the intermediate state is expected to be excitonically ordered. []{data-label="XXZfig"}](xxz.eps "fig:"){width="3.0in"}]{} (0.0,2.8)[${\cal H}/{\cal J}_z$]{} (2.8,1.7)[${\cal J}_\perp/{\cal J}_z$]{} (2.6,2.1)[**EI**]{} (1.0,1.7)[**MPS**]{} (1.0,2.6)[**MI $\uparrow$**]{} (1.0,0.5)[**BI $\downarrow$**]{}

Since ${\cal J}_\perp > {\cal J}_z$, we expect transverse pseudo-spin polarization, $\langle {\cal T}^{\mu 5} \rangle \neq 0$, provided $|{\cal H}| < {\cal H}_c = d{\cal J}_\perp$. Remarkably, the transverse components of the pseudo-spin operator are exactly the excitonic order parameters. In particular, straightforward algebra shows ${\cal T}^{\mu 5}  = (-\Delta_{\uparrow\downarrow},
\Delta_{\uparrow\uparrow}, -\Delta_{\downarrow\downarrow},
\Delta_{\downarrow\uparrow})$. Thus for ${\cal J}_\perp > {\cal
  J}_z$, MFT predicts an excitonic insulator.

We now turn to the evolution of the ground state in this regime on introducing the symmetry-breaking terms in $H_{\rm eff}^s$. In their absence, the excitonic order parameter can “point” in any direction which is equivalent under the broken SU(4) symmetry. Within MFT, this amounts to complete freedom to choose the four complex components of $\Delta_{\alpha\beta}$, subject to the constraint ${\rm Tr} \,
\Delta^\dagger \Delta = {1 \over 4}(1- {\cal H}^2/{\cal H}_c^2) \equiv
\Delta_0^2/2$. In term of singlet and triplet components defined by Eq. \[ddecomp\], this constraint simply implies $|\Delta_s|^2 +
\vec{\Delta}_t^*\cdot\vec{\Delta}_t = \Delta_0^2$. The perturbations in $H_{\rm eff}^s$ can be viewed as “anisotropies” favoring sub-manifolds within this space.

To clarify 