states. To build this into the strong coupling model thus requires small $J_{H}$. For the majority of the paper, therefore, we will neglect $J_{H}$ or treat it as a small perturbation.

Infinite interaction limit
--------------------------

The analysis of the strong-coupling limit begins by first considering the on-site Hamiltonian, $H_{\rm site} = H_0+H_1$, in the absence of electron hopping between adjacent unit cells. This may be thought of as the analog of the $U = \infty$ analysis of an ordinary Hubbard model. In this case, the occupation of each orbital is a good quantum number, and the states can be straightforwardly enumerated. Assuming $E_{\rm \scriptscriptstyle G}>V>0$, and at first also $J_{ab}=0$, one obtains the phase diagram shown in Fig. \[fig:ultralocal\].

![Strong coupling (ultra-local) phase diagram in the $E_{\rm
    \scriptscriptstyle G}$–$\mu$ plane, neglecting exchange and all inter-cell hopping and interactions. Regions with zero and four electrons per unit cell are not shown. In each phase, the lowest-energy states are pictured, with $a$ orbital above and $b$ orbital below.[]{data-label="fig:ultralocal"}](pds4.eps){width="2.75in"}

For the present study, we are particularly interested in densities near two electrons per unit cell. Note that the doping behavior (i.e. on increasing $\mu$) in this regime depends crucially on the relative strength 