We investigate possible phase transitions among the different quantum anomalous Hall (QAH) phases in single-layer graphene 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 (antichiral) edge state ν = +2 (ν = −2) and the pseudoquantum 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 hyperchiral edge state of high Chern number ν = +4 or antihyperchiral 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.

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The classical dynamics for a charged spin particle is governed by the Lorentz force equation for orbital motion and by the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation for spin precession. In static and homogeneous electromagnetic fields, it has been shown that the Foldy-Wouthuysen (FW) transform of the Dirac-Pauli Hamiltonian, which describes the relativistic quantum theory for a spin-1/2 particle, is consistent with the classical Hamiltonian (with both the orbital and spin parts) up to the order of 1/m 14 (m is the particle's mass) in the low-energy/weak-field limit. In this paper, we extend this correspondence to the case of inhomogeneous fields. Regardless of the field gradient (e.g., Stern-Gerlach) force, the T-BMT equation is unaltered and thus the classical Hamiltonian remains the same, but subtleties arise and need to be clarified. For the relativistic quantum theory, we apply Eriksen's method to obtain the exact FW transformations for the two special cases, which in conjunction strongly suggest that, in the weak-field limit, the FW transformed Dirac-Pauli Hamiltonian (except for the Darwin term) is in agreement with the classical Hamiltonian in a manner that classical variables correspond to quantum operators via a specific Weyl ordering. Meanwhile, the Darwin term is shown to have no classical correspondence.