  We investigated the reduction of the spin stiffness and the appearance of the
spiral spin density waves when the electric field is applied on the zigzag
graphene nanoribbons for the ferromagnetic and antiferromagnetic edge states.
For that purpose, we exploited the generalized Bloch theorem combined with a
constraint method to keep the direction of the magnetic moment of the carbon
atom at the edges. We found that the ground state of ferromagnetic
configuration is unstable and leads to the spiral ground state while the ground
state of the antiferromagnetic configuration is robust. We also showed that the
spin stiffness in the ferromagnetic and antiferromagnetic configurations
reduces as the electric field increases. Thus, we justified that not only the
spin stiffness but also the ground state of the zigzag graphene nanoribbons can
be controlled by the electric field.
