\nonumber \\
    & & - {\cal T}^{15}_i {\cal T}^{45}_j - {\cal
      T}^{45}_i {\cal T}^{15}_j + 
    {\cal T}^{52}_i {\cal T}^{52}_j + {\cal T}^{53}_i
    {\cal T}^{53}_j \nonumber \\
    & & - {\cal T}^{51}_i {\cal T}^{54}_j - {\cal
      T}^{54}_i {\cal T}^{51}_j \bigg].
    \label{Ising}\end{aligned}$$ The coupling ${\cal J}_I \propto y$. While it is perhaps not completely transparent in this notation (a better notation for this term will be introduced in next subsection – see Eq. \[Ising2\]), the effect of $H_{\rm eff}^I$ is to further break the SU(2)$\times$U(1) symmetry down to SU(2)$\times$Z$_2$. The Z$_2$ invariance is the remnant of the physical parity symmetry discussed in the previous subsection.

Mean-field theory and undoped phase diagram
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We expect that a simple Weiss mean field theory (MFT) gives qualitatively correct results for the stoichiometric phase diagram, as it does for the ordinary XXZ+Zeeman model.[@largeNnote]Neglecting $H_{\rm eff}^s$, the MFT consists in replacing $$T_i^{\mu 5} T_j^{5 \mu} \rightarrow \langle T_i^{\mu5 }\rangle
  T_j^{5 \mu} + T_i^{\mu 5} 
  \langle T_j^{5 \mu}\rangle - \langle T_i^{\mu 5}\rangle \langle
  T_j^{5 \mu}\rangle,$$ for each bond $i,j$ on the lattice, and similarly for the $T^z_i
T^z_j$ interaction. With this replacement, the Hamiltonian decouples on different lattice sites, and the problem reduces to solving self-consistently the appropriate 