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I am trying to get into graphene and I have a question about the Hamiltonian. As I read, the TB Hamiltonian for graphene is $$ \mathcal{H} = -t \left[ a_{i,\sigma}^\dagger b_{i,\sigma} + b_{i, \sigma}^\dagger a_{i,\sigma} \right]. $$

I understand that we have two terms due to the fact that a primitive cell contains two carbon atoms. They are identical, but we have to distinguish between them because of the differences in the neighbors.

But I also found that sometimes this Hamiltonian is presented as follows $$ \mathcal{H} = v_F \hat{\sigma} \circ \textbf{k}. $$

Do I understand correctly that the second Hamiltonian is obtained from the first one by applying k·p perturbation theory? Or what's the difference? In which cases do we use the first one and for which do we use the second one?

As I understand the k·p perturbation theory, we approximate the Hamiltonian near K/K' points, so do we somehow have a Hamiltonian that describes a particular region of our system?

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