Interplay of Coulomb and electron-phonon interactions in graphene

We consider the mutual effect of the electron-phonon and strong Coulomb interactions on each other by summing up leading logarithmic corrections via the renormalization group approach. We find that the Coulomb interaction enhances electron coupling to the intervalley $A_1$ optical phonons, but not to the intravalley $E_2$ phonons.

Figure 1

Phonon displacements for $E_2$, $A_1$, and $B_1$ modes.

Figure 2

Analytical expressions of the diagrammatic technique.

Figure 3

Electron self-energy due to the screened Coulomb interaction and the EPC.

Figure 4

Logarithmic corrections to the EPC vertex $F_{\mu }$ of the order $O(1∕\mathcal{N},{\lambda }_{\mu }^2)$. Diagrams (c)–(f) vanish.

Figure 5

Logarithmic diagrams of the order $O\left({\lambda }_{\mu }^2\right)$ not reduced to a renormalization of the EPC vertex $F_{\mu }$.

Figure 6

Flow of the dimensionless coupling constants ${\lambda }_{A_1}$ (three upper curves, starting from the bare value 0.04 at $10\mathrm{eV}$) for three values of the bare Coulomb coupling $g\left(0\right)=3.4$, 1.5, and 0.5 (solid, dashed, and dotted curves). The constant ${\lambda }_{E_2}=0.035$ is unchanged in the order $O\left(\lambda \right)$.