Quantum critical transport in clean graphene

We describe electrical transport in ideal single-layer graphene at zero applied bias. There is a crossover from collisionless transport at frequencies larger than kBT/! (T is the temperature) to collision-dominated transport at lower frequencies. The d.c. conductivity is computed by the solution of a quantum Boltzmann equation. Due to a logarithmic singularity in the collinear scattering amplitude (a consequence of relativistic dispersion in two dimensions) quasi-particles and -holes moving in the same direction tend to an effective equilibrium distribution whose parameters depend on the direction of motion. This property allows us to find the non-equilibrium distribution functions and the quantum critical conductivity exactly to leading order in 1/| log(α)| where α is the coupling constant characterizing the Coulomb interactions.