electrons in the $xz/yz$ orbitals on all other sites unchanged and equal to two. It follows that the only nonzero matrix elements of the two-site Hamiltonians (\[hamt\]-\[hamt2\]) describe processes in which a hole hops between a doubly occupied site and a singly occupied site. Furthermore, these hoppings obey two rules:

1.  The hole can hop to the neighbouring site, albeit only to an unoccupied orbital.

2.  The hole can hop to the neighbouring site, albeit only if the resulting on-site wavefunction is not a spin singlet.

The first rule is simply the ‘no double occupancy’ constraint and the second rule follows from the exclusion of the $S=0$ sector of the Hilbert space discussed in the main text.

All of the above means that on each site we have either the spin triplet or one spin $S=1/2$ fermion on the $xz/yz$ orbitals. This in turn implies that for each pair of neighbouring sites ${\bf i,j}$ we can use the following basis $$\label{site basis}
\left\{ \Big| {^{\uparrow}_{\_}} \Big>_{\bf i},  \Big| {^{\downarrow}_{\_}} \Big>_{\bf i}, \Big| {^{\_}_{\uparrow}} \Big>_{\bf i}, \Big| {^{\_}_{\downarrow}} \Big>_{\bf i} \right\} \; \otimes \; \left\{ \Big| {^{\downarrow}_{\downarrow}} \Big>_{\bf j}, \Big| {^{\uparrow}_{\uparrow}} \Big>_{\bf j}, \frac{1}{\sqrt{2}} \left( \Big| {^{\uparrow}_{\downarrow}} \Big> + \Big| {^{\downarrow}_{\uparrow}} 