(a) (b) Without Exchange Field  With Exchange Field  With Exchange Field and SOC FIG. 3: Evolution of the sub-bands structure upon increasing the exchange field. The solid (dashed) lines denote the sub-band which have even (odd) parity at Γ point, and the blue (red) color denotes the spin down (up) electrons. (a) The initial sub-bands are not inverted. When a exchange field is introduced, the spin down sub-bands (blue lines) are pushed down and the spin up sub-bands (red lines) are lifted up. When the exchange field is strong enough, a pair of inverted sub-bands appear (the red dash line and the blue solid line). (b) The initial sub-bands are already inverted at Γ point. The exchange field reverts the band inversion in one pair of sub-bands (the red solid line and the blue dashed line) and increase the band inversion in the other pair of sub-bands (the red dash line and the blue solid line). The SOC opens a gap at the cross points of spin-up and spin-down subbands as shown on the right most panel of (a) and (b), leading to the QAH state by the mechanism proposed in Ref. 12,15 17