Magnetoelectric effect in bilayer graphene controlled by valley-isospin density

We show that bilayer graphene exhibits magneto-electric effects that are formally similar to those commonly seen in band insulators with broken inversion and time-reversal symmetries. Three unusual features characterize the magneto-electric responses exhibited by bilayer graphene: (i) unlike most other magneto-electric media, bilayer graphene is a conductor, (ii) bilayer graphene has a non-quantized magneto-electric coupling even though its electronic structure does not break parity and time-reversal symmetry, and (iii) the magnitude of the magneto-electric coupling in bilayer graphene is determined by the valley-isospin density, which can be manipulated experimentally. This last property also enables a purely electric measurement of valley-isospin densities. While our theoretical arguments use bilayer graphene as an example, they are generally valid for any material with similar symmetries.