Intrinsic optical absorption in Dirac metals

A Dirac metal is a doped (gated) Dirac material with the Fermi energy ($E_\text{F}$) lying either in the conduction or valence bands. In the non-interacting picture, optical absorption in gapless Dirac metals occurs only if the frequency of incident photons ($\Omega$) exceeds the direct (Pauli) frequency threshold, equal to $2E_\text{F}$. In this work, we study, both analytically and numerically, the role of electron-electron ($ee$) and electron-hole ($eh$) interactions in optical absorption of two-dimensional (2D) and three-dimensional (3D) Dirac metals in the entire interval of frequencies below $2E_\text{F}$. We show that, for $\Omega\ll E_\text{F}$, the optical conductivity, $\Re\sigma(\Omega)$, arising from the combination of $ee$ and certain $eh$ scattering processes, scales as $\Omega^2\ln\Omega$ in 2D and as $\Omega^2$ in 3D, respectively, both for short-range (Hubbard) and long-range (screened Coulomb) interactions. Another type of $eh$ processes, similar to Auger-Meitner (AM) processes in atomic physics, starts to contribute for $\Omega$ above the direct threshold, equal to $E_\text{F}$. Similar to the case of doped semiconductors with parabolic bands studied in prior literature, the AM contribution to $\Re\sigma(\Omega)$ in Dirac metals is manifested by a threshold singularity, $\Re\sigma(\Omega)\propto (\Omega-E_\text{F})^{d+2}$, where $d$ is the spatial dimensionality and $0<\Omega-E_\text{F}\ll E_\text{F}$. In contrast to doped semiconductors, however, the AM contribution in Dirac metals is completely overshadowed by the $ee$ and other $eh$ contributions. Numerically, $\Re\sigma(\Omega)$ happens to be small in almost the entire range of $\Omega<2E_\text{F}$. This finding may have important consequences for collective modes in Dirac metals lying below $2E_\text{F}$.