Induced fractional valley number in graphene with topological defects

We report on the possibility of valley number fractionalization in graphene with a topological defect that is accounted for in the Dirac equation by a pseudomagnetic field. The valley number fractionalization is attributable to an imbalance in the number of one-particle states in one of the two Dirac points with respect to the other. The difference in the number of one- particle states is manifest and can be exactly evaluated thanks to an external uniform magnetic field. Although the external magnetic field is precluded, the net valley number results in being dependent only on the flux of the pseudomagnetic field. We also discuss the analogous effect that the topological defect might lead to the induced spin polarization of the charge carriers in graphene.