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arXiv:0906.3721 · Physical Review B

Overhauser’s spin-density wave in exact-exchange spin-density functional theory

Stefan Kurth, F. G. Eich

Published Sep 23, 2009• 8 citations
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Abstract

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.

DOI
10.1103/physrevb.80.125120

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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.
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.
We investigate possible spin instabilities of integer quantum Hall systems as a function of the density within the framework of the self-consistent Hartree-Fock theory. We have found a spin density wave (SDW) state that is lower in energy than the paramagnetic and ferromagnetic states for the filling factor $\ensuremath{\nu}=2$ in the density region 2.01<rs<2.152.01<{r}_{s}<2.152.01<rs​<2.15. The new SDW phase is the very one that has been expected by the softening of excitation spectra of the spin density excitation. We have examined the SDW phase in detail using the density matrix.
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.