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High Chern number quantum anomalous Hall phases in graphene ribbons with Haldane orbital coupling

We investigate possible phase transitions among the different quantum anomalous Hall (QAH) phases in a zigzag graphene ribbon under the influence of the exchange field. The effective tight-binding Hamiltonian for graphene is made up of the hopping term, the Kane-Mele and Rashba spin-orbit couplings as well as the Haldane orbital term. We find that the variation of the exchange field results in bulk gap-closing phenomena and phase transitions occur in the graphene system. If the Haldane orbital coupling is absent, the phase transition between the chiral (anti-chiral) edge state $ν=+2$ ($ν=-2$) and the pseudo-quantum spin Hall state ($ν=0$) takes place. Surprisingly, when the Haldane orbital coupling is taken into account, an intermediate QSH phase with two additional edge modes appears in between phases $ν=+2$ and $ν=-2$. This intermediate phase is therefore either the hyper-chiral edge state of high Chern number $ν=+4$ or anti-hyper-chiral edge state of $ν=-4$ when the direction of exchange field is reversed. We present the band structures, edge state wave functions and current distributions of the different QAH phases in the system. We also report the critical exchange field values for the QAH phase transitions.