half of $\pi \alpha \epsilon_0 c$. The cases $N=1$ and $2$ are shown in Fig. \[fig:sigma\](left panel). When $\gamma_1=0$, the mass vanishes and $\sigma^N_{j=1,\cdots,N} = \pi \alpha \epsilon_0 c$ for arbitrary $N$.

Bunched Conductivity
--------------------

When the electromagnetic fields are assumed to be sufficiently uniform in all layers, the dynamical conductivity of an $N$-layer graphene is well approximated by $$\begin{aligned}
 \sigma_{N}(\omega) = g_s g_v
 \frac{\hbar}{iV} \sum_{s,s'} \sum_{\bf p} \sum_{r,r'}
 \frac{f(\varepsilon_{r'{\bf p}}^{s'})-f(\varepsilon_{r{\bf p}}^{s})}{(\varepsilon_{r'{\bf p}}^{s'}-\varepsilon_{r{\bf p}}^{s})(\varepsilon_{r'{\bf p}}^{s'}-\varepsilon_{r{\bf p}}^{s}+\hbar \omega + i\epsilon)} 
 |\langle \Psi^{s}_{r{\bf p}} | (-e\hat{v}) | \Psi^{s'}_{r'{\bf p}} \rangle|^2.\end{aligned}$$ As regards this bunched conductivity, there is no concept of the local conductivity at a layer, as opposed to the layered conductivity. Rather, we treat an $N$-layer graphene as a whole.

In the following, we consider $x$-polarization only, because it can be shown that there is no polarization dependence of the dynamical conductivity. Using Eq. (\[eq:wf\]) and the following equalities $$\begin{aligned}
 & \sum_{j=1}^N \sin(jk_{r'}) \sin(jk_r) = \frac{N+1}{2} \delta_{r',r}, \\
 & \sum_{j=1}^N (-1)^j \sin(jk_{r'}) \sin(jk_r) = - \frac{N+1}{2} \delta_{r',N+1-r},\end{aligned}$$ we obtain absolute square of the matrix element of the current operator as $$\begin{aligned}
 |\langle \Psi^{s}_{r{\bf p}} | (-e\hat{v}^+) | \Psi^{s'}_{r'{\bf p}} \rangle|^2 = e^2
 \frac{\varepsilon_{r'{\bf p}}^{s'}}{2s'E_{r'}} \frac{\varepsilon_{r{\bf p}}^{s}}{2sE_r}
 \left|{\rm Re} \left( 
 \frac{\Pi}{\varepsilon_{r'{\bf p}}^{s'}} 