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Hartree-Fock Ground State Phase Diagram of Jellium

We calculate the ground state phase diagram of the homogeneous electron gas in three dimensions within the Hartree-Fock approximation and show that broken symmetry states are energetically favored at any density against the homogeneous Fermi gas state with isotropic Fermi surface. At high density, we find metallic spin-unpolarized solutions where electronic charge and spin density form an incommensurate crystal having more crystal sites than electrons. For $r_s \to 0$, our solutions approach pure spin-density waves, whereas the commensurate Wigner crystal is favored at lower densities, $r_s > 3.4$. Decreasing the density, the system undergoes several structural phase transitions with different lattice symmetries. The polarization transition occurs around $r_s \approx 8.5$.