Robustness of the optical conductivity sum rule in bilayer graphene

We calculate the optical sum associated with the in-plane conductivity of a graphene bilayer. A bilayer asymmetry gap generated in a field-effect device can split apart valence and conduction bands, which otherwise would meet at two K points in the Brillouin zone. In this way, one can go from a compensated semimetal to a semiconductor with a tunable gap. However, the sum rule turns out to be ``protected'' against the opening of this semiconducting gap, in contrast to the large variations observed in other systems where the gap is induced by strong correlation effects.