= Dirac valleys in graphene
{c}
{title2=$\mathbf K_\tau=(\tau4\pi/(3\sqrt3b),0),\quad\tau=\pm1$}
At either displayed <wave vector>, the three complex hopping amplitudes add to zero: $1+2\cos(2\pi/3)=0$. Therefore the <energy bands> touch. There are two inequivalent <electronic valleys>; other equivalent corners are related by reciprocal-lattice translations. Setting $\mathbf k=\mathbf K_\tau+\mathbf q$ and retaining first-order terms gives
$$
H_\tau(\mathbf q)\simeq\hbar v_F(\tau q_x\sigma_x+q_y\sigma_y),\qquad
v_F=\frac{3tb}{2\hbar},\qquad E_\pm\simeq\pm\hbar v_F|\mathbf q|,
$$
where $\sigma_x,\sigma_y$ are <Pauli matrices>. For example, at $\mathbf K_+$, $f\simeq-3b(q_x-iq_y)/2$. The matrix has the form of a massless two-dimensional <Dirac Hamiltonian>. Here $v_F$ is a material-model <velocity>, rather than the vacuum <speed of light>; this description does not change an <Electron>'s vacuum <rest mass>. It describes the <dispersion relation> of electronic excitations near a band crossing.
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