= Solution
At a valley $K$, $\gamma(K)=1+2\cos(2\pi/3)=0$. For <hexagonal boron nitride>[h-BN] the inequivalent sublattice energies therefore leave the direct two-<band gap>
$$
\boxed{E_g=|\epsilon_B-\epsilon_N|.}
$$
<Graphene>'s equivalent carbon sites instead have $\epsilon_B=\epsilon_N$ in this model, so the two bands touch at the valleys. Near a valley, $|\gamma(K+\mathbf q)|\simeq(\sqrt3a/2)|\mathbf q|$: <graphene> has approximately linear, gapless <Dirac cones> while h-BN has gapped band edges. The model panel in the original figure in <past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2018/1/c/ii/solution> displays this distinction.
First, undoped h-BN is an <electrical insulator> with thermally suppressed carriers, whereas neutral <graphene> is a zero-gap semimetal whose transport is readily changed by gating. Disorder, <temperature> and contacts still affect its measured conductivity. Second, h-BN suppresses interband absorption below its large gap and is transparent over much of the visible range, whereas <graphene> permits low-<energy> interband absorption unless occupations block it. A quantitative optical threshold in real h-BN also involves additional bands and strong excitonic effects; a two-orbital model should not be presented as a numerical prediction of its complete optical spectrum.
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