Graphene has two atoms in its primitive unit cell. This makes it intuitive to see that the tight binding Hamiltonian can be constructed as a $ 2 \times 2 $ matrix $H$ acting on a spinor $S$ that consists of the wavefunction from an atom in sublattice A and B.
$H_{monolayer}=\gamma \cdot \begin{pmatrix} 0 & k_x-ik_y \\ k_x+ik_y & 0 \end{pmatrix}$
$S_{monolayer}=\begin{pmatrix} |\psi_A\rangle\\ |\psi_B\rangle \end{pmatrix}$
Bilayer Graphene has four atoms in a primitive unit cell and its tight binding Hamiltonian is a 4x4 matrix whose matrix elements represent the hopping between said lattice sites (depending on how it is stacked and what hopping parameters you wish to involve in the calculation). An example might be as follows:
$H_{bilayer}=\begin{pmatrix} 0 & 0 & 0 & v(k_x-ik_y)\\ 0 & 0 & v(k_x+ik_y) & 0\\ 0 & v(k_x-ik_y) & 0 & \gamma'\\ v(k_x+ik_y) & 0 & \gamma' & 0 \end{pmatrix}$
$S_{bilayer}=\begin{pmatrix} |\psi_{A1}\rangle\\ |\psi_{B2}\rangle\\ |\psi_{A2}\rangle\\ |\psi_{B1}\rangle \end{pmatrix}$
Where the basis is chosen is in an arbitrary order (1 and 2 indices refer to layer number).
How does one write this in a "two-component basis" and what does that mean? Also, what is this hamiltonian acting on in this case? The bilayer hamiltation in this basis (which I do not know what it represents) is written as follows:
$H'_{bilayer}=-\dfrac{\hbar^2}{2m}\begin{pmatrix} 0 & (k_x-ik_y)^2 \\ (k_x+ik_y)^2 & 0 \end{pmatrix}$