(i k_x+k_y) & (t_{\pi} +t_{\delta}) C+\frac{\xi}{2} & 0 & (t_{\pi} -t_{\delta}) D & 0 \\

 \sqrt{2}\gamma_2 (i k_x+k_y) & 0 & \sqrt{2}\gamma_3 (i k_x-k_y) & -\frac{1}{\sqrt{2}} \xi  
 & \sqrt{2} i \gamma_1 (i k_x-k_y) & \frac{1}{\sqrt{2}} i \xi & 0 & (t_{\pi} +t_{\delta}) C+\frac{\xi}{2} & 0 & (t_{\pi} -t_{\delta}) D \\

 -\sqrt{\frac{3}{2}} \xi & \sqrt{2}\gamma_2 (i k_x+k_y) & 0 & \sqrt{2}\gamma_3 (i k_x-k_y) 
 & 0 & \sqrt{2} i \gamma_1 (i k_x-k_y) & (t_{\pi} -t_{\delta}) D & 0 & (t_{\pi} +t_{\delta}) C-\frac{\xi}{2} & 0 \\

 \sqrt{2}\gamma_2 (i k_x-k_y) & \sqrt{\frac{3}{2}} \xi & \sqrt{2}\gamma_3 (i k_x+k_y) & 0 
 & -\sqrt{2} i \gamma_1 (i k_x+k_y) & 0 & 0 & (t_{\pi} -t_{\delta}) D & 0 & (t_{\pi} +t_{\delta}) C-\frac{\xi}{2}
 \end{matrix} \right),
 }}}}
\label{eq:ham_k_neargamma}
\nonumber\end{aligned}$$ with $C=\cos k_x + \cos k_y \approx 2-\frac{k_x^2}{2}-\frac{k_y^2}{2}$, $D=\cos k_x - \cos k_y \approx -\frac{k_x^2}{2}+\frac{k_y^2}{2}$, where we performed a unitary transformation to diagonalize the $d_{xz/yz}$ subspace in the limit that the $d_{xz/yz}$ states are sufficiently far from other manifolds and $\vec{k} \rightarrow 0$.

The effective Hamiltonian can be obtained by projection onto the concerned manifold $$\begin{aligned}
 \mathcal{H}_{\mathrm{eff}}=\mathcal{P}\mathcal{H}\mathcal{P}+\mathcal{P}\mathcal{H}\mathcal{Q}
 \frac{1}{\epsilon-\mathcal{Q}\mathcal{H}\mathcal{Q}}\mathcal{Q}\mathcal{H}\mathcal{P},
\label{eq:ham_eff}\end{aligned}$$ where $\mathcal{P}$ is the projection operator onto the relevant manifold and $\mathcal{Q}=1-\mathcal{P}$. For the $d_{xy}$ bands, the effective Hamiltonian is $$\begin{aligned}
 