single-site problems. As an antiferromagnetic solution is expected, this amounts to equations for the ($8$ component) transverse staggered magnetization, defined by $\langle T_i^{\mu 5}\rangle = (-1)^i [n_\perp^{2\mu-1} + i
n_\perp^{2\mu}]$, and the uniform and staggered $z$-axis magnetizations, defined by $\langle T_i^z\rangle = m_z + (-1)^i n_z$. Because of SU(4) symmetry, all orientations of $n_\perp^k$ are degenerate, and it is sufficient to assume $n_\perp^k \equiv n_\perp \delta^{k 1}$. In this subspace, the equations of MFT become [*identical*]{} to those of the conventional spin-$1/2$ XXZ antiferromagnet in a Zeeman field. These equations were solved in Ref. [@XXZ]. The resulting phase diagram is shown in Fig. \[XXZfig\].

(4.0,3.0)(0,0) (0.4,0.1)[![Mean-field phase diagram of the fully SU(4)-invariant pseudo-spin model. A large pseudo-Zeeman field (which scales linearly with the orbital splitting $E_G$) stabilizes either the Mott insulating (MI) or band insulating (BI) state, depending upon its sign. For “in-plane” anisotropy (${\cal J}_\perp >{\cal J}_z$), the intervening phase is an excitonic insulator (EI). In the opposite limit (“Ising” anisotropy) it consists of a micro-phase-separated state with a checkerboard pattern of alternating band and Mott insulating configurations at the lattice scale. In obtaining the pseudo-spin model from the strong-coupling limit of Eqs. \[tb1\]-\[tb3\], one finds in-plane anisotropy, and 