\mathcal{H}_{\mathrm{eff}}
 \approx h_{xy}(\vec{k})I_{2\times 2}+\frac{-2\gamma_1 \xi}{\Delta_{xy,uxz/yz}}
 (\vec{\sigma}\times \vec{k})\cdot \hat{z},
\label{eq:ham_eff_gamma_dxy}\end{aligned}$$ where $\Delta_{xy,uxz/yz}=4t_{\pi}+\delta_1-\{2(t_{\pi}+t_{\sigma})+\frac{\xi}{2}\}$, and the Pauli matrices describe the subspace defined by $\{ |d_{xy}\uparrow\rangle, |d_{xy}\downarrow\rangle \}$. For the lower $d_{xz/yz}$ bands, $$\begin{aligned}
 \mathcal{H}_{\mathrm{eff}}
 \approx h_{lxz/yz}(\vec{k})I_{2\times 2}
 +\frac{2\sqrt{3}\gamma_2 \xi}{\Delta_{lxz/yz,z^2}}(\vec{\sigma}\times \vec{k})\cdot \hat{z},
\label{eq:ham_eff_gamma_lxz/yz}\end{aligned}$$ where $\Delta_{lxz/yz,z^2}=2(t_{\pi}+t_{\delta})-\frac{\xi}{2}-\{t_{\sigma}+3t_{\delta}+\delta_2\}$, and the Pauli matrices describe the subspace defined by $\{ \frac{1}{\sqrt{2}}(|d_{xz}\downarrow\rangle +i |d_{yz}\downarrow\rangle), 
\frac{1}{\sqrt{2}}(|d_{xz}\uparrow\rangle -i |d_{yz}\uparrow\rangle) \}$. For the upper $d_{xz/yz}$ bands, $$\begin{aligned}
 \mathcal{H}_{\mathrm{eff}}
 \approx h_{uxz/yz}(\vec{k})I_{2\times 2}
 +\left[\frac{-2\gamma_3 \xi}{\Delta_{uxz/yz,x^2-y^2}}+\frac{-2\gamma_1 \xi}{\Delta_{uxz/yz,xy}}\right]
 (\vec{\sigma}\times \vec{k})\cdot \hat{z},
\label{eq:ham_eff_gamma_uxz/yz}\end{aligned}$$ where $\Delta_{uxz/yz,x^2-y^2}=2(t_{\pi}+t_{\delta})+\frac{\xi}{2}-\{3t_{\sigma}+t_{\delta}+\delta_3\}$, $\Delta_{uxz/yz,xy}=2(t_{\pi}+t_{\delta})+\frac{\xi}{2}-\{4t_{\pi}+\delta_1\}$, and the Pauli matrices describe the subspace defined by $\{ \frac{1}{\sqrt{2}}(|d_{xz}\downarrow\rangle -i|d_{yz}\downarrow\rangle), 
\frac{1}{\sqrt{2}}(|d_{xz}\uparrow\rangle +i|d_{yz}\uparrow\rangle) \}$.

The angular momentum (AM) texture can be calculated using the eigenstates with the lowest perturbative correction in $\xi$. For the $d_{xy}$ manifold in close vicinity of the $\Gamma$ point, the dominant spin AM expectation value $\langle S_y \rangle \approx \hbar/2$ for an eigenstate in $x$ direction comes from the original $d_{xy}$ manifold. The remnant orbital AM $\langle L_y \rangle \approx\hbar \xi/\Delta_{xy,uxz/yz}$ is due to the inter-band coupling to the upper $d_{xz/yz}$, which can be calculated using the eigenstate with the first-order correction in $\xi$ that hybridizes the $d_{xy}$ manifold with the upper $d_{xz/yz}$ and the $d_{x^2-y^2}$ manifolds. As for the lower $d_{xz/yz}$ manifold, both the orbital and 