$H^{\prime\prime}$) is included. Far to the right of the vertical line, the unique low energy state consists of a doubly occupied $b$ orbital in each unit cell, and hopping is unimportant. As the vertical line is approached from either side, virtual hopping processes can induce interactions involving all [*five*]{} low energy states.

Bosonic $t$–$J$ model
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In this subsection, we develop an effective model for the interesting region near the vertical line. In this region, it is necessary and sufficient to truncate the Hilbert space to just the five low-energy states in each unit cell (although higher energy states must be kept in virtual processes). The physics is amusingly similar to a sort of generalized bosonic $t$–$J$ model. On the left-hand side of the thick vertical line, each unit cell is occupied by two spins. At second order in $H^\prime$, these interact via effective exchange interactions, $$H_{\rm eff}^{s}  =  \sum_{\langle ij \rangle} J\left(
    \vec{S}_{ia}\cdot\vec{S}_{ja} +
    \vec{S}_{ib}\cdot\vec{S}_{jb}\right) - \sum_i J_H
  \vec{S}_{ia}\cdot\vec{S}_{ib},
  \label{tJ1}$$ where $J = t^2/(V+2E_{\rm \scriptscriptstyle G})$. This exchange constant may be obtained by computing perturbatively the energy difference between singlet and triplet states on a bond to second order in the hoppings, and neglecting the deviation from the vertical 