we introduce the $t$-$J$ model that describes the motion of a single hole in the ground state of Ca$_2$RuO$_4$. Next, in Sec. \[sec:fromto\] we perform a mapping of the $t$-$J$ model onto a S=1 spin polaron model. The latter is solved using the self-consistent Born approximation (SCBA) in Sec. \[sec:mandr\]. Finally, we discuss the obtained results in Sec. \[sec:discussion\] and end the paper with conclusions in Sec. \[sec:conclusions\]. The paper is supplemented by an Appendix which contains some details of the mapping from the $t$-$J$ to the polaron model.

![Comparison between the experimental and theoretical spectral functions of Ca$_2$RuO$_4$: (a) Angle resolved photoemission (ARPES) spectrum obtained for Ca$_2$RuO$_4$ and published in Ref. ; (b) Hole spectral function $A({\bf k}, \omega)$ calculated for the spin $S=1$ $t$–$J$ Hamiltonian (\[model1\]-\[model2\]) with ${\bf e}_{xz} = \hat{x}$, ${\bf e}_{yz}=\hat{y}$ and using the mapping onto the spin-polaronic model (\[finalhamiltonian\]) and the SCBA method (see text); model (\[finalhamiltonian\]) parameters: $t = 22 J$, $\epsilon = 5.6 J$, $\gamma = 0.25 J$, $J = 5.6 $ meV, numerical broadening of $A({\bf k}, \omega)$ $\delta = 1.1 J $; (c) Hole spectral function $A({\bf k}, \omega)$ calculated as in (b) but convoluted with a Gaussian with the half-width 