Exchange interaction in a double quantum dot ($H_{\mathrm{ex}}=J{\mathbf{\sigma }}_1\cdot {\mathbf{\sigma }}_2/4$)

An effective coupling that separates the two-electron singlet state and triplet states by energy $J$. For a symmetric two-site Hubbard model with weak tunnelling $t_c$ and charging cost $U$, $J\simeq 4t_c^2/U$. Gate-controlled detuning and tunnelling change the interaction; the simple expression does not hold at arbitrary detuning or strong charge hybridization.