= a_\downarrow^\dagger b_\downarrow^\dagger |v\rangle,
|5\rangle = b_\uparrow^\dagger b_\downarrow^\dagger |v\rangle$. The Hamiltonian can be rewritten in terms of $5\times 5$ spin matrices ${\cal T}^{\mu\nu}$, where $\langle \mu'|{\cal T}^{\mu\nu}|\nu'\rangle
= \delta^{\mu\mu'}\delta^{\nu\nu'}$. Neglecting for the moment the hole non-conserving terms in Eq. \[fugacity\], $H_{\rm eff}^h =
H_{\rm eff}^{ps} + {\rm const.}$, where $$H_{\rm eff}^{ps}\! =\! \sum_{\langle ij\rangle} {{\cal J}_\perp
    \over 2}
  \! \sum_{\mu=1}^4 \! \left({\cal T}^{\mu 5}_i {\cal T}^{5\mu}_j  \!
    + \!  i \leftrightarrow j\right) \! +\! {\cal J}_z {\cal T}_i^z
  {\cal T}_j^z \! - \!
  {\cal H} \sum_i {\cal T}_i^z,
  \label{pseudospin}$$ and ${\cal T}^z_i = (\sum_{\mu=1}^4 {\cal
  T}_i^{\mu\mu}- {\cal T}^{55}_i )/2$. The generalized exchange constants ${\cal J}_\perp = 2t_h$, ${\cal J}_z=V_{hh}$, and Zeeman field ${\cal H}=dV_{hh}/2 - \mu_h$.

This form of the Hamiltonian exposes a strong similarity to the spin-$1/2$ XXZ model in a Zeeman field. In particular, the “boson hopping” ${\cal J}_\perp$ is analogous to an antiferromagnetic in-plane exchange ($S_i^+ S_j^-$ terms ), spin-boson interaction ${\cal J}^z$ to an antiferromagnetic Ising exchange, and ${\cal H}$ to a $z$-axis field. For ${\cal J}_\perp \gg {\cal J}_z$ and ${\cal H}$ not too large, one expects the analog of canted XY antiferromagnetism, while for ${\cal J}_z \gg {\cal J}_\perp$, one expects instead $z$-axis Ising 