# What is the definition of the pseudo spin of bilayer graphene

The Hamiltonian for graphene at $\vec{k}$ away from the $K$ point is

$$H=\vec{\sigma} \cdot \vec{k} =\begin{pmatrix} 0 & k_x - i k_y \\ k_x + i k_y & 0 \\ \end{pmatrix} = k \begin{pmatrix} 0 & e^{-i \theta} \\ e^{i \theta} & 0 \\ \end{pmatrix}$$

and eigen vectors are $(e^{-i \theta/2},\pm e^{i \theta/2})$. And we often call this vector pseudo spin (in terms of the another quesiton). We can discuss the direction of the pesudospin (like that question ).

How about bilayer graphene? The eigenvectors of bilayer graphene are four-dimensional because four atoms are in the unit cell and the Hamiltonian of the bilayer graphene is four-dimensional. Can we define the eigenvectors of the bilayer graphene as a pseudospin and discuss the direction of the pseudospin of bilayer graphene?

I think the answer should be YES. In arXiv:1109.0307v3, they discuss the pseudospin of double-layer graphene. But I could not understand the definition of the pseudospin of double-layer graphene.