Magnetoplasmons for the $\alpha$-T$_3$ model with filled Landau levels

Using the $\alpha$-T$_3$ model, we carried out analytical and numerical calculations for the static and dynamic polarization functions in the presence of a perpendicular magnetic field. These results were employed to determine the longitudinal dielectric function and the magnetoplasmon dispersion relation. The magnetic field splits the continuous valence, conduction and flat energy subband into discrete Landau levels which present significant effects on the polarization function and magnetoplasmon dispersion. We present results for a doped layer in the integer quantum Hall regime for fixed hopping parameter $\alpha$ and various magnetic fields as well as chosen magnetic field and different $\alpha$ in the random phase approximation.