content_image/417592.jpg content_image/417593.jpg content_image/417599.jpg content_image/417600.jpg content_image/417598.jpg content_image/417601.jpg content_image/417602.jpg content_image/417603.jpg content_image/417589.jpg content_image/417590.jpg content_image/417591.jpg content_image/417594.jpg content_image/417588.jpg content_image/417596.jpg content_image/417597.jpg content_image/417595.jpg
into 

the secular equation for a nontrivial solution yields four roots of λ ( E ), denoted as βλ$_{α}$ , with β ∈ { + , −} and a ∈ { 1 , 2 } . Note that here λ$_{α}$ is short for λ$_{α}$ ( k = 0) in Eq. (5 ). Each βλ$_{α}$ corresponds to a two-component vector 

The general solution is a linear combination of the four linearly independent two-component vectors 

Applying the boundary conditions Eq. (16) to this gen- eral solution, we obtain two transcendental equations, 

and 

The solutions to Eqs. (30) and (31) give two energies at the Γ point, designated as E 0 + ≡ E$_{+}$ ( k = 0) and E 0 − ≡ E$_{−}$ ( k = 0), respectively. The eigen wavefunctions for E 0 + and E 0 − are, respectively, 

where C$_{±}$ are the normalization factors. The superscripts of f ± and η ± 1 , 2 stand for E 0 $_{±}$, and the subscripts of f ± for parity, respectively. The expressions for f ± and η ± 1 , 2 are given by 

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The energy spectra and wavefunctions of the lower block h ( − A$_{1}$ ) of H$_{0}$ can be obtained directly by replacing A$_{1}$ by A$_{1}$ . Based on the above discussions, the four eigen- states of H$_{0}$ can be given by 

with Φ$_{1}$ → Φ$_{3}$ and Φ$_{2}$ → Φ$_{4}$ under the time-reversal op- eration. We should emphasize that these four solutions are for the surface states, and the solutions for the bulk states are not presented here. We use the four states as the basis states, and other states are discarded (except that in Fig. 3 where four extra bulk states are also in- cluded by the same approach), because of a large gap between the valence and conduction bands. 

With the help of the four states Eq. (38) at the Γ point, we can expand Hamiltonian Eq. (1) to obtain a new effective Hamiltonian 

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where for convenience, we organize the sequence of the basis states following { Φ$_{1}$, Φ$_{4}$, Φ$_{2}$, Φ$_{3}$ } . Under the reor- ganized basis, the effective Hamiltonian is found as 

with 

and 

We find that ˜ A$_{2}$ here can be either real or purely imagi- nary (see Appendix VII for details), classifying the model into two cases: 

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## B. Effective model for 3DTI films 

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