The paper by E. H. Hwang and S. Das Sarma investigates the dynamical dielectric function of two-dimensional (2D) graphene at arbitrary wave vector \( q \) and frequency \( \omega \), denoted as \( \epsilon(q, \omega) \), using the self-consistent field approximation. The results are used to determine the plasmon mode dispersion and the electrostatic screening of the Coulomb interaction in 2D graphene within the random phase approximation (RPA). Key findings include: 1. **Plasmon Mode Dispersion**: At long wavelengths (\( q \to 0 \)), the plasmon dispersion shows a local classical behavior \( \omega_{cl} = \omega_0 \sqrt{q} \), but the density dependence of the plasma frequency \( \omega_0 \propto n^{1/4} \) differs from the usual 2D electron system (\( \omega_0 \propto n^{1/2} \)). 2. **Static Screening**: The static screening function shows significant differences from the usual 2D case. The intrinsic interband contributions to static screening can be effectively absorbed into a background dielectric constant. 3. **Polarizability**: The polarizability \( \Pi(q, \omega) \) is calculated for both intrinsic and doped graphene, with detailed expressions provided for the real and imaginary parts. 4. **Plasmon Behavior**: For single-layer and bilayer graphene, the plasmon mode dispersion is derived, showing that the long-wavelength plasmons have the same dispersion as normal 2D plasmons but with different density dependences. 5. **Static Screening at Large Wave Vectors**: The static screening behavior at large wave vectors is different from that of normal 2D systems, with a linear increase in the static polarizability with \( q \). 6. **Effective Dielectric Constant**: The intrinsic screening contribution from interband transitions can be absorbed into an effective background dielectric constant \( \kappa^* \), enhancing the effective background dielectric constant to \( \kappa \kappa^* \). The paper concludes by discussing the implications of these findings for experimental verification and the potential applications in understanding transport properties and collective modes in 2D graphene.The paper by E. H. Hwang and S. Das Sarma investigates the dynamical dielectric function of two-dimensional (2D) graphene at arbitrary wave vector \( q \) and frequency \( \omega \), denoted as \( \epsilon(q, \omega) \), using the self-consistent field approximation. The results are used to determine the plasmon mode dispersion and the electrostatic screening of the Coulomb interaction in 2D graphene within the random phase approximation (RPA). Key findings include: 1. **Plasmon Mode Dispersion**: At long wavelengths (\( q \to 0 \)), the plasmon dispersion shows a local classical behavior \( \omega_{cl} = \omega_0 \sqrt{q} \), but the density dependence of the plasma frequency \( \omega_0 \propto n^{1/4} \) differs from the usual 2D electron system (\( \omega_0 \propto n^{1/2} \)). 2. **Static Screening**: The static screening function shows significant differences from the usual 2D case. The intrinsic interband contributions to static screening can be effectively absorbed into a background dielectric constant. 3. **Polarizability**: The polarizability \( \Pi(q, \omega) \) is calculated for both intrinsic and doped graphene, with detailed expressions provided for the real and imaginary parts. 4. **Plasmon Behavior**: For single-layer and bilayer graphene, the plasmon mode dispersion is derived, showing that the long-wavelength plasmons have the same dispersion as normal 2D plasmons but with different density dependences. 5. **Static Screening at Large Wave Vectors**: The static screening behavior at large wave vectors is different from that of normal 2D systems, with a linear increase in the static polarizability with \( q \). 6. **Effective Dielectric Constant**: The intrinsic screening contribution from interband transitions can be absorbed into an effective background dielectric constant \( \kappa^* \), enhancing the effective background dielectric constant to \( \kappa \kappa^* \). The paper concludes by discussing the implications of these findings for experimental verification and the potential applications in understanding transport properties and collective modes in 2D graphene.