  In the presence of the Hubbard interaction, graphene zigzag nanoribbons have
spontaneous edge magnetism with anti-parallel configuration, whose amplitude
can be tuned by a transversal electric field. As the electric field increases
or decreases across a critical value, the edges are demagnetized or
re-magnetized, respectively. A magnetic field at each edge determines the
orientation of the re-magnetization. Thus, a combination of slowly varying
transversal electric field and magnetic field in monolayer graphene zigzag
nanoribbon could drive the quantum system into a bistability loop. The same
phenomenon can be induced in a bilayer/monolayer zigzag nanoribbon without the
magnetic field, because the non-symmetry superexchange interaction controls the
orientation of the re-magnetization. By this way, the quantum system is
switched between ground state and quasi-stable excited state with different
magnetism, band structures and conductance. This feature could be used to
develop graphene-based spintronic nano-devices without magnetic field.
