a贝利相位

Effective Lagrangian of wave packet
d d L i H variaton i H dt dt
W : wave packetcentered position rc and momentumkc at d Leff W i H W variation EOMof rc and kc dt
E
doubly degenerate positive energy states.
E
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)
Berry Phase Phenomena
Optical Hall effect and Ferroelectricity as quantum charge pumping
Naoto Nagaosa CREST, Dept. Applied Physics, The University of Tokyo
Newton's equation of motion [turnin thedirectonof lower V (rc )] d d m rc V (rc ) dt dt m : mass, V (rc ) potential :
What determine the equations of motion? Historically, experiments and observations Any fundamental principles? (Fermat’s principle, principle of least action)
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
a k : creationoperator circularlypolarizedphoton of W RH W Conditionfor thecenterof gravity rc W H W
Leff
k ( z | | z ) i ( z | z ) v(r )k c kc rc c c k c c c c
c
zc 1 | zc ) , v(rc ) , k (rc ) (rc ) zc


iek k e k
Equations of motion of optical packet
rc : posit ion k c : momentum , v( rc ) : light speed | zc ) : stat eof polarizaton i k : Berryconnect ion k : Berrycurvat ure
parametersadiabatic change
X2
C
X1
eigenvalue and eigenstate for each parameter set X
(T ) e
i n ( C ) ( i / ) 0 dtEn ( X ( t ))
e

T
(0)
n (C ) i dX n ( X ) | X n ( X )
M.V.Berry, Proc. R.Soc. Lond. A392, 45(1984)
H (X )
X ( X1 , X 2 , , , X n ) it (t ) H ( X (t )) (t ) H ( X )n ( X ) En ( X )n ( X )
Hamiltonian, Transitions between eigenstates are forbidden during the adiabatic change Projection to the sub-space of Hilbert space constrained quantum system
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) eikrunk (r)
R : positionoperator Conditionrc 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)
Chiao-Wu, Tomita-Chiao, Haldane, Berry
out in in | zc t [ei z , ei z ]
dk [ k ] dS k [ k ]
S
Spin 1 Berry phase
Reflection and refraction at an interface
Trajectories of light and particle
GeometricaOptics [turnin thedirectionof larger n(rc )] l d d n(rc ) ds rc n(rc ) ds n(rc ) : refractive index， n(rc )ds dt
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 2 W w(k kc , rc ) zc ak 0 , zc 1 (2 )3
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
xi ri An (k ) i ki An (k )
[ x, y] i( kx Any (k ) k y Anx (k )) iBnz (k )
k dx(t ) V k x V i[ x, H ] x i[ x, y ] Bnz (k ) dt m y m y
Shift perpendicular to both of incident axis and gradient of refractive index
No polarization
Circularly polarized
Conservation law of angular momentum
Anomalous velocity

k k k k i k k 3 3 k
Neglecting polarization → Conventional geometrical optics
k iek k e k ek : polarizati vector on
C
dX An ( X ) dS Bn ( X )
C
Berry Phase
Connection of the wavefunction in the parameter spaceBerry phase curvature
Electrons with ”constraint”




Degeneracy point Monopole in momentum space
SrRuO3 Z.Fang
Fermat’s principle and principle of least action
Goal Path 4 Path 3 Path 2 Path 1 Start Path 5
Bloch wavefucntion Berry phase connection in k-space covariant derivative Curvature in k-space Anomalous Velocity and Anomalous Hall Effect
An (kkc k ( z | | z ) rc c c c c kc 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