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arXiv:1507.06884 · Physical Review B

Upper bounds of spin-density wave energies in the homogeneous electron gas

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

Published Dec 14, 2015• 8 citations
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Abstract

Studying the jellium model in the Hartree-Fock approximation, Overhauser has shown that spin-density waves (SDWs) can lower the energy of the Fermi gas, but it is still unknown whether these SDWs are actually relevant for the phase diagram. In this paper, we give a more complete description of SDW states. We show that a modification of the Overhauser ansatz explains the behavior of the jellium at high density compatibly with previous Hartree-Fock simulations.

DOI
10.1103/physrevb.92.235124

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Studying the jellium model in the Hartree-Fock approximation, Overhauser has shown that spin-density waves (SDWs) can lower the energy of the Fermi gas, but it is still unknown whether these SDWs are actually relevant for the phase diagram. In this paper, we give a more complete description of SDW states. We show that a modification of the Overhauser ansatz explains the behavior of the jellium at high density compatibly with previous Hartree-Fock simulations.
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.
The spin-density wave (SDW) state of the uniform electron gas is investigated in the exact-exchange approximation of noncollinear spin-density functional theory (DFT). Unlike in Hartree-Fock theory, where the uniform paramagnetic state of the electron gas is unstable against formation of the spin-density wave for all densities, in exact-exchange spin DFT this instability occurs only for densities lower than a critical value. It is also shown that, although in a suitable density range it is possible to find a noninteracting SDW ground-state Slater determinant with energy lower than the corresponding paramagnetic state, this Slater determinant is not a self-consistent solution of the optimized effective potential (OEP) integral equations of noncollinear spin DFT. A self-consistent solution of the OEP equations which gives an even lower energy can be found using an excited-state Slater determinant where only orbitals with single-particle energies in the lower of two bands are occupied while orbitals in the second band remain unoccupied even if their energies are below the Fermi energy.
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 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.
The noncollinear spin-spiral density wave of the uniform electron gas is studied in the framework of reduced-density-matrix-functional theory. For the Hartree-Fock approximation, which can be obtained as a limiting case of reduced-density-matrix-functional theory, Overhauser showed a long-time ago that the paramagnetic state of the electron gas is unstable with respect to the formation of charge- or spin-density waves. Here we not only present a detailed numerical investigation of the spin-spiral density wave in the Hartree-Fock approximation but also investigate the effects of correlations on the spin-spiral density wave instability by means of a recently proposed density-matrix functional.