# Symmetry Class of Graphene with Spin-orbit Coupling

I know that tight-binding graphene with $$\text{p}_\text{z}$$-, $$\text{d}_\text{xz}$$- and $$\text{d}_\text{yz}$$-orbitals and spin-orbit coupling is a $$\mathbb{Z}_2$$-topological insulator. But I want to categorise it according to the periodic table of topological invariants. My problem now is that I don't know which of the ten Altland-Zirnbauer classes is the correct one. Based on my logic, it has to be AII:

• Graphene satisfies time-reversal symmetry, and $$T^2=-1$$. So it has to be one of the symmetry classes AII, DIII and CII.
• Graphene is a two-dimensional $$\mathbb{Z}_2$$-topological insulator. So, it must be either AII or DIII.
• If I'm not mistaken, the d-orbitals break particle-hole symmetry as the band structure is no longer symmetric around the Fermi energy. So it has to be AII.

I've also computed $$\sigma_z H(-\vec{k})\sigma_z \neq -H(\vec{k})$$ (with $$\sigma_z$$ acting on the sublattices). As far as I know, equality here would imply particle-hole symmetry in graphene. The two terms $$\sigma_z H(-\vec{k})\sigma_z$$ and $$-H(\vec{k})$$ differ by twice the d-orbital energy, which I consider reassuring.

My question can now be summarised as: Is graphene, as I described it, in the symmetry class AII? Furthermore, did I make any mistakes with my arguments above?