(half-filled = two electrons per unit cell), we obtain a novel quantum pseudo-spin model (Eqs. \[tJ1\]-\[Ising\], Sec. \[sec:undoped\]). Within this model, the excitonic insulator (EI) appears as an intermediate state separating not a metal and a semiconductor but a [*Mott insulator*]{} and a semiconductor (or band insulator). In some respects, the behavior is argued to be quite similar to that of a quantum spin-$1/2$ XXZ antiferromagnet in a magnetic field, with excitonic ordering analogous to XY antiferromagnetism. The “spins” of the model, however, can take on [*five*]{} distinct states per site: one singlet state with both electrons in the lower-energy $b$ orbital, and four different spin states with one $a$ and one $b$ electron. This is in contrast to the two states of a single spin-$1/2$ particle.

In the strong coupling limit, this large Hilbert space is “unified” by several approximate symmetries valid at different energy scales. At the largest energy scales this is an enormous SU(4) group, corresponding to arbitrary complex rotations of the four components of $\Delta_{\alpha\beta}$. The approximate SU(4) symmetry fully unifies all possible excitonic states, including singlet, triplet, and singlet-triplet coexistences. These are described by the general decomposition $$\Delta = {1 \over 2} \left( \Delta_s {\cal 