Uniform Stoner decoupling of the Hubbard interaction

For spin imbalance $s=n_{\uparrow }-n_{\downarrow }$ per site, the uniform mean-field interaction energy is $U(n^2-s^2)/4$. With noninteracting density of states $D$ per spin per site, the kinetic cost is $s^2/(4D)$. The quadratic coefficient changes sign at $UD=1$, producing the Stoner mean-field instability. This does not describe every strong-coupling Hubbard phase.