectricity as quantum charge pumpingNaoto NagaosaCREST, Dept. Applied Physics, The University of TokyoM. 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)Hamiltonian, parametersadiabatic changeBerry PhaseConnection of the wavefunction in the parameter spaceBerry phase curvatureeigenvalue and eigenstate for each parameter set XTransitions between eigenstates are forbidden during the adiabatic changeProjection to the sub-space of Hilbert space constrained quantum system

Electrons with constraintProjection onto positive energy stateSpin-orbit interaction as SU(2) gauge connectionDirac electronsdoublydegeneratepositive energy states.Bloch electronsProjection onto each bandBerry phase of Bloch wavefunction

Spin Hall Effect (S.C.Zhangs talk)Anomalous Hall Effect (Haldanes talk)

Berry Phase Curvature in k-spaceBloch wavefucntion

Berry phase connection in k-spacecovariant derivativeCurvature in k-spaceAnomalous Velocity andAnomalous Hall EffectNon-commutative Q.M.

Duality between Real and Momentum Spacesk- space curvaturer- space curvature

Z.FangSrRuO3Degeneracy point Monopole in momentum space

Fermats principle and principle of least actionPath 1Path 2Path 3Path 4Path 5Every path has a specific optical path length or action.Fermat:stationary optical path length actual trajectoryLeast action : stationary action actual trajectoryStartGoalSearching stationary value ~ Solving equations of motion

Trajectories of light and particleWhat determine the equations of motion?Historically, experiments and observations

Any fundamental principles?(Fermats principle, principle of least action)

Geometrical phase (Berry phase)Principle of least actionPhase factor Equations of motionAlthough 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 packetR. 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

Equations of motion of optical packetAnomalous velocityNeglecting polarization Conventional geometrical optics

Berry Phase in OpticsPropagation of light and rotation of polarization plane in the helical optical fiberChiao-Wu, Tomita-Chiao, Haldane, BerrySpin 1 Berry phase

Reflection and refraction at an interfaceNo polarizationCircularly polarizedShift perpendicular to both of incident axis and gradient of refractive index

Conservation law of angular momentumConservation of total angular momentum as a photonEOM are derived under the condition of weak inhomogeneity.Application to the case with a sharp interface?

Comparison with numerical simulationV0: 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 phaseKnowledge about electrons in solidsPeriodic structure without a symmetryBloch wave with Berry phaseExample of 2D photonic crystal without inversion symmetryPhotonic crystal without a symmetry Bloch wave of light with Berry phase

Enhancement of optical Hall effect ?!Shift in reflection and refractionSmall Berry curvaturesmall 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 inclinehttp://ppprs1.phy.tu-dresden.de/~rosam/kurzzeit/main/bloch/bo_sub.htmlStrength of periodic structureEnergyMeaning of the height of periodic structureElectron : electrical potentialLight : (phase) velocity of light

For low energy Bloch waveLarge amplitude at low pointSmall amplitude at high pointBloch waveAn intermediate between traveling wave and standing wave

Dielectric function and photonic bandWe shall consider wave ribbons with kz=0.Note: Eigenmodes with kz=0 are classified into TE or TM mode.

Berry curvature of optical Bloch waveFor simplicity, we consider the case in which the spin degeneracy is resolved due to periodic structure.

Berry curvature in photonic crystalBerry curvature is large at the region whereseparation between adjacent bands is small.c.f. Haldane-Raghu Edge mode

Trajectory of wave packet in photonic crystalLarge shift of several dozens of lattice constantSuperimposed 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 polarizationpolarization due to displacements of rigid ionsIonic polarization+ It is not well-defined in general. It depends on the choice of a unit cell. It is not a bulk polarization.Polarization of a unit cell RAveraged polarization at rCharge determines pol.Ionicity is needed !!

quantum theory of polarizationCovalent ferroelectric: polarization without ionicityr 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 !!

Ferroelectricity in Hydrogen Bonded Supermolecular ChainS.Horiuchi et al 2004Neutraland covalentPolarization is huge compared with the classical estimate

Ferroelectricity in Phz-H2caS. Horiuchi @ CERC et al. Hydrogen bond ( covalency) Polarization as a Berry phaseFirst-principles calculationIsolated molecule 0.1 C/cm2 (too small !)

