of $E_{\rm \scriptscriptstyle G}$ and $U$. In particular, for $2E_{\rm \scriptscriptstyle G} >U-V$, the preferred charge $Q=2e$ state is one with both electrons in the lower orbital, corresponding to the band insulator. For $U-V>2E_{\rm \scriptscriptstyle G}$, by contrast, the two-electron ground state has one electron in each orbital, and hence a net spin on each site. This is the ultra-strong coupling (i.e. local) version of a Mott insulator. Note that [ *neither*]{} of these two states exhibits [*excitonic*]{} order. This can be seen by directly computing $\langle a^\dagger
b^{\vphantom\dagger}\rangle = 0$ in either state. In fact, the operators $a^\dagger b^{\vphantom\dagger}$ and $b^\dagger a$ act to transform the two phases into one another, i.e. move an electron from the lower to upper orbital or vice versa.

Hopping terms
-------------

To investigate further, we must introduce hopping between adjacent cells. We will principally consider the simplest such term, $$H^\prime = \sum_{\langle ij\rangle} t \left( a_i^\dagger
    a_j^{\vphantom\dagger} + b_i^\dagger b_j^{\vphantom\dagger} + {\rm 
      h.c.}\right),
  \label{tb3}$$ where $\langle ij\rangle$ indicates that the sum is over nearest neighbor pairs of sites. Different hopping integrals $t_a$ and $t_b$ between the two orbitals can also be easily included, but do not change the results significantly, so we will 