High Chern number quantum anomalous Hall phases in single-layer graphene with Haldane orbital coupling

Center for Condensed Matter Sciences, National Taiwan University, Taipei 106, Taiwan(Dated: March 22, 2011)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 Rashbaspin-orbit couplings as well as the Haldane orbital term. We find that the variation of the exchangefield results in bulk gap-closing phenomena and phase transitions occur in the graphene system. Ifthe Haldane orbital coupling is absent, the phase transition between the chiral (anti-chiral) edgestate 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 twoadditional edge modes appears in between phases ν = +2 and ν = −2. This intermediate phase istherefore either the hyper-chiral edge state of high Chern number ν = +4 or anti-hyper-chiral edgestate 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. Wealso report the critical exchange field values for the QAH phase transitions.