Valley-isospin-density-controlled magneto-electric effect in bilayer graphene

We show that bilayer graphene (BLG) exhibits magneto-electric effects that are formally similar to those usually seen in band insulators with broken inversion and time-reversal symmetries. Three unusual features characterize the magneto-electric responses exhibited by BLG: (i) unlike most other magneto-electric media, BLG is a conductor, (ii) BLG has a non-quantized finite magneto-electric coupling even though it does not break parity and time-reversal symmetry, and (iii) the magnitude of the magneto-electric coupling in BLG is determined by an electronic degree of freedom - the valley-isospin density - and, therefore, can be manipulated experimentally. This last property also enables a purely electric measurement of valley-isospin densities.