+ \frac{\Pi^*}{\varepsilon_{r{\bf p}}^{s}} \right) \delta_{r',r}
 + i {\rm Im} \left( 
 \frac{\Pi}{\varepsilon_{r'{\bf p}}^{s'}} + \frac{\Pi^*}{\varepsilon_{r{\bf p}}^{s}} \right)
 \delta_{r',N+1-r}
 \right|^2.
 \label{eq:mat2}\end{aligned}$$ We focus on the interband transitions ($s=+1$ and $s'=-1$) with the assumption that $E_F=0$ and $T=0$. Two different interband transitions contribute to $\sigma_{N}(\omega)$. One originates from ${\rm Re}(\cdots)$ in Eq. (\[eq:mat2\]) which is associated with the direct transitions that preserve the wavenumber $(r'=r)$. The other originates from ${\rm Im}(\cdots)$ which is associated with the indirect transitions that satisfy $r'=N+1-r$ (i.e., $k_r'+k_r=\pi$) and the indirect interband transitions are associated with the change in the sign of the mass; $m_{N+1-r} = - m_r$. \[When $N$ is an odd number, the indirect transitions contain a direct transition as a special case of $r'=r$, for which the mass vanishes.\] The former gives $$\begin{aligned}
 \sigma_{N}^{\rm direct}(\omega) 
 &=g_s g_v
 \frac{e^2\hbar}{V} \sum_{\bf p} \sum_{r}
 \frac{\pi}{2E_r}
 \delta (\hbar \omega - 2 E_r)
 \frac{\varepsilon_{r{\bf p}}^{-}}{2E_r}
 \frac{\varepsilon_{r{\bf p}}^{+}}{2E_r}
 (p_x)^2 \left( 
 \frac{1}{m_r -E_r} + \frac{1}{m_r + E_r} \right)^2 \nn \\
 &=g_s g_v
 \frac{S}{V}\frac{e^2}{16\hbar} \sum_{r}
 \left( \frac{2m_r}{\hbar \omega} \right)^2 \Theta (\hbar \omega - 2 |m_r|).
 \label{eq:direct}\end{aligned}$$ The step function represents the fact that direct interband transitions exist for the limited photon energy above the bandgap, $\hbar \omega > 2 |m_r|$. Meanwhile the latter 