@Wang2015] and for which such a broad and flat incoherent band has not been observed. The most apparent are the two anisotropies. The spin anisotropy reflects the distortion of the lattice and leads to the $\gamma$ and $\epsilon$ terms in Eq. (\[model1\]). The perfect hopping anisotropy, on the other hand, which has its origin in the nominal valence of the ruthenium ions and the geometry of the ruthenium-oxide plane, leads to an effectively one-dimensional hole motion, cf. Eq. (\[model2\]). On top of that, a more subtle distinction is related to the larger value of the spin $S=1$ in the studied model. The latter leads to the onset of additional projection operators in the hopping part of the Hamiltonian.

![The hole spectral functions $A({\bf k}, \omega)$ obtained for distinct versions of the relevant $t$–$J$ models and calculated by mapping onto the spin-polaronic model and using the SCBA method (see text): (a) the ‘standard’ spin $S=1/2$ $t$–$J$ model, cf. Ref. ; (b) spin $S=1$ $t$–$J$ model with neither the spin nor the hopping anisotropy, i.e. model (\[model1\]-\[model2\]) with $\varepsilon=\gamma \equiv 0$ and ${\bf e}_{xz}\equiv{\bf e}_{yz} \in \{\hat{x}, \hat{y}\}$; (c) the spin $S=1$ $t$–$J$ model with the hopping anisotropy as suggested for Ca$_2$RuO$_4$ 