Quantum anomalous Hall effect on a square lattice with spin–orbit couplings and an exchange field

We investigate a tight-binding model on a two-dimensional square lattice with three terms: the Rashba spin–orbit coupling, the real amplitude next-nearest spin–orbit coupling, and an exchange field. We calculate the first Chern number to identify band topology. It is found that the Chern number takes the quantized values of C 1 = 1, 2 and the chiral edge modes can be obtained. Therefore our model realizes the quantum anomalous Hall (QAH) effect. The Rashba coupling is positive for the QAH phase while the next-nearest coupling is detrimental to it. By increasing the exchange field intensity, the Chern number changes from quantized value 2 to 0. The behavior of the edge states is also studied. Particularly for C 1 = 2 case, there are two gapless spin-polarized edge states with the same spin polarization moving in the same spatial direction. This indicates that their appearance is topological rather than accidental.