文档介绍：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 , Proc. . Lond . A392, 45(1984) )(XH Hamiltonian, ),,,,( 21nXXXX? parameters ? adiabatic change )( ))(()(ttXHti t????? 1X 2XC )0()( 0 ))(()/()(?????? Tn ntX dtE iCieeT ?)()()()(XXEXXH nnn????????????????)()( )(|)()(XBdS XAdX XXdX iC n C n n C Xn n??? 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 Electrons with ” constraint ” Projection onto positive energy state Spin-orbit interaction as SU(2) gauge connection Dirac electrons doubly degenerate positive energy states . Ek Bloch electrons Projection onto each band Berry phase of Bloch wavefunction k E Spin Hall Effect ( ’s talk) Anomalous Hall Effect ( Haldane ’s talk) Berry Phase Curvature in k-space Bloch tion )()(ruer nk ikr nk??????? nk knk nuuikA||)( Berry phase connection in k-space )()(kAikArx nknii i????? covariant derivative )( ))()((],[k iB kAkAiyx nz nx kny k y x????? Curvature in k-space y VkBm ky Vyxim kHxidt tdx nz x x??????????)(],[],[ )(xk yk zk????knku|?? nku| k? Anomalous Velocity and Anomalous Hall Effect mutative . dt tkdkBk kdt trd n n)()( )()( ????????????dt trdrBr rVdt tkd)()( )()( ??????????? Duality between Real and Momentum Spaces k- space curvature r- space curvature SrRuO3 Degeneracy point ? Monopole in momentum space Fermat ’ s principle and principle of least action Path 1 Path 2 Path 3 Path 4 Path 5 Every path has a specific optical p