  Borrowing ideas from tight binding model, we propose a board class of Lattice
QFT models that are classified by the ADE Lie algebras. In the case of su(N)
series, we show that the couplings between the quantum states living at the
first nearest neighbor sites of the lattice $\mathcal{L}_{su(N)}$ are governed
by the complex fundamental representations \underline{${{\mathbf{N}}}$} and
$\bar{{\mathbf{N}}}$ of $su(N)$; and the second nearest neighbor interactions
are described by its adjoint $\underline{\mathbf{N}} \otimes \bar{\mathbf{N}}$.
The lattice models associated with the leading su(2), su(3) and su(4) cases are
explicitly studied and their fermionic field realizations are given. It is also
shown that the su(2) and su(3) models describe respectively the electronic
properties of the acetylene chain and the graphene. It is established as well
that the energy dispersion of the first nearest neighbor couplings is
completely determined by the $A_{N}$ roots $ \mathbf{\alpha}$ through the
typical dependence $N/2+\sum_{roots}\cos(\mathbf{k}.\alpha) $ with $\mathbf{k}$
the wave vector. Other features such as DE extension and other applications are
also discussed.
  Keywords: Tight Binding Model, Graphene, Lattice QFT, ADE Symmetries.
