Coulomb interaction at the metal-insulator critical point in graphene

We compute the renormalization-group flow of the long-ranged electron-electron interaction at the Gross-Neveu quantum critical point between the semimetal and the excitonic insulator in graphene, perturbatively in the small parameter $ϵ=d-1$, with $d$ as the spatial dimension. The $O\left(ϵ\right)$ correction to the usual beta function makes the long-range interaction only more irrelevant at the critical than at the Gaussian fixed point. A weak long-range tail of the Coulomb interaction is found to be marginally irrelevant also in arbitrary dimension when the number of Dirac fermions is large. Its ultimate irrelevancy notwithstanding, it is shown that the metal-insulator transition may still be induced by increasing only the long-range tail of the Coulomb interaction.

Figure 1

Schematic renormalization-group flow in the infrared for the metal-insulator transition in graphene. $\widehat{g}$ is the (dimensionless) short-range component and ${\widehat{e}}^2$ is the long-range component of the Coulomb electron-electron interaction. GN and G are the critical Gross-Neveu and the Gaussian fixed points, respectively.

Figure 2

One- and two-loop diagrams that renormalize the Fermi velocity. The wavy line is the long-range interaction and the dashed line is the contact fixed-point interaction.

Figure 3

The remaining three two-loop diagrams that do not contribute to the Fermi velocity renormalization.

Figure 4

Renormalization of the Gross-Neveu coupling due to the long-range Coulomb interaction.