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The Hartree-Fock phase diagram of the two-dimensional electron gas

We calculate the ground state phase diagram of the homogeneous electron gas in two dimensions within the Hartree-Fock approximation. At high density, we find stable solutions, where the electronic charge and spin density form an incommensurate crystal having more crystal sites than electrons, whereas the commensurate Wigner crystal is favored at lower densities, rs> 1.22. Our explicit calculations demonstrate that the homogeneous Fermi liquid state -- though being an exact stationary solution of the Hartree-Fock equations -- is never the Hartree-Fock ground state of the electron gas.