the orbitally-resolved hole spectral function: (a) Constant-energy cut of the spectral function $A_{xz} ({\bf k}, \omega_c)$ for a hole introduced into the $xz$ orbital ($\omega_c = -1.9 $ eV); (b) Constant-energy cut of the spectral function $A_{yz} ({\bf k}, \omega_c)$ for a hole introduced into the $yz$ orbital ($\omega_c = -1.9 $ eV); (c-d) A schematic view of the ruthenium-oxygen plane explaining the dominant one-dimensional character of the electronic hopping processes on the single-particle level that is also inherited by the many-body hopping processes of Eq. (\[model2\]): For the $xz$ ($yz$) orbital, only hopping in the $\hat{x}$ ($\hat{y}$) direction is possible, cf. panel (c) \[(d)\] [@Slater1954; @Harris2004]. The oxygen (ruthenium) orbitals are shown in blue (red).[]{data-label="oned"}](fig3.png){width="50.00000%"}

![A schematic view of the possible nearest neighbor hole hoppings in the $S=1$ and $S=1/2$ antiferromagnet (AF): (Top panel) A hopping process in the $S=1$ antiferromagnet that, according to the here studied $t$–$J$ Hamiltonian (\[model1\]-\[model2\]), leads to the creation of one magnon in the effective $S=1$ spin polaron model (\[finalhamiltonian\]); (Middle panel) An analogous hopping process as above but in the $S=1/2$ antiferromagnet which, according to the ‘standard’ $t$–$J$ Hamiltonian [@Chao1977], leads to the creation of one magnon in the spin polaron model of 