per unit cell, so as to give rise to two bands in a non-interacting limit (the actual situation in the hexaborides is more complex, with orbital degeneracy leading to multiple electron and hole pockets). A strong-coupling limit is obtained by first considering only [*local*]{} interactions within a unit cell: $$\begin{aligned}
  H_0 & = & \sum_i E_{\scriptscriptstyle G} \left( a_i^\dagger
    a_i^{\vphantom\dagger} - b_i^\dagger b_i^{\vphantom\dagger}\right) - \mu 
  \left(a_i^\dagger a_i^{\vphantom\dagger} + b_i^\dagger
    b_i^{\vphantom\dagger}\right) 
  \nonumber \\
  & + & \!\!\!\! U\left( a_{i\uparrow}^\dagger a_{i\uparrow}^{\vphantom\dagger}
    a_{i\downarrow}^\dagger a_{i\downarrow}^{\vphantom\dagger} \!+\!
    b_{i\uparrow}^\dagger b_{i\uparrow}^{\vphantom\dagger}
    b_{i\downarrow}^\dagger b_{i\downarrow}^{\vphantom\dagger}\right) + V
  a_i^\dagger a_i^{\vphantom\dagger} b_i^\dagger
  b_i^{\vphantom\dagger}, 
  \label{tb1}\end{aligned}$$ where $a,b$ are electron annihilation operators for the “conduction” and “valence” states, respectively, obeying $\{a_{i\alpha}^{\vphantom\dagger},a_{j\beta}^\dagger\} =
\{b_{i\alpha}^{\vphantom\dagger},b_{j\beta}^\dagger\} =
\delta_{ij}\delta_{\alpha\beta}$. Here and throughout the paper, we use Latin indices $i,j,\ldots$ to denote the lattice site, and Greek indices $\alpha,\beta,\ldots=\uparrow,\downarrow$ to denote the spin state. Labels will be suppressed and implicit wherever clarity allows. The parameters $E_{\scriptscriptstyle
  G}$,$\mu$,$U$,$V$ describe the “band gap” (orbital energy difference), chemical potential, on-site “Hubbard” repulsion, and nearest neighbor repulsion, respectively, within the unit cell.

A crucial feature of $H_0$ is the absence of direct hopping between the $a$ and $b$ orbitals within the unit cell. For excitonic ordering to be well-defined, it is necessary 