= Hubbard kinetic exchange
{c}
{title2=$H_{\rm eff}=\frac{4t^2}{U}(\mathbf S_1\cdot\mathbf S_2-1/4)$}
At one <electron> per site and $U\gg|t|$, the low-energy singlet of a repulsive <two-site Hubbard dimer> mixes virtually with doubly occupied configurations. Its <energy> is $-4t^2/U$ to leading order, while the three triplets have no such shift. With dimensionless spin-half operators this yields the displayed effective antiferromagnetic <Hamiltonian>. Restoring angular-momentum operators with units of $\hbar$ requires the corresponding $1/\hbar^2$ in the <spin> product.
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