In order to analytically capture and identify peculiarities in the electronic structure of silicene, Weaire-Thorpe(WT) model, a standard model for treating three-dimensional (3D) silicon, is applied to silicene with the buckled 2D structure. In the o
riginal WT model for four hybridized $sp^3$ orbitals on each atom along with inter-atom hopping, the band structure can be systematically examined in 3D, where flat (dispersionless) bands exist as well. For examining silicene, here we re-formulate the WT model in terms of the overlapping molecular-orbital (MO) method which enables us to describe flat bands away from the electron-holesymmetric point. The overlapping MO formalism indeed enables us to reveal an important difference: while in 3D the dipersive bands with cones are sandwiched by doubly-degenerate flat bands, in 2D the dipersive bands with cones are sandwiched by triply-degenerate and non-degenerate (nearly) flat bands, which is consistent with the original band calculation by Takeda and Shiraishi. Thus emerges a picture for why the whole band structure of silicene comprises a pair of dispersive bands with Dirac cones with each of the band touching a nearly flat (narrow) band at $Gamma$. We can also recognize that, for band engineering, the bonds perpendicular to the atomic plane are crucial, and that a ferromagnetism or structural instabilities are expected if we can shift the chemical potential close to the flat bands.
Chiral symmetry, fundamental in the physics of graphene, guarantees the existence of topologically stable doubled Dirac cones and anomalous behaviors of the zero-energy Landau level in magnetic fields. The crucial role is inherited in the optical res
ponses and many-body physics in graphene, which are explained in this paper. We also give an overview of multilayer graphene from the viewpoint of the optical properties and their relation with the chiral symmetry.
