Division of Materials Sciences and Engineering under Contract No. DE-AC02-07CH11358. Computations were performed through the support of the Korea Institute of Science and Technology Information (KISTI) and the National Energy Research Scientific Computing Center (NERSC) in Berkeley, CA.

**Author contributions**

All authors contributed extensively to the work presented in this paper.

Tight-binding model for the TaO$_2$ film
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We consider the tight-binding Hamiltonian for a TaO$_2$ film ($d$-orbitals in a square lattice) [-@Shanavas2014a], $$\begin{aligned}
 \mathcal{H}=\mathcal{H}_\mathrm{hop}+\mathcal{H}_\mathrm{SOC}+\mathcal{H}_\mathrm{E}+\mathcal{V}_\mathrm{sf},\end{aligned}$$ where $\mathcal{H}_\mathrm{hop}$ describes the hopping between the nearest neighbors, $\mathcal{H}_\mathrm{SOC}$ is the atomic spin-orbit coupling of Ta, $\mathcal{H}_\mathrm{E}$ describes the orbital mixing due to the inversion symmetry breaking field near the surface, and $\mathcal{V}_\mathrm{sf}$ describes onsite potential changes due to the surface field. Specifically, the hopping term is given by $$\begin{aligned}
 \mathcal{H}_\mathrm{hop}= 
 {
 \left( \begin{matrix}
 \frac{t_{\sigma}+3t_{\delta}}{2}(c_x+c_y) & -\frac{\sqrt{3}}{2}(t_{\sigma}-t_{\delta})(c_x-c_y) & 0 & 0 & 0 \\

 -\frac{\sqrt{3}}{2}(t_{\sigma}-t_{\delta})(c_x-c_y) & \frac{3t_{\sigma}+t_{\delta}}{2}(c_x+c_y) & 0 & 0 & 0 \\

 0 & 0 & 2t_{\pi}(c_x+c_y) & 0 & 0 \\

 0 & 0 & 0 & 2(t_{\pi} c_x +t_{\delta} c_y) & 0 \\

 0 & 0 & 0 & 0 & 2(t_{\delta} c_x +t_{\pi} c_y) \\
 \end{matrix} \right),
 }
\label{eq:ham_hopping}
\nonumber\end{aligned}$$ where the basis is $\{ |d_{z^2}\rangle, |d_{x^2-y^2}\rangle, |d_{xy}\rangle, |d_{xz}\rangle, 