CT-Invariant Quantum Spin Hall Effect in Ferromagnetic Graphene

We predict a quantum spin Hall effect (QSHE) in ferromagnetic graphene under a magnetic field. Unlike the previous QSHE, this QSHE appears in the absence of spin-orbit interaction and thus, is arrived at from a different physical origin. The previous QSHE is protected by the time-reversal (T) invariance. This new QSHE is protected by CT invariance, where C is the charge conjugation operation. Because of this QSHE, the longitudinal resistance exhibits quantum plateaus. The plateau values are at 1/2, 1/6, 3/28, …, (in units of h/e²), depending on the filling factors of the spin-up and spin-down carriers. The spin Hall resistance is also investigated and is found to be robust against the disorder.