1. 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)

2. Berry phase M.V.Berry, Proc. R.Soc. Lond. A392, 45(1984) Hamiltonian,parameters  adiabatic change Berry Phase Connection of the wavefunction in the parameter space  Berry phase curvature 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

3. Electrons with ”constraint” Projection onto positive energy state Spin-orbit interaction as SU(2) gauge connection Dirac electrons doubly degenerate positive energy states. Bloch electrons Projection onto each band Berry phase of Bloch wavefunction Spin Hall Effect (S.C.Zhang’s talk)Anomalous Hall Effect (Haldane’s talk)

4. 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.

5. Duality between Real and Momentum Spaces k- space curvature r- space curvature

6. Z.Fang SrRuO3 Degeneracy point  Monopole in momentum space

7. Fermat’s principle and principle of least action Path 1 Path 2 Path 3 Path 4 Path 5 Every path has a specific optical path length or action. Fermat : stationary optical path length → actual trajectory Least action : stationary action → actual trajectory Start Goal Searching stationary value ~ Solving equations of motion

8. Trajectories of light and particle What determine the equations of motion? Historically, experiments and observations Any fundamental principles? (Fermat’s principle, principle of least action)

9. Geometrical phase (Berry phase) Principle of least action Phase factor → Equations of motion Although light has spin, no effect of Berry phase in conventional geometrical optics. Berry phase “Wave functions with spin obtain geometrical phase in adiabatic motion.” Topological effects (wave optics) in trajectory of light (geometrical optics) → wave packet

10. 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)

11. Light in weakly inhomogeneous medium

12. Equations of motion of optical packet Anomalous velocity Neglecting polarization → Conventional geometrical optics

13. 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

14. Reflection and refraction at an interface No polarization Circularly polarized Shift perpendicular to both of incident axis and gradient of refractive index

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

16. 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.

17. Photonic crystal and Berry phase Knowledge about electrons in solids Periodic structure without a symmetry →Bloch wave with Berry phase Example of 2D photonic crystal without inversion symmetry Photonic crystal without a symmetry → Bloch wave of light with Berry phase Enhancement of optical Hall effect ?! Shift in reflection and refraction Small Berry curvature →small shift of the order of wave length

18. Wave in periodic structure -- Bloch wave -- 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 Strength of periodic structure Energy 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 Bloch wave An intermediate between traveling wave and standing wave

19. 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.

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

21. 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

22. Trajectory of wave packet in photonic crystal Large shift of several dozens of lattice constant 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.

23. classical theory of polarization 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. Polarization of a unit cell R Averaged polarization at r Charge determines pol. Ionicity is needed !!

24. 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 !!

25. Ferroelectricity in Hydrogen Bonded Supermolecular Chain S.Horiuchi et al 2004 Neutral and covalent Polarization is “huge” compared with the classical estimate

26. Ferroelectricity in Phz-H2ca S. Horiuchi @ CERC et al. Hydrogen bond ( covalency) Polarization as a Berry phase First-principles calculation Isolated molecule → 0.1 μC/cm 2 (too small !) Large polarization with covalency With F. Ishii @ERATO-SSS Isolated molecule Bulk

27. Geometrical meaning of polarization in 1D two-band model dP : Solid angle of the ribon Generalized Born charge

28. Strings as trajectories of band-crossing points 1. only along strings (trajectories of band-crossing points) with k in [  a  a   -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] flux density: C×[  /a,  /a] B C Band-crossing point

29. Biot-Savart law, asymptotic behavior & charge pumping Transverse part of the polarization current A Biot-Savart law: L : strings Asymptotic behavior (leading order in 1/E g ) string EgEg Strength ~ 1/E g Direction: same as a magnetic field created by an electric current Quantum charge pumping due to cyclic change of Q around a string ne

30. Specific models Simplest physically relevant models Different choices of f and g Geometrically different structures of strings B and polarization current A

31. Quantum Charge Pumping in Insulator Electron(charge)flow Large polarization even in the neutral molecules or Pressure

32. Dimerized charge-ordered systems TTF-CA (TMTTF) 2 PF 6 (DI-DCNQI) 2 Ag TTF-CA: polarization perpendicular to displacement of molecules.   triggers the ferroelectricity.

33. 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

34. Motivation of this study Goal : dissipationless functionality of electrons in solids Key concept : topological effects of wave phenomena of electrons What is corresponding phenomena in optics? 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 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

35. Polarization and Angular momentum Linear S = 0Right circular S = +1Left circular S = -1 http://www.physics.gla.ac.uk/Optics/projects/singlePhotonOAM/ Polarization and spin Rotation and angular momentum Rotation of center of gravityRotation around center of gravity http://www.expocenter.or.jp/shiori/ ugoki/ugoki1/ugoki1.html

36. 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 Similar relation holds between geometrical and wave optics.

37. “Wave and geometrical optics”, “Quantum and classical mechanics” Wave optics → Eikonal → Fermat’s principle → Geometrical optics Quantum mechanics → Path integral → Principle of least action → Classical mechanics Optical path, Action ~ Phase factor Roughly speaking, Trajectory is determined by the phase factor of a wave function.

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

39. Optical path length and action Particle in inhomogeneous potential Action = Sum of (kinetic energy – potential) x (infinitesimal time) along a trajectory 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) Point Optical path length and action can be defined for any trajectories, regardless of whether realistic or unrealistic.

40. Why is it interpreted as the optical Hall effect ? 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)] 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.) QHE, AHE, spin HE ~ optical HE NOTE: spin is not indispensable in QHE

41. 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.

42. 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

43. 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

44. Internal Angular momentum of light Linear S=0Right circular S=1Left circular S=-1 http://www.physics.gla.ac.uk/Optics/projects/singlePhotonOAM/ Spin angular momentum Orbital angular momentum L=0L=1L=2L=3 The above OAM is interpreted as internal angular momentum when optical packets are considered. More generally, Berry phase → internal rotation ?

45. Rotation of optical packet Non-zero Berry curvature ~ Rotation Periodic structure without inversion → rotating wave packet

46. Molecular orbitals(extended Huckel ） Transfer integral t is estimated by ｔ = ES, E~10eV （ S: overlap integral ）

47. Phz stack H 2 ca stack LUMO HOMO LUMO HOMO Transfer integrals along the stacking direction （ b-axis ） -4.95.5 1.5-1.4 -5.2 -2.2 (x10 -3 ) 2.7 -1.6

48. Polarization is “huge” compared with the classical estimate neutral

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

50. Simple example (electron in periodic potential)

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