connection between the bosons before and after the Bogoliubov transformation and are needed to correctly express the kinetic part of the Hamiltonian in the polaronic language (see next section), are expressed through the magnon energies $ {\Omega}_{\bf q}$ via the standard formulae, cf. Ref. .

Mapping onto a polaronic model: Kinetic Hamiltonian ($\mathcal{H}_t$)
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### Dealing with projection operators

The implicit projection operators in Eq. (\[model2\]) in the main text have a relatively complex form. The easiest way to deal with them is to construct the projected form of the two single orbital, two-site kinetic Hamiltonians $$\begin{aligned}
 \label{hamt}
    (H_t^{xz})_{{\bf i},{\bf i}+{\bf e}_{xz}} &= -t \sum\limits_{\sigma} \tilde{c}_{{\bf i},xz,\sigma}^{\dag}\tilde{c}_{{\bf i}+{\bf e}_{xz},xz,\sigma} + h.c.,\\
    \label{hamt2}
    (H_t^{yz})_{{\bf i},{\bf i}+{\bf e}_{yz}} &= -t \sum\limits_{\sigma} \tilde{c}_{{\bf i},yz,\sigma}^{\dag}\tilde{c}_{{\bf i}+{\bf e}_{yz},yz,\sigma} + h.c.,\end{aligned}$$ step by step, that is by considering every possible hopping process and the matrix element associated with it.

We consider a single hole introduced into the half-filled ground state and $H_t$ conserves the number of electrons in the system. Moreover, the ‘no double occupancy’ constraint implies that there can never be 3 electrons on one site in the $xz/yz$ orbitals. As a result, a single hole introduced into the system propagates leaving the number of 