at a minimum that the $a$ and $b$ states be distinguished by a discrete symmetry operation, e.g. parity. When this is the case, direct hopping between these orbitals is prohibited. It may be helpful to imagine an artificial situation in which the $a$ and $b$ orbitals represent $s$ and $d_{xy}$ orbitals on a single site of a square lattice (see Fig. \[fig:orbs\]).

In this situation, $a$ and $b$ orbitals are orthogonal both on the same site and on nearest neighbor sites. An overlap [*is*]{} possible, though for next-nearest neighbor pairs, i.e. on a diagonal. In general, an exchange interaction [*is*]{} allowed by symmetry, and takes the form $$H_1 = -J_H \sum_i \vec{S}_{ia}\cdot \vec{S}_{ib},
  \label{tb2}$$ where $\vec{S}_{ia} = {1 \over 2}a_i^\dagger
\vec{\sigma}a_i^{\vphantom\dagger}$, $\vec{S}_{ib} = {1 \over 2}
b_i^\dagger \vec{\sigma} b_i^{\vphantom\dagger}$. Here and in the following, the Pauli matrices $\vec{\sigma}$ act in the spin space. On physical grounds, a ferromagnetic exchange ($J_{H}>0$) is most appropriate due to Hund’s rule effects for orthogonal orbitals. For pedagogical purposes, we may wish to consider instead the opposite antiferromagnetic sign for this exchange. From the discussion in Sec. 1, it is clear that an essential ingredient for excitonic ferromagnetism is the near-degeneracy of singlet and triplet 