p}}^{-}-\varepsilon_{r{\bf p}}^{+})(\varepsilon_{r'{\bf p}}^{-}-\varepsilon_{r{\bf p}}^{+}+\hbar \omega + i\epsilon)} \nn \\
 & \times \frac{1}{E_{r}} \frac{\sin^2(jk_r)}{N+1} \times
 \frac{1}{E_{r'}} \frac{\sin^2(jk_{r'})}{N+1} \times
 p^2 \left(\frac{\varepsilon_{r{\bf p}}^{+}}{\varepsilon_{r'{\bf p}}^{-}} + \frac{\varepsilon_{r'{\bf p}}^{-}}{\varepsilon_{r{\bf p}}^{+}} \right).\end{aligned}$$ By taking the collisionless limit ($\epsilon \to 0$), we use $\left( \varepsilon_{r'{\bf p}}^{-}-\varepsilon_{r{\bf p}}^{+}+\hbar \omega + i\epsilon \right)^{-1} = -i\pi 
\delta \left( \varepsilon_{r'{\bf p}}^{-}-\varepsilon_{r{\bf p}}^{+}+\hbar \omega \right)$. Moreover, because the Dirac delta function may be rewritten as $$\begin{aligned}
 \delta \left( \frac{\varepsilon_{r'{\bf p}}^{-}-\varepsilon_{r{\bf p}}^{+}}{\partial p}|_{p=p_{rr'}} (p-p_{rr'}) \right),\end{aligned}$$ where $p_{rr'}$ satisfies $\varepsilon_{r'{\bf p}_{rr'}}^{-}-\varepsilon_{r{\bf p}_{rr'}}^{+}+\hbar \omega=0$, we obtain after integrating over $p$, $$\begin{aligned}
 \sigma^N_j(\omega) &= - g_s g_v\frac{e^2}{4\hbar}
 \sum_{r,r'} \frac{p_{rr'}^2}{\hbar \omega |E^2_{r'}(r) + E^1_{r}(r')|}
 \frac{\sin^2(jk_r)}{N+1}
 \frac{\sin^2(jk_{r'})}{N+1} \nn \\
 &\times
 \left( \frac{m_r + E^1_{r}(r')}{m_{r'} - E^2_{r'}(r)} + \frac{m_{r'} - E^2_{r'}(r)}{m_r + E^1_{r}(r')} \right)
 \Theta(E^1_{r}(r')-m_r)\Theta(E^2_{r'}(r)+m_{r'}),
 \label{eq:stepsigma}\end{aligned}$$ where $$\begin{aligned}
 & 
 E^1_{r}(r') \equiv E_r(p_{rr'}) = \frac{(\hbar \omega + m_{r'} - m_r )^2 - (m_{r'}-m_r)(m_{r'}+m_r)}{2(\hbar \omega + m_{r'} - m_r )},
 \\
 &
 E^2_{r'}(r) \equiv E_{r'}(p_{rr'}) = \frac{(\hbar \omega + m_{r'} - m_r )^2 + (m_{r'}-m_r)(m_{r'}+m_r)}{2(\hbar \omega + m_{r'} - m_r )}.\end{aligned}$$ The appearance of the step functions in Eq. (\[eq:stepsigma\]), satisfying $\Theta(x) = 1$ for $x \ge 0$ and 0 otherwise, needs explanations. They represent the conditions of the existence of $p_{rr'}$. Since $\varepsilon_{r{\bf p}}^{+} \ge 0$ for the conduction band, 