to introduce a term which violates $h$–particle conservation: $$H_{\rm eff}^{nnn} = \sum_{\langle\langle ij\rangle\rangle} y \bigg[
  h_i h_j \big(\vec{O}^{t\dagger}_{a;ij}\cdot
  \vec{O}^{t\dagger}_{b;ij} + O^{s\dagger}_{a;ij}
  O^{s\dagger}_{b;ij}\big) + {\rm h.c.} \bigg].
  \label{fugacity}$$ Here $\vec{O}^{t\dagger}_{a/b;ij}$ creates a triplet of spin one states of $a/b$ particles on the pair of sites $ij$, $O^{s\dagger}_{a/b;ij}$ creates a singlet of $a/b$ particles on this pair, and the “fugacity” $y= 2t_{ab}^2/V$. Note that although Eq. \[fugacity\]  violates conservation of the number of “holes”, it creates and annihilates them only in pairs. There thus remains a conserved Ising charge or parity ($= \sum_i h_i^\dagger h_i^{\vphantom\dagger} {\rm
  (mod 2)}$), signifying whether the number of holes is even or odd. This parity can be traced back to fact that the two orbitals on each site transform differently under spatial reflections.

Pseudo-spin description
-----------------------

To understand the behavior of this model, we now introduce a useful reformulation. Formally, the five possible states on each site can be viewed as different quantized values of a generalized pseudo-spin, and the above terms then take the form of nearest-neighbor interactions between these spins. In particular, we define five states per site via $|1\rangle = a_\uparrow^\dagger b_\uparrow^\dagger |v\rangle,
|2\rangle = a_\uparrow^\dagger b_\downarrow^\dagger |v\rangle,
|3\rangle = a_\downarrow^\dagger b_\uparrow^\dagger |v\rangle,
|4\rangle 