Ref. ; (Bottom 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 three magnons and is [*neglected*]{} in the $S=1$ spin polaron model (\[finalhamiltonian\]) for it goes beyond the linear spin wave approximation.[]{data-label="hoppingprocesses"}](fig4.png){width="50.00000%"}

Secondly, one can see that the one-dimensional hole motion completely changes the character of the spectral function, cf. Fig. \[fourplots\](b) and (c). In order to better understand why this happens, in Fig. \[oned\] we present the constant energy cuts of two spectral functions–one describing a hole in the $xz$ orbital, the other a hole in the $yz$ orbital. We see that the one-dimensional hole motion—a consequence of the geometry of the ruthenium-oxygen plane and the vanishing of the transfer integrals between the oxygen $p$ orbitals and some of the $t_{2g}$ orbitals [@Harris2004]—is reflected in the hole spectral functions. They both show a manifestly one-dimensional dispersion, very much unlike what we see, for instance, in the copper oxides [@Wells1995; @LaRosa1997; @Kim1998; @Damascelli2003; @Shen2007]. We note that, while including a finite spin-orbit coupling for holes in the $xz$ or $yz$ orbitals would naturally lead to the ‘mixing’ between the one-dimensional bands, a good agreement between the 