Interaction-disorder-driven characteristic momentum in graphene, approach of multi-body distribution functions

Multi-point probability measures along with the dielectric function of Dirac Fermions in mono-layer graphene containing particle-particle and white-noise (out-plane) disorder interactions on an equal footing in the Thomas-Fermi-Dirac approximation is investigated. By calculating the one-body carrier density probability measure of the graphene sheet, we show that the density fluctuation (ζ^-1) is related to the disorder strength (n_i), the interaction parameter (r_s) and the average density ([Formula: see text]) via the relation [Formula: see text] for which [Formula: see text] leads to strong density inhomogeneities, i.e. electron-hole puddles (EHPs), in agreement with the previous works. The general equation governing the two-body distribution probability is obtained and analyzed. We present the analytical solution for some limits which is used for calculating density-density response function. We show that the resulting function shows power-law behaviors in terms of ζ with fractional exponents which are reported. The disorder-averaged polarization operator is shown to be a decreasing function of momentum like ordinary 2D parabolic band systems. It is seen that a disorder-driven momentum q_ch emerges in the system which controls the behaviors of the screened potential. We show that in small densities an instability occurs in which imaginary part of the dielectric function becomes negative and the screened potential changes sign. Corresponding to this instability, some oscillations in charge density along with a screening-anti-screening transition are observed. These effects become dominant in very low densities, strong disorders and strong interactions, the state in which EHPs appear. The total charge probability measure is another quantity which has been investigated in this paper. The resulting equation is analytically solved for large carrier densities, which admits the calculation of arbitrary-point correlation function.