$b$ charges. The perturbation $H^{\prime\prime}$ reduces the continuous symmetries of the model down to $SU(2)\times
U(1)$ corresponding to spin and total charge, which are required by the physics of the system. Eq. \[nnn\] actually still respects a number of discrete symmetries, such as $b \rightarrow -b$ simultaneously with a $\pi/2$ rotation. These symmetries, which in fact comprise the point-group operations of the square lattice, can be viewed as a residual discrete subgroup of the original $U(1)$ present in the absence of $H^{\prime\prime}$. We will see in the next section that this gives rise to an Ising symmetry under which the excitonic order parameters transform.

Effective Theory for the Undoped System {#sec:undoped}
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In the central region of Fig. 2, e.g. for $(U+V)/2 < \mu < (U+3V)/2$, all sites are doubly occupied in the strong coupling limit. Nevertheless, for $2E_G \leq U-V$, the low energy states are highly degenerate. Well to the left of the thick vertical line, each $a$ and $b$ orbital is singly occupied, so that there are effectively two spin-$1/2$ degrees of freedom in each unit cell. In the infinite coupling limit these are completely free, but they will interact due to virtual hopping processes when $H^\prime$ (and 