of the top and bottom surface states, and ↓↑ indicating the spin, the low energy effective model for the thin film can be written in the form of H$_{2}$$_{D}$ as in Eq. 3, with h ( k ) = Δ$_{k}$ σ$_{z}$ + v$_{F}$ ( k$_{x}$σ$_{y}$ − k$_{y}$σ$_{x}$ ), Δ$_{k}$ = Δ$_{0}$ + Bk 2 $_{‖}$, similar to the Bernevig-Hughes-Zhang model describing the low energy physics in HgTe/CdTe quantum well $^{10}$. When Δ$_{0}$ B < 0, band inversion occurs, and the system will be in the QSH phase if this is the only band inversion between two subbands with opposite parity, which is exactly the situation in HgTe/CdTe quantum well system. The situation for Bi$_{2}$Se$_{3}$ film is somewhat different in the sense that more pairs of subbands can get inverted at the Γ point, and the system may still remain in the topologically trivial state if the band inversion occurs even number of times $^{19}$. Nevertheless, as we shall show below, regardless of whether the 4-bands system is originally in the topologically non-trivial phase or not, a strong enough exchange field will induce QAH effect in this system. The exchange coupling term induced by the finite magnetization in the FM phase can be written as, with M denoting the strength of the exchange field and g$_{eff}$ being the effective g-factor of the surface states. Added to the 2D model (3), the exchange field increases the mass term of the upper block and reduces it for the lower block, breaking the time reversal symmetry. More importantly, a sufficiently large exchange field can change the Chern number of one of the two blocks. As schematically illustrated in Fig.3, if the 4-band system is originally in the topologically trivial phase, the exchange field will induce a band inversion in the upper block and push the two sub-bands in the lower block even farther away from each other. Therefore the 2D model with a negative mass in the upper block contributes e $^{2}$/h for the Hall conductance. On the other hand, if the system is originally in the topologically non-trivial phase, both blocks have inverted band structures. In this case, a sufficiently large exchange field can increase the band inversion in the upper block and revert the band inversion in the lower block. Again the negative mass in the upper block contributes e $^{2}$/h for the Hall conductance. These two scenarios are illustrated in Fig.3. To be realistic, we carried out quantitative calculations for Bi$_{2}$Se$_{3}$ films, based on the density functional theory and linear response formalism $^{33}$. A spacial uniform exchange field is included to take into account the effect of magnetization in the FM state at the mean 10 H$_{exchange}$ =   $^{g$_{eff}$}$$^{Mσ$_{z}$}$ 0 0 − g$_{eff}$Mσ$_{z}$    (6)