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This is actually exactly the same question I was asking myself a while ago, and it took me quite some time to figure it out. What I ended up doing was: diagonalise the BdG Hamiltonian in Mathematica solve the expressions for the eigenvalues for $\kappa$ neglect terms proportional to $\kappa^2$, $\kappa \Delta$ in the expressions for the eigenvectors, ...

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The hexagonal Graphene lattice can be considered as a superposition of two identical sub-lattices set off by one one carbon-carbon bond length. As a result, it has two sets of wavevectors k,that are picked out by the lattice, inequivalent (since the two sublattices really are distinct) but otherwise identical (since it's semantics to say which sublattice is ...

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When you write down the formulas A15-A18, apply this assumption to these definitions and put it into wave functions, which are first calculated by solving BdG equation. you can reach the Beenakker results.

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