Dissipationless spin-valley current in zigzag-edge graphene ribbons with a net magnetization

We have investigated the spin-valley degree of freedom characterized by the product of spin and valley indices in zigzag-edge graphene ribbons with a net magnetization by using a tight-binding model. We found that the CT invariance leads to the spin-valley current which is neither spin polarized nor valley polarized but fully spin-valley polarized, where C and T are the charge conjugation and the time-reversal operation, respectively. In the quantum Hall regime, the gapless edge states protected by the CPT invariance carry the spin-valley degree of freedom leading to a dissipationless spin-valley current, where P is the inversion operation.