Effect of electron-electron interactions on the conductivity of clean graphene

Minimal conductivity of a single undoped graphene layer is known to be of the order of the conductance quantum, independent of the electron velocity. We show that this universality does not survive electron-electron interaction, which results in nontrivial frequency dependence. We begin with analyzing the perturbation theory in the interaction parameter g for the electron self-energy and observe the failure of the random-phase approximation. The optical conductivity is then derived from the quantum kinetic equation, and the exact result is obtained in the limit when g « 1 « g |nω|.