Contact-interaction Hartree-Fock spin energy ($\mathcal{E}=A(n_{\uparrow }^{5/3}+n_{\downarrow }^{5/3})-\gamma (n_{\uparrow }^2+n_{\downarrow }^2)/2$)

In a uniform three-dimensional Hartree-Fock electron gas with repulsive contact coupling $\gamma$, a spin-independent neutralizing background can cancel the Hartree energy. The remaining same-spin exchange term is negative and quadratic in each spin density. Integrating the filled single-spin Fermi sphere gives $A=3{\hslash }^2(6{\pi }^2{)}^{2/3}/(10m)$. At fixed total density the exchange term favors spin polarization, whereas the kinetic term opposes it.

Let $r=(\gamma /A)(n/2{)}^{1/3}$ and $P=(n_{\uparrow }-n_{\downarrow })/n$. In units $2A(n/2{)}^{5/3}$ the spin-dependent energy is $[(1+P{)}^{5/3}+(1-P{)}^{5/3}]/2-r(1+P^2)/2$. Comparing only $P=0$ and $P=1$ gives the endpoint crossing $r=2^{5/3}-2$. Expanding near $P=0$ gives a quadratic coefficient $5/9-r/2$, so the unpolarized state first loses local stability at $r=10/9$. The endpoint crossing is therefore not the actual continuous onset; intermediate polarizations matter.