“mass” of Dirac fermions, $$\begin{aligned}
 m_r = \gamma_1 \cos(k_r).
 \label{eq:mass}\end{aligned}$$ The energy dispersion relation of the massive Dirac fermions becomes $$\begin{aligned}
 \varepsilon_{r{\bf p}}^{s} = m_r + s \sqrt{p^2 + m_r^2},\end{aligned}$$ where the positive and negative energy eigenstates are separated by the band index $s=\pm$; i.e., $\varepsilon_{r{\bf p}}^{+}\ge 0$ and $\varepsilon_{r{\bf p}}^{-}\le 0$. The mass appears as a (normal) mass that creates an energy bandgap $s\sqrt{p^2 + m_r^2}$ and also as a potential that shifts the band center by $m_r$. The “mass” shift can be positive and negative depending on the $k_r$ value. We assume a positive value of $\gamma_1$ in this paper, though the negative one is more plausible. No result is changed by this convention because the difference is removed by the replacement $k_r \to k_r + \pi$.

Dirac fermions acquire different kinds of masses, depending on the patterns of the symmetry breaking of the equivalence between two carbon atoms in the hexagonal unit cell (A and B atoms, known as pseudospin). By combining the concept of the “mass” with valley and spin degrees of freedoms, we can argue in a unified manner various aspects of physics from intriguing phenomena such as the quantum Hall effect [@haldane88] and quantum 