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.

This paper presents a theoretical analysis on bulk and edge states in honeycomb lattice photonic crystals with and without time-reversal and/or space-inversion symmetries. Multiple Dirac cones are found in the photonic band structure and the mass gaps are controllable via symmetry breaking. The zigzag and armchair edges of the photonic crystals can support novel edge states that reflect the symmetries of the photonic crystals. The dispersion relation and the field configuration of the edge states are analyzed in detail in comparison to electronic edge states. Leakage of the edge states to free space is inherent in photonic systems and is fully taken into account in the analysis. A topological relation between bulk and edge, which is analogous to that found in quantum Hall systems, is also verified.

We construct a semiclassical theory for propagation of an optical wave packet in a nonconducting medium with a periodic structure of dielectric permittivity and magnetic permeability, i.e., a nonconducting photonic crystal. We employ a quantum-mechanical formalism in order to clarify its link to those of electronic systems. It involves the geometrical phase, i.e., Berry's phase, in a natural way, and describes an interplay between orbital motion and internal rotation. Based on the above theory, we discuss the geometrical aspects of the optical Hall effect. We also consider a reduction of the theory to a system without periodic structure and apply it to the transverse shift of an optical beam at an interface reflection or refraction. For a generic incident beam with an arbitrary polarization, an identical result for the transverse shift of each reflected or transmitted beam is given by the following different approaches: (i) analytic evaluation of wave-packet dynamics, (ii) total angular momentum (TAM) conservation for individual photons, and (iii) numerical simulation of wave-packet dynamics. It is consistent with a result by classical electrodynamics. This means that the TAM conservation for individual photons is already taken into account in wave optics, i.e., classical electrodynamics. Finally, we show an application of our theory to a two-dimensional photonic crystal, and propose an optimal design for the enhancement of the optical Hall effect in photonic crystals.

We study the effect of disorder on the anomalous Hall effect (AHE) in two-dimensional ferromagnets. The topological nature of AHE leads to the integer quantum Hall effect from a metal, i.e., the quantization of σxy induced by the localization except for the few extended states carrying Chern number. Extensive numerical study on a model reveals that Pruisken's two-parameter scaling theory holds even when the system has no gap with the overlapping multibands and without the uniform magnetic field. Therefore the condition for the quantized AHE is given only by the Hall conductivity σxy without the quantum correction, i.e., |σxy| > e 2 /(2h).PACS numbers: 72.15. Rn, The origin of the anomalous Hall effect (AHE) has been a subject of extensive controversy for a long term. One is based on the band picture with the relativistic spin-orbit interaction [1], while the other is due to the impurity scatterings [2]. Most of the succeeding theories follows the idea that the AHE occurs due to the scattering events modified by the spin-orbit interaction, i.e., the skew scattering or the side jump mechanism [3]. Recently several authors recognized the topological nature of the AHE discussed in Refs. [4,5,6]. In this formalism, the Hall conductivity σ xy is given by the Berry phase curvature in the momentum ( k-) space integrated over the occupied states [7]. Also there appeared some experimental evidences supporting it [8]. Therefore it is very important to study the effect of the scatterings by disorder, which makes k ill-defined, to see the topological stability of this mechanism for AHE.This issue is closely related to the integer quantum Hall effect (IQHE) [9] but there are several essential differences. Usually the topological stability which guarantees the quantization of some physical quantity, e.g., σ xy , has been discussed in the context of the adiabatic continuation [9]. Therefore it appears that the gaps between Landau levels in pure system are needed to start with even though the disorder potential eventually buries it. In the IQHE system without disorder, the periodic potential is irrelevant because the carrier concentration is much smaller than unity per atom. Although numerical simulations [10] use lattice models, the main concern is put on the limit of dispersionless Landau levels separated by the gaps. In the present case, i.e., in ferromagnetic metals, there are multiple bands overlapping without the gaps in the density of states. The periodicity of the lattice remains unchanged, which prohibits the uniform magnetic field and also gives a large energy dispersion. In * Electronic address: m.onoda@aist.go.jp † Electronic address: nagaosa@appi.t.u-tokyo.ac.jp the language of the effective magnetic field for electrons, it reaches a huge value of the order of ∼ 10 4 Tesla, i.e., the magnetic cyclotron length is of the order of the lattice constant, but the net flux is zero when averaged over the unit cell. Therefore these two cases belong to quite different limits although the symmetries of the systems are common, ...

We study the anomalous Hall effect (AHE) for the double exchange model with the exchange coupling |JH | being smaller than the bandwidth |t| for the purpose of clarifying the following unresolved and confusing issues: (i) the effect of the underlying lattice structure, (ii) the relation between AHE and the skyrmion number, (iii) the duality between real and momentum spaces, and (iv) the role of the disorder scatterings; which is more essential, σH (Hall conductivity) or ρH (Hall resistivity)? Starting from a generic expression for σH , we resolve all these issues and classify the regimes in the parameter space of JH τ (τ : elastic-scattering time), and λs (length scale of spin texture). There are two distinct mechanisms of AHE; one is characterized by the real-space skyrmion-number, and the other by momentum-space skyrmion-density at the Fermi level, which work in different regimes of the parameter space.PACS numbers: 72.15. Eb,75.50.Pp, The anomalous Hall effect (AHE) is a phenomenon where the Hall resistivity has an additional contribution due to the spontaneous magnetization in ferromagnets. This anomalous contribution has been attributed to the spin-orbit interaction, and various mechanisms has been proposed [1,2,3,4]. Recently it has been recognized that the original expression by Karplus and Luttinger [1], i.e., the intrinsic contribution, has the geometrical meaning in terms of the Berry-phase curvature in momentum space [5,6,7]. This is analogous to the the integer quantum Hall effect (IQHE) with the strong external magnetic field [8,9]. It was also proposed that AHE arises even without the spin-orbit interaction if the spin configuration is non-coplanar with finite spin chirality, i.e., the solid angle subtended by the spins where the electron hops successively [10,11,12,13,14]. Consider the double-exchange modelwhere r, r ′ runs the nearest neighbor sites, cr↓ ) is the annihilation (creation) operator at the site r, and S r is the classical spin localized at the site r. Assuming a strong Hund coupling |J H |(≫ |t|) between the conduction electrons and the localized spins, the Berry phase of the localized spins acts as a fictitious magnetic field for the conduction electron [15,16,17]. Ye et al. assumed that this fictitious magnetic field has a uniform component due to the spin-orbit interaction in the presence of the uniform magnetization [10]. However there is a subtle issue concerning the definition of the skyrmion number when the spins are defined on the discrete points and/or the underlying lattice is relevant. This is related to the length scale with respect to the spin texture and/or the lattice structure. Furthermore, the effect of the spin-orbit interaction can not be represented by the spatially uniform magnetic field; it induces the effective "magnetic field", i.e., the Berry phase curvature, in momentum space. In real systems, the disorder is also relevant and often the following question arises: Which is more essential, the Hall conductivity σ H or the Hall resistivity ρ H ? Therefore it is...