Third harmonic generation of undoped graphene in Hartree-Fock approximation

We theoretically investigate the effects of Coulomb interaction, at the level of unscreened Hartree-Fock approximation, on third harmonic generation of undoped graphene in an equation of motion framework. The unperturbed electronic states are described by a widely used two-band tight binding model, and the Coulomb interaction is described by the Ohno potential. The ground state is renormalized by taking into account the Hartree-Fock term, and the optical conductivities are obtained by numerically solving the equations of motion. The absolute values of conductivity for third harmonic generation depend on the photon frequency $\Omega$ as $\Omega^{-n}$ for $\hbar\Omega<1$, and then show a peak as $3\hbar\Omega$ approaches the renormalized energy of the $M$ point. Taking into account the Coulomb interaction, $n$ is found to be $5.5$, which is significantly greater than the value of $4$ found with the neglect of the Coulomb interaction. Therefore the Coulomb interaction enhances third harmonic generation at low photon energies -- for our parameters $\hbar\Omega<0.8$~eV -- and then reduces it until the photon energy reaches about $2.1$~eV. The effect of the background dielectric constant is also considered.