line (i.e. setting $U=2E_{\rm \scriptscriptstyle G}+V$) in the denominators. The latter approximation is valid provided $|U-V-2E_{\rm \scriptscriptstyle G}| \ll V$. Well to the left of the vertical line (in particular when $U-V-2E_{\rm \scriptscriptstyle G}
\gg t$), no doubly-occupied $b$ states are present, and Eq. \[tJ1\] is a complete model. It describes two ferromagnetically bulk coupled Heisenberg spin-$1/2$ antiferromagnets. On a hypercubic lattice (square or cubic in two or three dimensions, respectively), one expects long-range antiferromagnetic order of spins on the same orbital sublattice, with $a$ and $b$ spins aligned parallel at each site.

As the vertical line is approached, the energy cost of a doubly-occupied $b$ orbital is reduced towards zero, and they must be introduced into the lattice. Unit cells with both electrons in the $b$ orbital act as “holes”, having no associated local moment. Unlike the usual $t$-$J$ model holes, they are, however, bosonic and neutral (relative to the magnetic state, they represent the removal of an $a$ electron and replacement with a $b$ electron). Hole hopping occurs at second order in $t$, : $$\begin{aligned}
  H_{\rm eff}^{h} & = & - \mu_h \sum_i
  h_i^\dagger h_i^{\vphantom\dagger} +  t_h \sum_{\langle ij \rangle}  \left(
    h_i^\dagger h_j^{\vphantom\dagger} + h_j^\dagger
    h_i^{\vphantom\dagger}\right) {\cal P}_{ij} 