superimposed to make the wavefunction vanish at every two layers, there is a state that does not feel the interlayer interaction. This degeneracy is possible when $N$ is an odd number, and such a state with the original massless linear dispersion exists.

Because $\sum_{j=1}^N \sin^2(j k_r) = \frac{N+1}{2}$ holds, the normalization constant $N_{r{\bf p}}^s$ in Eq. (\[eq:wf\]) is given by $$\begin{aligned}
 (N_{r{\bf p}}^{s})^2 
 = \frac{(\varepsilon_{r{\bf p}}^{s})^2}{(\varepsilon_{r{\bf p}}^{s})^2+p^2} \frac{2}{N+1} 
 = \frac{\varepsilon_{r{\bf p}}^{s}}{sE_r} \frac{1}{N+1}.\end{aligned}$$ In the last equation, we have introduced the energy of a (normal) massive Dirac fermions $E_r \equiv \sqrt{p^2 + m_r^2}$, which will be used frequently in the subsequent calculations.

Conductivity calculation {#sec:for}
========================

Layered Conductivity
--------------------

We define the layered dynamical conductivity of the $j$th layer in an $N$-layer graphene, within a linear response theory, as $$\begin{aligned}
 \sigma_j^N(\omega) = g_s g_v \frac{\hbar}{iS} \sum_{s,s'} \sum_{\bf p} \sum_{r,r'}
 \frac{f(\varepsilon_{r'{\bf p}}^{s'})-f(\varepsilon_{r{\bf p}}^{s})}{(\varepsilon_{r'{\bf p}}^{s'}-\varepsilon_{r{\bf p}}^{s})(\varepsilon_{r'{\bf p}}^{s'}-\varepsilon_{r{\bf p}}^{s}+\hbar \omega + i\epsilon)}
|\langle \Psi_{r{\bf p}}^{s}| (-e\hat{v}_j) |\Psi_{r'{\bf p}}^{s'}\rangle|^2.
 \label{eq:layeredsigma}\end{aligned}$$ Here, $g_s(=2)$ and $g_v(=2)$ represents spin and valley degeneracy, respectively, and $S$ the area of the graphene layer. We take the limit $S\to \infty$ in the following calculations. Multiplying this conductivity with the electric field at the $j$th layer gives the local current in an 