Large polarizationwith covalencyWith F. Ishii @ERATO-SSSIsolated moleculeBulk

Geometrical meaning of polarization in 1D two-band modeldP : Solid angle of the ribonGeneralized Born charge

Strings as trajectories of band-crossing points 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-free3. Total flux of the string is quantized to be an integer (Pontryagin index, or wrapping number):[c.f. Thouless]flux density:C[-p/a,p/a]BCBand-crossing point

Biot-Savart law, asymptotic behavior & charge pumpingTransverse part of the polarization current A

Biot-Savart law:

L : stringsAsymptotic behavior (leading order in 1/Eg)stringEgStrength ~ 1/EgDirection: same as a magnetic field created by an electric currentQuantum charge pumping due to cyclic change of Q around a string

ne

Specific modelsSimplest physically relevant modelsDifferent choices of f and gGeometrically differentstructures of strings Band polarization current A

Quantum Charge Pumping in InsulatorElectron(charge)flowLarge polarization even in the neutral molecules orPressure

Dimerized charge-ordered systemsTTF-CA(TMTTF)2PF6(DI-DCNQI)2AgTTF-CA: polarization perpendicular to displacement of molecules. D2 triggers the ferroelectricity.

ConclusionsGeneralized 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 studyGoal : dissipationless functionality of electrons in solidsKey concept : topological effects of wave phenomena of electronsWhat is corresponding phenomena in optics?Example of our studyTopological interpretation of quantization in quantum Hall effectIntrinsic anomalous Hall effect and spin Hall effect due to the geometrical phase of wave functionGeometrical optics : simple and useful for designing optical devicesWave optics : complicated but capable of describing specific phenomena for waveTopological effects of wave phenomena Photonic crystals as media with eccentric refractive indices Extended geometrical optics

Polarization and Angular momentumLinear S = 0Right circular S = +1Left circular S = -1http://www.physics.gla.ac.uk/Optics/projects/singlePhotonOAM/Polarization and spinRotation and angular momentumRotation of center of gravityRotation around center of gravityhttp://www.expocenter.or.jp/shiori/ugoki/ugoki1/ugoki1.html

Action and quantum mechanicsQuantum mechanicsWave-particle dualityEverything 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 approximatelySimilar relation holds between geometrical and wave optics.

Wave and geometrical optics, Quantum and classical mechanicsWave optics Eikonal Fermats principle Geometrical opticsQuantum mechanics Path integral Principle of least action Classical mechanicsOptical path, Action ~ Phase factorRoughly speaking,Trajectory is determined by the phase factor of a wave function.

Hall effect of 2DES in periodic potentialM.-C. Chang and Q. Niu, Phys. Rev. B 53, 7010 (1996)

Optical path length and actionParticle in inhomogeneous potentialAction = Sum of (kinetic energy potential) x (infinitesimal time) along a trajectoryLight 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 Studies1. 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.

SummaryTopological effects in wave phenomena of electrons What are the corresponding phenomena of light?Equations of motion of optical packet with internal rotationDeflection of light due to anomalous velocityQHE, Intrinsic AHE, Intrinsic spin HE ~ Optical HEPhotonic 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 measurementOptical Hall effect of packet with internal OAM (Sasada)Localization in photonic band with Berry phaseSurface mode of photonic crystal and Berry curvatureMagnetic 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 lightLinear S=0Right circular S=1Left circular S=-1http://www.physics.gla.ac.uk/Optics/projects/singlePhotonOAM/Spin angular momentumOrbital angular momentumL=0L=1L=2L=3The above OAM is interpreted as internal angular momentum when optical packets are considered.More generally, Berry phase internal rotation ?

Rotation of optical packetNon-zero Berry curvature ~ RotationPeriodic structure without inversion rotating wave packet

Molecular orbitals(extended HuckelTransfer integral t is estimated by = ES,E~10eV S: overlap integral

Phz stackH2ca stackLUMOHOMOLUMOHOMOTransfer integrals along the stacking directionb-axis-4.95.51.5-1.4-5.2-2.2 (x10-3)2.7-1.6

Polarization is huge compared with the classical estimate neutral

Wave packetWave packet (Green) in potential (Red)http://mamacass.ucsd.edu/people/pblanco/physics2d/lectures.htmlImage 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)

Magnetic field by circuit(i)(ii)Case (ii) can not explain the obs. value energy perturbation due to atomic displacement