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The single-band current-dipole Kubo formula for the dynamical conductivity of heavily doped graphene from Kup\v{c}i\'{c} [Phys. Rev. B 91, 205428 (2015)] is extended to a two-band model for conduction $\pi$ electrons in lightly doped graphene. Using a posteriori relaxation-time approximation in the two-band quantum transport equations, with two different relaxation rates and one quasi-particle lifetime, we explain a seemingly inconsistent dependence of the dc conductivity $\sigma^{\rm dc}_{\alpha \alpha}$ of ultraclean and dirty lightly doped graphene samples on electron doping, in a way consistent with the charge continuity equation. It is also shown that the intraband contribution to the effective number of conduction electrons in $\sigma^{\rm dc}_{\alpha \alpha}$ vanishes at $T=0$ K in the ultraclean regime, but it remains finite in the dirty regime. The present model is shown to be consistent with a picture in which the intraband and interband contributions to $\sigma^{\rm dc}_{\alpha \alpha}$ are characterized by two different mobilities of conduction electrons, the values of which are well below the widely accepted value of mobility in ultraclean graphene. The dispersions of Dirac and $\pi$ plasmon resonances are reexamined to show that the present, relatively simple expression for the dynamical conductivity tensor can be used to study simultaneously single-particle excitations in the dc and optical conductivity and collective excitations in energy loss spectroscopy experiments., Comment: 13 pages, 11 figures

The frequency and wave-vector-dependent memory function in the longitudinal conductivity tensor of weakly interacting electronic systems is calculated by using the approach based on the quantum transport equations. It is shown, for the first time consistently, that there is a close relation between the single-electron self-energy, the electron-hole pair self-energy, and the memory function. It is also shown in which way singular long-range Coulomb interactions, together with other q approx 0 scattering processes, drop out of both the memory function and the related transport equations. The theory is illustrated on heavily doped graphene, which is the prototype of weakly interacting single-band electron-phonon systems. A steplike increase of the width of the quasiparticle peak in angle-resolved photoemission spectra at frequencies of the order of the frequency of in-plain optical phonons is shown to be consistent with similar behavior of the intraband plasmon peak in energy loss spectroscopy spectra. Both anomalies can be understood as a direct consequence of weak electron scattering from in-plane optical phonons.

A first-principle analysis is presented of the effect on transport phenomena of the relaxation of the semiclassical assumptions of energy conservation and point-like nature in space/time of the scattering processes. Quantitative estimates for the quantum case have been obtained within the Wigner-function approach with the use of the concept of Wigner paths. This formulation of the transport problem, although rigorous from the point of view of quantum mechanics, is very close to the classical language, so that comparisons are very straightforward. Results of the analysis of phonon scattering show that multiple collisions reduce collisional broadening and contribute to understanding the success of the semiclassical approximation. An analogous formulation in terms of the density matrix confirms the obtained results.

A systematic method of calculating the dynamical conductivity tensor in a general multiband electronic model with strong boson-mediated electron-electron interactions is described. The theory is based on the exact semiclassical expression for the coupling between valence electrons and electromagnetic fields and on the self-consistent Bethe--Salpeter equations for the electron-hole propagators. The general diagrammatic perturbation expressions for the intraband and interband single-particle conductivity are determined. The relations between the intraband Bethe--Salpeter equation, the quantum transport equation and the ordinary transport equation are briefly discussed within the memory-function approximation. The effects of the Lorentz dipole-dipole interactions on the dynamical conductivity of low-dimensional $sp_\alpha$ models are described in the same approximation. Such formalism proves useful in studies of different (pseudo)gapped states of quasi-one-dimensional systems with the metal-to-insulator phase transitions and can be easily extended to underdoped two-dimensional high-$T_c$ superconductors., Comment: 13 pages, 10 figures