by the $d^2$ or $d^1$ multiplets on the single ruthenium ions. \[Since the $xy$ orbital is considered to be ‘always’ occupied by two electrons in the studied model [@Mizokawa2001; @Gorelov2010; @Sutter2017], the $xy$ electrons are integrated out and effectively the nominal occupancy of the ruthenium ions is not $d^4$ ($d^3$) but $d^2$ ($d^1$) in the undoped (single-hole) case, respectively.\] As typical to any $t$–$J$-like model [@Chao1977] to formally denote such a constraint we use the tildas above the electron creation and annihilation operators. Second, just as in the case of the ground state (see discussion above), we project the spin $S=0$ states out of the Hilbert space and, formally, Hamiltonian (\[model2\]) contains such projections. We will not, however, keep them explicit in the formulae below.

Finally, as we are interested in the photoemission spectrum, we define the following orbitally-resolved hole spectral function, $$\begin{aligned}
A_{\alpha}({\bf k}, \omega)\!= \!- \frac{1}{\pi} 
{\rm Im} \Big\langle 0 \Big| \tilde{c}^\dag_{{\bf k}, \alpha, \sigma} \frac{1}{\omega - {\mathcal H} + E_0 +i \delta } \tilde{c}_{{\bf k}, \alpha, \sigma}  \Big| 0 \Big\rangle,\end{aligned}$$ where $| 0 \rangle$ is the ground state of the undoped $t$–$J$ model (\[model1\]-\[model2\]) with energy $E_0$, $\delta$ is the infinitesimally small broadening that is nevertheless finite 