Usually the surface states are located dominantly near the top and bottom surfaces. The signal intensity for these two branches of energy spectra of ARPES are dif- ferent. The SIA will cause the coupling between two surface states near their crossing point. That is why the Rashba-like splitting of the ARPES spectra has a bright crossing point near the Γ point, with one branch bright and the other almost invisible. Thus the SIA term can be used to describe the ARPES measurements on the thin film Bi$_{2}$Se$_{3}$ very well. Our effective model demonstrates that the 3DTI can be reduced to an two-dimensional quantum spin Hall sys- tem due to the spatial confinement. Strictly speaking, the system is no longer a 3DTI in the original sense once the energy gap opens in the surface bands, since the Z2 invariant for the bulk states becomes zero. However the surface bands themselves may contribute a non-trivial one in the Z2 invariant even when the SIA term is in- cluded. Our calculation demonstrates that a strong SIA always intends to destroy the quantum spin Hall effect. A critical value for SIA exists, at the point there is a transition from a topological trivial to non-trivial phases. Based on the model parameters fitted from the experi- mental data of ARPES, we conclude that the thin film Bi$_{2}$Se$_{3}$ should exhibit quantum spin Hall effect once the energy gap opens in the surface spectra due to the spatial confinement of the thin film. We thank Ke He and Qi-Kun Xue for providing ex- perimental data prior to publication, and Qian Niu for helpful discussions. This work was supported by the Re- search Grant Council of Hong Kong under Grant No. HKU 7037/08P and HKU 10/CRF/08. For arbitrary energy, Eq. (5) requires that the values of λ$_{1}$ and λ$_{2}$ can only be one of the combinations shown in Tab. III. By putting these combinations into Eq. (34)- (37), one can show that for the first entry, η$_{1}$ and η$_{2}$ are real while f$_{+}$ and f$_{−}$ are pure imaginary, referred as the case A; while for entries 2 - 4, all of η$_{1}$ , η$_{2}$ , f$_{+}$ , and f$_{−}$ are real, referred as the case B. For an arbitrary group of E$_{+}$ and E$_{−}$ , each of them belongs to either the case A or B, leading to four possibilities, as shown in Tab. IV. In particular, according to Eq. (15), when A 2 $_{1}$/ ( − D$_{+}$D$_{−}$ ) > 4 M/B$_{1}$ , there is no complex λ$_{1}$$_{,}$$_{2}$ , corresponding to the last row of Tab. IV, i.e., ˜ A$_{2}$ is pure imaginary while ˜ V is real. Eqs. (32) and (33) into the definitions in Eqs. (42) and (52), we have In this appendix, we demonstrate that both the pa- rameters ˜ A$_{2}$ and ˜ V in the effective model (40) can be either real or purely imaginary, and the product of ˜ A$_{2}$ ˜ V must be pure imaginary. By putting the wavefunctions 11 ∗ Electronic address: sshen@hkucc.hku.hk 1 For introductions to topological insulators, see Kane C L References Acknowledgments VII. APPENDIX: MODEL PARAMETERS ˜ A 2 AND ˜ V TABLE III: Four possible combinations of λ 1 and λ 2 , accord- ing to Eq. (5), and resulting f ± and η 1$_{,}$2 according to Eq. (38). According to Eq. (5), λ 2 1 < λ 2 $_{2}$, so there does not exist a case when λ 1 is real and λ 2 is pure imaginary. TABLE IV: Four possible groups of E + and E − , and resulting values of ˜ A 2 and ˜ V . ˜ A$_{2}$ = iA$_{1}$A$_{2}$D$_{+}$C$_{+}$C$_{−}$ ∫ L 2 − L 2 dz [ η − 2 f + ∗ + f − + + η + ∗ 1 f + ∗ − f − $_{−}$] ˜ V = C$_{+}$C$_{−}$ ∫ L 2 − L 2 dzV ( z )[ D 2 $_{+}$η + ∗ 1 η − 2 f + ∗ − f − + + A 2 $_{1}$f + ∗ + f − $_{−}$] . (68) E + E − ˜ A 2 ˜ V case A case A imaginary real case A case B real imaginary case B case A real imaginary case B case B imaginary real λ 1 λ 2 f ± η 1$_{,}$2 case A λ ∗ 2 λ ∗ 1 imaginary real real real real real case B imaginary real real real imaginary imaginary real real