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

摘要

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