We derive the semiclassical equation of motion for the wave-packet of light taking into account the Berry curvature in momentum space. This equation naturally describes the interplay between orbital and spin angular-momenta, i.e., the conservation of total angular-momentum of light. This leads to the shift of wave-packet motion perpendicular to the gradient of dielectric constant, i.e., the polarization-dependent Hall effect of light. An enhancement of this effect in photonic crystals is also proposed.PACS numbers: 42.15. Eq, 03.65.Sq, 03.65.Vf, The similarity between geometrical optics and particle dynamics has been the guiding principle to develop the quantum mechanics in its early stage. One can consider a trajectory or ray of light at the length scale much larger than the wavelength λ. By setting = c = 1, the equation for the eikonal ψ in the geometrical optics reads [∂ µ ψ(x)] 2 = [∇φ(r)] 2 − n(r) 2 = 0, where x µ = (r, t) is the four-dimensional coordinates, ψ(x) = φ(r) − n(r)t, and n(r) is the refractive index slowly varying within the wavelength [1]. This equation is identical to the Hamilton-Jacobi equation for the particle motion by replacing ψ by the action S. The equation for the ray of light can be derived from the eikonal equation in parallel to the Hamiltonian equation of motion. Deviation from this geometrical optics is usually treated in terms of the diffraction theory. However, most of the analyses have been done in terms of the scalar diffraction theory, neglecting its vector nature. However, the light has the degree of freedom of the polarization, which is represented by the spin S = 1 parallel or anti-parallel to the wavevector k. It is known that this spin produces the Berry phase when the light is guided by the optical fiber with torsion [2]. Also the adiabatic change of the polarization even without the change of k produces the phase change called Pancharatnam phase [3]. In this Letter, we will show that the trajectory or ray of light itself is affected by the Berry phase [4], which leads to various nontrivial effects including the Hall effect.For simplicity, we focus on an isotropic, nonmagnetic medium; the refractive index n(r) is real and scalar, but is not spatially uniform. Let us consider the wavefunction for the wave-packet with the wavevector centered at k c and position centered at r c . Because of the conjugate relation of the position and wavevector, both of them inevitably have finite width of distribution. Therefore, the relative phase of the wavefunctions at k and k + dk matters, which is the Berry connection or gauge field Λ k defined by [Λ k , where e λk is the polarization vector of λ-polarized photon, and λ = ± correspond to the right-and left-circular polarization. Here, it is noted that the SU(2) gauge field Λ k is 2 × 2 matrix. The corresponding Berry curvature (BC) or field strength is given byDue to the masslessness of a photon, it is diagonal in the basis of the right and left circular polarization as given by Ω k = σ 3 k/k 3 [a massive case will be discussed late...

The anomalous Hall effect in two-dimensional ferromagnets is discussed to be the physical realization of the parity anomaly in (2+1)D, and the band crossing points behave as the topological singularity in the Brillouin zone. This appears as the sharp peaks and the sign changes of the transverse conductance σ xy as a function of the Fermi energy and/or the magnetization. The relevance to the experiments including the three dimensional systems is also discussed.

The localization problem of electronic states in a two-dimensional quantum spin Hall system (that is, a symplectic ensemble with topological term) is studied by the transfer matrix method. The phase diagram in the plane of energy and disorder strength is exposed, and demonstrates "levitation" and "pair annihilation" of the domains of extended states analogous to that of the integer quantum Hall system. The critical exponent nu for the divergence of the localization length is estimated as nu congruent with 1.6, which is distinct from both exponents pertaining to the conventional symplectic and the unitary quantum Hall systems. Our analysis strongly suggests a different universality class related to the topology of the pertinent system.

An effective theory is constructed for analyzing a generic phase transition between the quantum spin Hall and the insulator phases. Occurrence of degeneracies due to closing of the gap at the transition are carefully elucidated. For systems without inversion symmetry the gap-closing occurs at ± k0( = G/2) while for systems with inversion symmetry, the gap can close only at wave-numbers k = G/2, where G is a reciprocal lattice vector. In both cases, following a unitary transformation which mixes spins, the system is represented by two decoupled effective theories of massive twocomponent fermions having masses of opposite signs. Existence of gapless helical modes at a domain wall between the two phases directly follows from this formalism. This theory provides an elementary and comprehensive phenomenology of the quantum spin Hall system.