antiferromagnetism up to a threshold value of $|{\cal H}|$. For large fields, $|{\cal H}| \gg {\cal J}_\perp,
{\cal J}_z$, one expects ultimately fully polarized states, which correspond to the Mott and band insulators for ${\cal H}>0$ and ${\cal 
  H}<0$, respectively.

Surprisingly, $H_{\rm eff}^{ps}$ displays an enormous SU(4) invariance under $T^{5\mu} \rightarrow \sum_{\nu=1}^4 U_{\mu\nu}T^{5\nu}$, $T^{\mu 5} \rightarrow \sum_{\nu=1}^4 U^*_{\mu\nu} T^{\nu 5}$, where $U$ is an SU(4) matrix. SU(4) symmetry is expected to be a good approximation over a range of energies, because in the physical limit $V \ll U \sim E_{\rm \scriptscriptstyle G}$, $J_H \ll J \ll
{\cal J}_\perp, {\cal J}_z, {\cal H}$. Thus we will take the approach of first solving the SU(4) invariant model, and considering successively the exchanges $J$ and $J_H$, which reduce the symmetry of $H_{\rm
  eff}$ to SU(2)$\times$SU(2) (independent [*physical*]{} spin rotations of the $a$ and $b$ moments) and SU(2)$\times$U(1), respectively.

Lastly, we consider the effects of the hole-pair creation and annihilation terms in Eq. \[fugacity\], which can also be transcribed into the pseudo-spin language. One finds $H_{\rm
  eff}^{ps} \rightarrow H_{\rm eff}^{ps} + H_{\rm eff}^I$, where $$\begin{aligned}
  H_{\rm eff}^I & = & \sum_{\langle\langle ij\rangle\rangle} {\cal
    J}_I \bigg[ {\cal T}^{25}_i {\cal T}^{25}_j + {\cal T}^{35}_i
    {\cal T}^{35}_j 