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We investigate the photoinduced effect to the silicene, which is a topological insulator, by the circularly polarized light in off-resonance regime with a frequency much larger than the critical value (also much larger than the frequency about the particle-hole pair creation), and with a perpendicular electric field. The anomalous Rabi frequency which is a non-linear optical, arised by the off-resonance circularly polarized light. The temporal behavior of the pseudospin, valley, and spin degrees of freedom, which are momentum-and quasienergy-dependent, are explored. The anomalous Rabi oscillation is also related to the photoinduced topological phase transition between the topological trivial state with zero Chern number and gapped edge state and the topological nontrivial state with nonzero Chern number and gapless edge state. The off-resonance laser can also induce the topological phase transition by manipulating the energy band structure, rather than excite the atoms to the high quantum-number states like the resonance light. The exchange between the radiation driving field and the two-component dynamical polarizations with the dipole oscillation, plays a important role in the determination of the out-of-plane spin polarization and the motion of the center of mass, which can induced a collapse-and-revival pattern under a certain condition. The Rabi oscillation observed in the motion of the center of mass in a laser-induced harmonic potential can be used to detect the time evolution of the atom polulation. Our results can also be applied to the other two-dimension low-energy Dirac models or the surface of the three-dimension topological insulators, and even the weyl semimetal with the photoinduced topological phase transition.

We investigate the interband and intraband transition of the monolayer and AB-stacked bilayer silicene in low-energy tight-binding model under the electric field, where we focus on the dynamical polarization function, screening due to the charged impurity, and the plasmon dispersion. We obtain the logarithmically divergen polarization function within the random-phase-approximation (RPA) whose logarithmic singularities corresponds to the discontinuities of the first derivative which is at the momentum q = 2kF in static case and indicate the topological phase transition point from the gapless semimetal to the gapped band insulator. We also obtain the power-law-dependent Friedel oscillation which can be enhanced by increasing the Rashba-coupling, that can contribute to the screened potential of the charged impurity which scale as ∼ r −1/2 in the short distance from the impurity and scale as ∼ r −1/3 in the long distance from the impurity. In the single-particle excitation regime with the electron-hole continuum, the interband and intraband transition happen, and the plasmon dispersion, which we mainly focus on the optical plasmon (which ∼ √ q in long-wavelength limit) in this paper, start to damped into the electron-hole pairs due to the nonzero imaginary part of the polarization function. In low-frequency regime where the collective behavior and optical properties of the Dirac material relys more on the frequency than the fine structure constant, the intraband transition is dominate and it's found that completely undamped in the static case (ω = 0), which is due to the absence of the imaginary dynamic polarization. We also observe the linear (weakly damped) plasmon model for the classical bilayer silicene which is similar to the high-energy π-plasmon or the case of conducting substrate which with strong metallic screening in the bulk semiconductor. For the large carrier density, we find the plasmon diapersion has ωp ∼ n 1/2 which consistent with the quadratic dispersion around the Dirac-point like the bilayer silicene with the effective mass about the interlayer hopping (esperially when taking the Rashba-coupling and exchange field into consider) or the normal two-dimension electron gas, while in the little concentration limit, ωp ∼ n 1/4 which consistent with the linear dispersion like the monolayer silicene. Under the nonmagnetic impurity scattering, the Thomas-Fermi decay and Friedel oscillation can easily be observed due to the strong spin-orbit couopling of the bilayer silicene even we don't take the Rashba-coupling into consider. *

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We discuss the dynamical polarization with finite momentum and frequency in the presence of many-electron effect, including the screened Coulomb interaction, self-energy and vertex correction. The longitudinal conductivity, screened Coulomb interaction, and the response function are calculated. The behavior of the Dirac Fermions, including the propagation of the charge density which exhibits the causality, affects largely the low-temperature physical properties of the Dirac semimetal, like the silicene. For the polarization-related quantities (like the dielectric function), the method of standard random phase approximation (RPA) provides the non-interaction results (ignore the many electron effect), for a more exact result, we discuss the selfenergy and the vertex correction for the two-dimension Dirac model. We found that, after the self-energy correction, the longitudinal conductivity increase compared to the noninteracting one in optical limit. For the renormalization treatment, the ultraviolet cutoff is setted as Λ = t in our calculations, i.e., within the range between two Van Hove singularities where the density of states divergent logarithmically. The (corrected) screened Coulomb interaction and the response function are also discussed. Our results are helpful to the application of the Dirac materials (or the Weyl semimetal) in spintronics or valleytronics.