$\hat{A}$ operator is diagonal and an explicit expression for it can be found (see Sec. \[sec:appendix\] for details). Second, we we rotate spins on one of the antiferromagnetic sublattices and express the spin operators through bosonic operators by way of the Holstein-Primakoff transformation. Finally, we use the linear spin wave approximation and the Bogoliubov transformation to diagonalize the resulting spin Hamiltonian–see Sec. \[sec:appendix\] for details.

In the end we are left with a diagonal magnon term and a vertex coupling spinless holes to magnons in the following $S=1$ spin polaron Hamiltonian: $$\begin{aligned}
 \label{finalhamiltonian}
    H &= H_t + H_J \approx \sum\limits_{\bf q} \: {\Omega}_{\bf q} \: {\beta}_{\bf q}^{\dag} {\beta}_{\bf q} + E_0 \nonumber \\
    &+\frac{\sqrt{2} \: t}{\sqrt{N}} \sum\limits_{\textbf{k},\textbf{q}}\Big[\left({\gamma}_{k_{x}}v_{\textbf{q}}+{\gamma}_{k_{x}-q_{x}}u_{\textbf{q}}\right)
    h_{\textbf{k},xz}^{\dag}h_{\textbf{k}-\textbf{q},xz} {\beta}_{\textbf{q}}\nonumber \\
    &+ \left({\gamma}_{k_{y}}v_{\textbf{q}}+{\gamma}_{k_{y}-q_{y}}u_{\textbf{q}}\right)
    h_{\textbf{k},yz}^{\dag}h_{\textbf{k}-\textbf{q},yz}{\beta}_{\textbf{q}}+h.c. \Big],\end{aligned}$$ where $\gamma_{k_i} = \cos(k_i)$ and $\beta_q$ are the Bogoliubov boson (magnon) annihilation operators. $u_{\bf q}, v_{\bf q}$ are the Bogoliubov coefficients - see Sec. \[sec:appendix\] for details. The above transformations also lead directly to the expression for the spectral function in terms of the spinless hole Green’s function (see Sec. \[sec:appendix\] for details): $$\begin{aligned}
A_{\alpha}({\bf k}, \omega)\! =\! - \frac{1}{\pi} 
{\rm Im} \Big\langle 0 \Big| {h}_{{\bf k}, \alpha} \frac{1}{\omega - H + E_0 +i \delta } 