  We present a general microscopic first-principles method to study the
Coulomb-interaction-mediated heat transfer in the near field. Using the
nonequilibrium Green's function formalism, we derive Caroli formulas for heat
transfers between materials with translational invariance. The central physical
quantities are the screened Coulomb potential and the spectrum function of
polarizability. Within the random phase approximation, we calculate the
polarizability using the linear response density functional theory and obtain
the screened Coulomb potential from a retarded Dyson equation. We show that the
heat transfer mediated by the Coulomb interaction is consistent with that of
the $p$-polarized evanescent waves which dominate the heat transfer in the near
field. We adopt single-layer graphene as an example to calculate heat transfers
between two parallel sheets separated by a vacuum gap $d$. Our results show a
saturation of heat flux at the extreme near field which is different from the
reported $1/d$ dependence for local response functions. The calculated heat
flux is up to $5\times10^4$ times more than the black-body limit, and a $1/d^2$
dependence is shown at large separations. From the spectrum of energy current
density, we infer that the near-field enhancement of heat transfer stems from
electron transitions around the Fermi energy. With a uniform strain, the heat
flux increases for most of the distances while a negative correlation is shown
at the moderate field. Our method is valid for inhomogeneous materials in which
the macroscopic response function used in conventional theory of fluctuational
electrodynamics would fail at the subnanometer scale.
