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Contributions to Plasma Physics

Periodic ground states of the electron gas in two and three dimensions

B. Bernu, Fran�ois Delyon, Lucas Baguet, Markus Holzmann

Published Nov 1, 2017• 4 citations
Publisher

Abstract

In this paper, we review the possibility of periodic ground states of the two‐ and three‐dimensional electron gas at high and intermediate densities. At the Hartree–Fock level, we obtain explicit solutions at all densities having lower energies than the homogeneous Fermi gas. In the high‐density region, the electrons only partially fill up the band approaching spin density waves as predicted by Overhauser. Preliminary variational Monte Carlo calculations on the polarized two‐dimensional electron gas confirm the existence of these new types of ground states.

DOI
10.1002/ctpp.201700139

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In this paper, we review the possibility of periodic ground states of the two‐ and three‐dimensional electron gas at high and intermediate densities. At the Hartree–Fock level, we obtain explicit solutions at all densities having lower energies than the homogeneous Fermi gas. In the high‐density region, the electrons only partially fill up the band approaching spin density waves as predicted by Overhauser. Preliminary variational Monte Carlo calculations on the polarized two‐dimensional electron gas confirm the existence of these new types of ground states.
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, ${r}_{s}\ensuremath{\gtrsim}1.22$. 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.
In 1962, Overhauser showed that within Hartree-Fock (HF) the electron gas is unstable to a spin-density wave state. Determining the true HF ground state has remained a challenge. Using numerical calculations for finite systems and analytic techniques, we study the unrestricted HF ground state of the three-dimensional electron gas. At high density, we find broken spin symmetry states with a nearly constant charge density. Unlike previously discussed spin wave states, the observed wave vector of the spin-density wave is smaller than 2k(F). The broken-symmetry state originates from pairing instabilities at the Fermi surface, a model for which is proposed.
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)→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)≈8.5.
We determine numerically the ground state of the two-dimensional fully polarized electron gas within the Hartree-Fock approximation without imposing any particular symmetries on the solutions. At low electronic densities, the Wigner crystal solution is stable, but for higher densities (rs{r}_{s}rs​ less than $\ensuremath{\sim}2.7$) we obtain a ground state of different symmetry: the charge density forms a triangular lattice with about 11% more sites than electrons. We prove analytically that this conducting state with broken translational symmetry has lower energy than the uniform Fermi-gas state in the high-density region giving rise to a metal to insulator transition.