GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s53> GLYPH<s54> GLYPH<s55> GLYPH<s56> GLYPH<s57> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32> GLYPH<s76>GLYPH<s32>GLYPH<s91>GLYPH<s81>GLYPH<s76>GLYPH<s93> GLYPH<s32> GLYPH<s91>GLYPH<s101>GLYPH<s86>GLYPH<s93> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s53> GLYPH<s54> GLYPH<s55> GLYPH<s56> GLYPH<s57> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s32> GLYPH<s110>GLYPH<s117>GLYPH<s109>GLYPH<s101>GLYPH<s114>GLYPH<s105>GLYPH<s99>GLYPH<s97>GLYPH<s108> GLYPH<s32> GLYPH<s50>GLYPH<s66> GLYPH<s49> GLYPH<s50> GLYPH<s47>GLYPH<s76> GLYPH<s50> GLYPH<s32> GLYPH<s32> GLYPH<s37> GLYPH<s32> GLYPH<s40>GLYPH<s63>GLYPH<s89> GLYPH<s41> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s53> GLYPH<s54> GLYPH<s55> GLYPH<s56> GLYPH<s57> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s49> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s50>GLYPH<s48> GLYPH<s51>GLYPH<s48> GLYPH<s40>GLYPH<s100>GLYPH<s41> GLYPH<s40>GLYPH<s99>GLYPH<s41> GLYPH<s66>GLYPH<s32>GLYPH<s91>GLYPH<s101>GLYPH<s86> GLYPH<s197> GLYPH<s50> GLYPH<s93> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s53> GLYPH<s54> GLYPH<s55> GLYPH<s56> GLYPH<s57> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s49> GLYPH<s45>GLYPH<s54>GLYPH<s48> GLYPH<s45>GLYPH<s53>GLYPH<s48> GLYPH<s45>GLYPH<s52>GLYPH<s48> GLYPH<s45>GLYPH<s51>GLYPH<s48> GLYPH<s45>GLYPH<s50>GLYPH<s48> GLYPH<s45>GLYPH<s49>GLYPH<s48> GLYPH<s48> GLYPH<s49>GLYPH<s48> 4 III. EFFECTIVE CONTINUOUS MODELS A. Basis states at Γ point H ( ⃗ k ) = H$_{0}$ ( k = 0) + Δ H, (22) H$_{0}$ = [ h ( A$_{1}$ ) 0 0 h ( − A$_{1}$ ) ] , (23) h ( A$_{1}$ ) = [ − D$_{−}$∂ 2 z + C + M − iA$_{1}$∂$_{z}$ − iA$_{1}$∂$_{z}$ − D$_{+}$∂ 2 z + C − M ] , (24) Δ H = [ D$_{2}$k 2 − B$_{2}$k $^{2}$σ$_{z}$ A$_{2}$k$_{−}$σ$_{x}$ A$_{2}$k$_{+}$σ$_{x}$ D$_{2}$k 2 − B$_{2}$k $^{2}$σ$_{z}$ ] . (25) a ⋍ A$_{1}$ 2 √ − D$_{+}$D$_{−}$ , (20) 2 b ⋍ M B$_{1}$ + A 1 4 D$_{+}$D$_{−}$ . (21) ψ ↑ = ψ ↑ $_{λ}$e λz (26) M A 1 A 2 B 1 B 2 C D 1 D 2 (eV) (eV ˚ A) (eV ˚ A) (eV ˚ A $^{2}$) (eV ˚ A $^{2}$) (eV) (eV ˚ A $^{2}$) (eV ˚ A $^{2}$) 0.28 3.3 4.1 1.5 -54.1 -0.0068 1.2 -30.1 0.28 2.2 4.1 10 56.6 -0.0068 1.3 19.6 TABLE I: Two sets of parameters for the 3D Dirac model. The first row is extracted from our effective model parameters for 4QL Bi 2 Se 3 film in table II, and the second row is adopted from the first principles calculation $^{19}$. FIG. 2: (Color online) [(a)(b)] The energy gap Δ ≡ E + − E − and [(c)(d)] the model parameter B [defined in Eq. (42)] as functions of the film thickness L . Circles correspond to the numerical results of the transcendental equations (30) and (31). Solid and dash lines correspond to the approximate formulas to Δ when L is finite [Eqs. (18) and (19)] or very small [Eq. (50)], respectively. All the parameters are adopted (a) by fitting experimental results of 4QL Bi 2 Se 3 , and (b) from the numerical fitting for the first principles calculation of Bi 2 Se 3 $^{19}$, as listed in Tab. I. The solutions of the surface states and thin film of 3DTI can be applied to calculate physical properties ex- plicitly. For instance, we can see whether the ground state of a thin film exhibits QSHE or not by calculating the Chern number or Z$_{2}$ invariant. It is also desirable to establish an effective continuous model to explore the properties of these surface states especially when other interactions have to be taken into account. For this pur- pose, in this section we derive an effective low-energy and continuous models for the surface states and thin film of 3DTI. Due to the low-energy long-wavelength nature of the Dirac cone of the surface electrons, we can use the solu- tions of the surface states at the Γ point as a basis to ex- pand the Hamiltonian H ( k ) in Eq.(1), which will be valid when the energy is limited within the band gap between the conduction and valence bands. This is equivalent to a truncation approximation as we exclude the solutions for the bulk states in the basis. In this approach, the Hamiltonian in Eq. (1) can be expressed as where with and The first term can be solved exactly, and the last term describes the behaviors of electrons near the Γ point. with H$_{0}$ in Eq. (23) is block-diagonal. Its solution can be found by solving each block separately, i.e., h ( A$_{1}$ )Ψ$_{↑}$ = E Ψ$_{↑}$ and h ( − A$_{1}$ )Ψ$_{↓}$ = E Ψ$_{↓}$. Because the lower block is the ”time” reversal of the upper block, the solutions satisfy Ψ$_{↓}$( z ) = ΘΨ$_{↑}$( z ), where Θ = − iσ$_{y}$ K is the time-reversal operator, with σ$_{y}$ the y component of the Pauli matrices and K the complex conjugation operation. Equivalently, we can replace A$_{1}$ by − A$_{1}$ in all the results for the upper block, to obtain those for the lower block. Therefore, we only need to solve h ( A$_{1}$ ). Following the same approach in Sec. II, we put a two-component trial solution According to this result, the oscillation period of the gap π/b becomes π √ B$_{1}$/M when A$_{1}$ = 0, in accordance with the result obtained by Liu et al $^{32}$. Fig. 2(b) shows the gap oscillation by using the model parameters listed in the second entry of Tab. I. Besides, the sine function implies that Δ may be negative. Later we will see that the sign of Δ can be found by solving E 0 + and E 0 − from Eqs. (30) and (31), respectively.