Interacting Electrons in Graphene: Fermi Velocity Renormalization and Optical Response

We have developed a Hartree-Fock theory for electrons on a honeycomb lattice aiming to solve a long-standing problem of the Fermi velocity renormalization in graphene. Our model employs no fitting parameters (like an unknown band cutoff) but relies on a topological invariant (crystal structure function) that makes the Hartree-Fock sublattice spinor independent of the electron-electron interaction. Agreement with the experimental data is obtained assuming static self-screening including local field effects. As an application of the model, we derive an explicit expression for the optical conductivity and discuss the renormalization of the Drude weight. The optical conductivity is also obtained via precise quantum Monte Carlo calculations which compares well to our mean-field approach.

Figure 1

Band structure (undoped) along high symmetry directions for various fine-structure constants $\alpha$ and self-screening (solid lines). For suspended graphene ($\alpha =2.2$, $t=3.1\mathrm{eV}$), also the unscreened dispersion is shown (dashed line). Inset: Enlargement of the dispersion around the Dirac cone.

Figure 2

The renormalized Fermi velocity $v_F^{*}$ for suspended ($\epsilon =1$ and $t=3.1\mathrm{eV}$) and hBN-encapsulated ($\epsilon =4.9$ and $t=2.6\mathrm{eV}$) graphene. Left-hand side for suspended graphene: The experimental data of Ref. [4] compared to $v_F^{*}$ at the Fermi surface based on the bare and self-consistent self-screened Coulomb interaction. Right-hand side for hBN-encapsulated graphene: The experimental data of Ref. [8] compared to $v_F^{*}$ at the Fermi surface in $K\mathrm{\Gamma }$ (black squares) and $KM$ (blue stars) directions based on the bare self-screened Coulomb interaction. In both cases, the result for $v_F^{*}$ at the neutrality point as function of the electronic density $n$ is also shown based on the unscreened interaction with $\alpha =\alpha /{\epsilon }^{\mathrm{RPA}}$.

Figure 3

Left: The correction to the optical conductivity $C(\alpha )$ of Eq. (5) compared to the result expected from the RPA, i.e., $[4{\epsilon }^{\mathrm{RPA}}{]}^{-1}=\{4[1+(\pi /2)\alpha ]{\}}^{-1}$. Right: The same for the conductivity of Eq. (5) with $v_F^{*}/v_F=1$ compared to the conductivity at $\hslash \omega =0.7t$ obtained from Hartree-Fock (blue triangle) and quantum Monte Carlo (red circle) calculations.