= Solution
Use a three-dimensional <electron gas> with a uniform positive background, the <jellium model>. The <Hartree-Fock method> minimizes the expectation of the interacting Hamiltonian over normalized <Slater determinants>. Antisymmetry is exact within that trial class, but correlations requiring a superposition of determinants are omitted. For orthonormal occupied <spin> orbitals $\phi_i$, the <energy> is
$$
E_{\rm HF}=\sum_i\langle i|h_0|i\rangle+\frac12\sum_{ij}(J_{ij}-K_{ij})+E_{\rm background},
$$
where $J_{ij}$ is the direct <Coulomb integral> and $K_{ij}$ the <exchange integral>; the <spin> overlap makes exchange vanish between opposite-spin orbitals. The self-interaction terms $J_{ii}$ and $K_{ii}$ cancel. Variation gives a self-consistent <Fock operator> $\hat f=h_0+\sum_j(\hat J_j-\hat K_j)$; exchange is nonlocal. Its <eigenfunctions> define occupied orbitals of a stationary determinant, and the determinant must be compared with competing self-consistent solutions to select a variational <ground state>.
The <eigenvalues> are orbital Lagrange multipliers. Their sum counts each interaction twice:
$$
E_{\rm HF}=\sum_{i\in\rm occ}\epsilon_i-\frac12\sum_{ij}(J_{ij}-K_{ij})+E_{\rm background}.
$$
Frozen-orbital removal energies may be estimated using <Koopmans' theorem>, but measured quasiparticle energies can contain relaxation and correlations absent from this approximation. Unoccupied orbital energies are not automatically an exact excitation spectrum. In particular, unscreened exchange in a uniform three-dimensional gas produces an unphysical logarithmic slope singularity at its <Fermi surface>; a one-determinant mean field does not reproduce the interacting <Fermi liquid>'s screened response.
For a spatially uniform plane-wave determinant, electron-background and background-background terms cancel the uniform direct Coulomb contribution. Retain the background when making this cancellation: an isolated infinite <electron> charge distribution would have a divergent direct <energy>. The surviving contributions per particle are kinetic <energy> and negative same-spin exchange <energy>. Let $(3/(4\pi n))^{1/3}=r_sa_B^*$ define the <Dimensionless Wigner-Seitz radius>, with $a_B^*=4\pi\varepsilon\hbar^2/(m^*e^2)$, $E_H^*=\hbar^2/(m^*a_B^{*2})$ and $\zeta=(n_\uparrow-n_\downarrow)/n$. Then
$$
\frac{E_{\rm HF}(r_s,\zeta)}{NE_H^*}=\frac{3(9\pi/4)^{2/3}}{10r_s^2}\frac{(1+\zeta)^{5/3}+(1-\zeta)^{5/3}}2-\frac{3(9\pi/4)^{1/3}}{4\pi r_s}\frac{(1+\zeta)^{4/3}+(1-\zeta)^{4/3}}2.
$$
For the unpolarized state this is approximately $1.105/r_s^2-0.4582/r_s$. The kinetic contribution penalizes <spin> <polarization> because it enlarges an occupied Fermi sphere; exchange favors <polarization> because equal-spin occupation gains more exchange <energy>. As density falls, the $r_s^{-1}$ exchange scale overtakes the $r_s^{-2}$ kinetic scale. The uniform fully polarized and unpolarized energies cross at
$$
r_s=\frac{[3(9\pi/4)^{2/3}/10](2^{2/3}-1)}{[3(9\pi/4)^{1/3}/(4\pi)](2^{1/3}-1)}\simeq5.45.
$$
This is a variational comparison within the homogeneous <Hartree-Fock electron gas>, not an exact physical ferromagnetic phase boundary. It demonstrates how Pauli kinetic <energy> and exchange compete and why the allowed trial states matter.
Allowing spatial symmetry breaking changes the result: the <Overhauser instability> lowers the paramagnetic determinant's Hartree-Fock <energy> through a <spin density wave>, even in the high-density limit. A calculation restricted to uniform Fermi spheres therefore does not give the unrestricted Hartree-Fock <ground state>.<Spin Density Waves in an Electron Gas by A. W. Overhauser>[abstract and p. 1437, instability of the uniform paramagnetic state to spin-density waves]{ref} At still lower density, localization and a <Wigner crystal> compete with fluid states. <electron-gas correlation energy> beyond a single determinant is important in assessing their competition. <Quantum Monte Carlo> provides a concrete route to comparing correlated fluid and <crystal> energies; the original stochastic calculation demonstrates why exchange-only <energy> differences are insufficient.<Ground State of the Electron Gas by a Stochastic Method by D. M. Ceperley and B. J. Alder>[abstract and pp. 566–569, correlation energies and competing fluid/crystal states]{ref}
\b[Hartree-Fock is a useful variational baseline with exact exchange, not a controlled complete low-density phase diagram.] At high density the kinetic scale dominates, but even a small symmetry-breaking <energy> gain can invalidate a restricted determinant as the exact Hartree-Fock minimizer. At low density neglected correlations, screening and localization can change both transition locations and the ordering of candidate phases. A numerical phase boundary should therefore state dimension, <spin> constraints, background model and whether correlation and spatially modulated states were included.
\Image[/past-exam-of-the-advanced-quantum-condensed-matter-physics-course-of-the-university-of-cambridge/2016/paper-1-electron-gas.png]
{title=Original homogeneous Hartree-Fock kinetic and exchange energy comparison; the crossing is a restricted variational result}
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