in the numerical calculations below, and we explicitly keep the orbital index $\alpha \in \{ xz, yz \}$ but suppress the spin index $\sigma$ (the spectral function is spin-independent). In what follows we are also interested in the orbitally-integrated spectral function which is defined in the usual way: $A({\bf k}, \omega)=\sum_{\alpha}A_{\alpha}({\bf k}, \omega)$.

From ${\bf t}$-${\bf J}$ to polaron model. {#sec:fromto}
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Stimulated by the successful description of the photoemission spectra of the undoped cuprates [@Wells1995; @LaRosa1997; @Kim1998; @Ramsak1998; @Damascelli2003; @Shen2007; @Manousakis2007; @Wang2015] and to gain a better insight into the physics of the photoemission problem, we perform a mapping of the $S=1$ $t$–$J$ problem onto an $S=1$ spin polaron problem. This is done in two steps:

First, we introduce the slave fermions, $$\label{slavefermions}
  \begin{array}{l}
    \tilde{c}_{{\bf i},\alpha,\uparrow} \rightarrow h_{{\bf i},\alpha}^{\dag}, \quad \quad
    \tilde{c}_{{\bf i},\alpha,\downarrow} \rightarrow \hat{A} \: h_{{\bf i},\alpha}^{\dag} \: S_{\bf i}^+,\\
  \end{array}$$ where $h_{{\bf i},\alpha}^{\dag}$ is the creation operator for a spinless hole on site $i$ and orbital $\alpha$, $S_{\bf i}^+$ is the spin $S=1$ operator on site ${\bf i}$ and $\hat{A}$ is an operator yet to be determined. It can be shown that in the Hilbert space being considered, that is with the $S=0$ states projected out, the 