spin-orbit coupling [@Das2018] and therefore the spin $S=0$ states having considerably higher energy than the spin $S=1$ states, see Fig. S1 of [@Jain2017], it has been postulated [@Khaliullin2013; @Akbari2014; @Jain2017; @Gretarsson2019] that nevertheless the ‘excitonic magnetism’ can be at play here.

The second part of the Hamiltonian, ${\mathcal H}_t$, is the ‘kinetic’ term. This term is introduced, in order to describe the motion of a single hole created in the ruthenium oxide plane in the photoemission experiment. We restrict our model to the $xz$ and $yz$ orbitals, for, as discussed in the introduction, we are solely interested in the part of the photoemission spectrum associated with these orbitals [@Sutter2017]. Altogether, we end up with, $$\begin{aligned}
 \label{model2}
  \begin{split}
    {\mathcal H}_t &= - t \sum\limits_{{\bf i},  \sigma} \left( \tilde{c}_{{\bf i}+{{{\bf e}_{xz}}},xz,\sigma}^{\dag} \tilde{c}_{{\bf i},xz,\sigma} +\tilde{c}_{{\bf i}+{{\bf e}_{yz}},yz,\sigma}^{\dag} \tilde{c}_{{\bf i},yz,\sigma} +h.c.\right),\\
  \end{split}\end{aligned}$$ where the first (second) term describes the hoppings of an electron with spin $\sigma$ between the nearest neighbor ruthenium $xz$ ($yz$) orbitals along the ${\bf e}_{xz}=\hat{x}$ (${\bf e}_{yz}=\hat{y}$) direction in the 2D square lattice, respectively. Such effectively one-dimensional (‘directional’) hoppings follow from the Slater-Koster scheme [@Slater1954] applied to the square lattice geometry of the ruthenium oxide plane and is, in fact, a 