arise. For example, a correction to $\gamma_1$ of the form $\delta \gamma_1 \cos(k_r)$ or $m_r \to \gamma_1 \cos(k_r) + \delta \gamma_1 \cos^2(k_r)$ in Eq. (\[eq:bunchedc\]) can be used for phenomenological explanations of the double peaks. An asymmetry between $m_1$ and $m_N$ is also indicative of a broken particle-hole symmetry of the band structure.

![(Color Online) (left) Comparison between calculated reflectance ($N=1500$) and the experiment on HOPG. (right) The peak structure that originates from interlayer interaction appears at $2\gamma_1=0.8$ eV irrespective of the calculation methods. For $\hbar \omega < 2\gamma_1$, the layered conductivity underestimates reflectance compared with the bunched conductivity. []{data-label="fig:Ref"}](69750Fig3.eps)

Nonlocal correction
-------------------

The underestimation of reflectance seen for the layered conductivity in Fig. \[fig:Ref\] at low photon energies below the peak may be related to a nonlocal effect. We define the nonlocal conductivity $\sigma^N_{jj'}$ as the corrections to the current of $j$th layer that are caused by the electric fields of the other $j'$th layers ($j' \ne j$) as $$\begin{aligned}
 J_j = \sigma^N_{j} E_{j}+ \sum_{j'\ne j} \sigma^N_{jj'} E_{j'}.\end{aligned}$$ The existence of a nonlocal conductivity is physically plausible because an electron-hole pair locally excited by an electric field at a layer may recombine at a different layer. Since such a 