Graphene, its Homologues and Their Classification

Using tight binding model, lattice QFT and group theory methods, we study a class of lattice QFT models that are cousins of graphene; and which are classified by finite dimensional ADE Lie groups containing the usual crystallographic symmetries as discrete subgroups. We show in particular that the electronic properties of the 1D lattice poly-acetylene chain are given by a SU(2) model and those of the well known 2D graphene by SU(3). We also give two other models classified by SU(4) and SO(6) symmetries; they respectively describe 3D diamond and 3D lattice with octahedral sites. It is shown as well that the dispersion energies of this set of models are completely characterized by the roots of the Lie algebras underlying the symmetry groups. Other features, such as SO(5) lattice involving sp^{3}d hybridization as well as the relation between the 4D hyperdiamond, having a SU(5) symmetry and the 4D lattice QCD, are also discussed.