Minimal conductivity of graphene: Nonuniversal values from the Kubo formula

The minimal conductivity of graphene is a quantity measured in the dc limit. It is shown, using the Kubo formula, that the actual value of the minimal conductivity is sensitive to the order in which certain limits are taken. If the dc limit is taken before the integration over energies is performed, the minimal conductivity of graphene is 4∕π (in units of e²∕h) and it is π∕2 in the reverse order. The value π is obtained if weak disorder is included via a small frequency-dependent self-energy. In the high-frequency limit, the minimal conductivity approaches π∕2 and drops to zero if the frequency exceeds the cutoff energy of the particles.