qualitative analysis and supplement it with quantitative results from maximally localized Wannier functions (MLWFs). In the TB model, we consider a Hamiltonian for all Ta $d$-orbitals, including the $e_g$ orbitals ($d_{z^2}$, $d_{x^2-y^2}$) in a square lattice [-@Shanavas2014; -@Shanavas2014a] to describe the TaO$_2$ layer 2D bands, $$\begin{aligned}
\mathcal{H}=\mathcal{H}_\mathrm{hop}+\mathcal{H}_\mathrm{SOC}+\mathcal{H}_\mathrm{E}+\mathcal{V}_\mathrm{sf},\end{aligned}$$ where the first term $\mathcal{H}_\mathrm{hop}$ describes the nearest-neighbor hopping, and the second term $\mathcal{H}_\mathrm{SOC} = \xi {\bf L} \cdot {\bf S}$ is the atomic SOC, with $\xi \approx 0.26$ eV for Ta. The third term $\mathcal{H}_\mathrm{E}$ includes the additional hoppings that would have been forbidden if not for the ISB: $$\begin{aligned}
 &\gamma_1&=\langle d_{xy}| \mathcal{H}_\mathrm{E} |d_{xz} \rangle_{\hat{y}} 
 =\langle d_{xy}| \mathcal{H}_\mathrm{E} |d_{yz} \rangle_{\hat{x}} \nonumber \\ 
 &\gamma_2&=\langle d_{xz}| \mathcal{H}_\mathrm{E} |d_{z^2} \rangle_{\hat{x}} 
 =\langle d_{yz}| \mathcal{H}_\mathrm{E} |d_{z^2} \rangle_{\hat{y}} \\
 &\gamma_3&=\langle d_{x^2-y^2}| \mathcal{H}_\mathrm{E} |d_{yz} \rangle_{\hat{y}} 
 =\langle d_{xz}| \mathcal{H}_\mathrm{E} |d_{x^2-y^2} \rangle_{\hat{x}} \nonumber,\end{aligned}$$ in which the vectors in the subscripts denote the relative position of the two orbitals with the lattice constant set to 1 for convenience (these ISB hoppings play a role analogous to the chiral orbital angular momentum effect in the $p$-orbital bands [-@SPark2011; -@BKim2012; -@CPark2012]). Here, $\gamma_1$ is intra-$t_{2g}$ ISB hopping while $\gamma_2$ and $\gamma_3$ describe $t_{2g}$-$e_g$ ISB hoppings. The fourth term $\mathcal{V}_\mathrm{sf}$ describes the potential difference originating 