$$\textbf{k}=\frac{2\pi}{3a}\left(1,\frac{1}{\sqrt{3}}\right)$$$$\textbf{k}'=\frac{2\pi}{3a}\left(1,-\frac{1}{\sqrt{3}}\right)$$
$$V(\mathbf q,\omega)=\frac{e^2}{\epsilon_0}\frac{1}{|\mathbf \mathbf q|^2+k_{\mathrm{TF}}^2}\left(1+\frac{\omega_{\mathrm{ph}}^2}{\omega^2-\omega^2_{\mathrm{ph}}}\right)$$
$$\textbf{q}=\textbf{k}-\textbf{k}'=\frac{2\pi}{3a}(0,2\sqrt{3})$$
If we have macroscopic quantum phenomena, why not low-speed relativistic phenomena?!
In fact, the first example already exists: Graphene! which has
$$E=pv_F,$$
where \(v_F\) is the Fermi velocity, and \(p=\hbar k\). This suggests that \(v_F\) plays the role of \(c\)
$$\gamma_F=\frac{1}{\sqrt{1-(\frac{v}{v_F})^2}}.$$
In graphene, bands meet at special points. Is it possible to create a material whose bands meet at special lines?
