theoretical and experimental spectra suggests that such an effect should be small in Ca$_2$RuO$_4$.

Finally, with all other parameters equal, the fact that we do not consider here a spin $S=1/2$ (which would be formed by a single hole or electron per site) but a spin $S=1$ antiferromagnet (two holes on each site) does not influence the spectral function qualitatively–thus, the difference between these two cases is purely quantitative, cf. Fig. \[fourplots\](a) and (b). To understand why it is so, let us compare the possible hole hopping processes in the $S=1/2$ and $S=1$ antiferromagnet, which are represented schematically in Fig. \[hoppingprocesses\]. What we can conclude by looking at the process represented on the bottom panel is that all the more complex processes, which have no analog in the single hole per site case, involve more than one magnon. In fact they involve either three or five magnons, which is why we exclude them in the spin wave approximation employed here and why they are absent from Hamiltonian (\[finalhamiltonian\]). Consequently, only the simplest process remains, the one analogous to the only process possible in the spin $S=1/2$ case, cf. the first two panels of Fig. \[hoppingprocesses\]. We stress that such a 