Low-speed Relativistic Phenomena

 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 \(c\)

$$\gamma_F=\frac{1}{\sqrt{1-(\frac{v}{v_F})^2}}.$$

For someone with QFT in mind, the next question about graphene would be it's magnetic properties, since that was one of the most important successes of the Dirac equation. 

To that purpose, one begins with $$\textbf{p}\to\textbf{p}-e\textbf{A}$$ which for graphene $$H=\pm v_F \textbf{p}\cdot\boldsymbol{\sigma},$$ becomes $$H=\pm v_F (\textbf{p}-e\textbf{A})\cdot\boldsymbol{\sigma}.$$

The new term $$\boxed{E=ev_F A}$$ tells us that, surprisingly, graphene responds to (external) magnetic vector potential, unlike the conventional para/diamagnetic materials that respond to an external magnetic field.