 Bosonic superexchange

Effective exchange of internal states through virtual boson hopping into energetically costly occupation sectors. For two bosons initially one per site and charging energy $U{\Delta }^2/4$, the second-order operator is $-(t^2/U)(3+4{\mathbf{s}}_a\cdot {\mathbf{s}}_b)$ for dimensionless spins. The triplet is lowered and the singlet is unchanged in this example; signs for other energy functions require their actual denominators.