Graphene Hamiltonian: Eq.(1) in PRB 81, 205444

[hokhani]

As far as I know, the Hamiltonian of graphene in the Bloch's sums $|A\rangle$ and $|B\rangle$ near the points K or K' is a $2 \times 2$ matrix with the components: $\langle A|H|A\rangle$, $\langle A|H|B\rangle$, $\langle B|H|A\rangle$,$\langle B|H|B\rangle$ which all are parameters (and not variables). But in the Eq.(1) in PHYSICAL REVIEW B 81, 205444 (2010) (http://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.205444) these components are functions and operators in terms of (x,y). Could anyone please tell me how to obtain this formula?

 

[fzero]

The tight-binding model to Dirac-Weyl equation derivation is outlined in the review by Castro Neto et al http://arxiv.org/abs/0709.1163 (Sect. II.B), as well as in these notes https://www.tfkp.physik.uni-erlangen.de/download/research/DW-derivation.pdf, which have a few more details of the steps.