the hole motion in the $xz$ and $yz$ orbitals [@Sutter2017]. The reason for leaving the $xy$ orbital out of our analysis is that the highly dispersive, quasiparticle-like part of the photoemission spectrum stretching from low- to high-binding energy \[see Fig. \[result\](a)\] can be easily understood as the free hole motion on the $xy$ orbitals. This follows straightforwardly from, on the one hand, the relatively large energy gap between the $xy$ and $xz/yz$ orbitals leading to the $xy$ orbital being fully occupied and, on the other hand, the absence of any mixing between $xy$ and $xz/yz$ orbitals, be it from hopping or the Coulomb interaction. Thus, including the $xy$ orbitals in our analysis would not much enrich our understanding of the physics at work in this compound.

Altogether, our aim here is twofold: First, we want to model the high-binding energy part of the photoemission spectrum of Ca$_2$RuO$_4$ by using a realistic [*$S=1$ spin polaron*]{} model, specifically derived for this case. Second, we wish to understand to what extent such an $S=1$ spin polaron problem is different from the ‘standard’ (i.e. $S=1/2$) spin polaron model that is well-known from the cuprate studies.

The paper is organized as follows. In Sec. \[sec:tjmodel\] 