# Graphene with a disclination and the spin-orbit coupling

I am trying to follow the methods used in this paper (http://arxiv.org/pdf/1208.3023.pdf) to construct the Hamiltonian of a graphene cone, but taking into account the spin-orbit coupling.

The paper constructs the Hamiltonian for a graphene cone with a Haldane mass term of the form $m \tau_z \sigma_z$.

I wish to take into account the spin. So, instead of the mass term having the form $m \tau_z \sigma_z$, it should look like $m \tau_z \sigma_z s_c$, where $s_c = \vec{n} \cdot \vec{s}$ accounts for the spin projection on the surface of the cone. The spin degree of freedom can easily be added to the massless term of the Hamiltonian by putting $s_0$ at the end (since it does not affect it).

(If this were flat graphene, the mass term would be $m \tau_z \sigma_zs_z$ like in the Kane-Mele model.)

The problem is that when I construct my Hamiltonian, I don't know "when" to add the mass term. Do I start from the standard 2D graphene Hamiltonian (with spin), add this mass term, and then use the required formalism to take into account the conical topology?

Or can I already start from the cone's massless Hamiltonian (which is already in block-diagonal form) and simply add the mass term to it?

Ideally I would like to obtain a Hamiltonian which is in block-diagonal form, which is attainable if we neglect the spin (as shown in the paper).

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