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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, parametersadiabatic 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 spaceBerry 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 ak : 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 | zc ) v(rc )kc c zc 1 | zc ) , v(rc ) , k (rc ) (rc ) zc iek 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 iek k e k ek : 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 | zc ) 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 zin , ei zin ] 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 AT ,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 rc 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 dl r l dl 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 n1 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 • • • • • • • • 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 rc Enk e (rc ) Leff c Enk : energy of nth band c rc k Enk c c kc e rc (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