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Journal of Applied Physics

Spin density waves in the uniform electron gas

Nathan E. Brener, John L. Fry

Published Mar 1, 1981• 1 citation
Publisher

Abstract

We have made a study of the effect of exchange and correlation on spin density waves in the uniform electron gas. In order to take these effects into account, we have used two new magnetic susceptibility formulas which we have recently derived. The first of these is the time-dependent Hartree-Fock (TDHF) formula which contains exchange but no correlation, while the second one is the generalized random phase approximation (GRPA) formula which contains both exchange and correlation. Both of these formulas were evaluated for the uniform electron gas. The TDHF susceptibility was found to have an absolute maximum at a nonzero value of q, while the GRPA susceptibility was found to have an absolute maximum at q = 0. This indicates that when exchange effects are included in the susceptibility, the ground state of the uniform electron gas contains a spin density wave, but when correlation effects are also included, then the spin density wave is no longer present in the ground state of the uniform electron gas.

DOI
10.1063/1.329659

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Journal of Physics Condensed Matter•Dec 7, 1992•9 citations
We have made a study of the effect of exchange and correlation on spin density waves in the uniform electron gas. In order to take these effects into account, we have used two new magnetic susceptibility formulas which we have recently derived. The first of these is the time-dependent Hartree-Fock (TDHF) formula which contains exchange but no correlation, while the second one is the generalized random phase approximation (GRPA) formula which contains both exchange and correlation. Both of these formulas were evaluated for the uniform electron gas. The TDHF susceptibility was found to have an absolute maximum at a nonzero value of q, while the GRPA susceptibility was found to have an absolute maximum at q = 0. This indicates that when exchange effects are included in the susceptibility, the ground state of the uniform electron gas contains a spin density wave, but when correlation effects are also included, then the spin density wave is no longer present in the ground state of the uniform electron gas.
The spin susceptibility of the electron gas is studied using a functional integral formulation. A Hubbard-Stratonovich transformation is used to decouple the electron-electron interaction. A uniform Fock mean field is chosen, and effects of small fluctuations about this mean field are retained up to second order. The term that gives the usual spin random-phase approximation susceptibility is identified, with the ladder sum contained in the propagator for the field fluctuations. Computations of the static spin susceptibility outline the region of stability as a function of electron-gas coupling parameter and wave vector. These computations parallel those of Hamann and Overhauser [Phys. Rev. 143, 183 (1966)] for the Yukawa interaction. We use the analytic continuation method of Staunton et al. [Phys. Rev. Lett. 82, 3340 (1999)] to extend the calculations to nonzero frequency. The resulting spectral density shows the increase in low-energy spin fluctuations as the spin-density-wave transition is approached, and the development of a pseudogap in the spin fluctuation spectrum near the wave vector of the transition.
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.
The calculation of the induced spin density of a normal fermion system, such as the electron gas, in the limit of small wave numbers, is carried beyond the random phase approximation (R.P.A.) by the method of canonical transformation so as to include the first non-R.P.A. corrections. An expression for the induced spin density in the same limit, exact to all orders of particle-particle coupling, is then derived by more general methods of many-body perturbation theory. It depends only on the knowledge, to all orders of interparticle coupling, of the effective mass and the forward scattering of quasi-particles at the Fermi surface, and also yields the results of the extended R.P.A. on appropriate expansion. In the limit of zero wave number, the resulting expression for the magnetic susceptibility is found identical with that deduced by Landau from a phenomenological basis and may be regarded as an additional confirmation of the microscopic validity of the theory of the Fermi liquid.
The problem of the many-body enhancement of the static spin susceptibility at long wavelengths and its relation to the quasiparticle effective mass is investigated for a normal electron gas in two-dimensional space as a function of the electronic density. We start from a discussion of the results of the simple Hartree-Fock approximation for various interaction potentials and proceed to develop a complete theory. We find that the effects of the electron-electron interaction are significantly larger than in the familiar three-dimensional case. Our approach is based on a new self-consistent scheme which goes beyond the simple random-phase approximation by explicitly allowing for charge- and spin-fluctuation-induced vertex corrections of the Hubbard type. We show that when the latter are neglected, the many-body enhancement of the spin susceptibility can be cast in a remarkably simple and elegant analytic form.
The Hubbard model in the limit of strong Coulomb interaction (the (t-J) model) is studied by the diagram technique for Hubbard operators. The generalized random-phase approximation (GRPA) is formulated as an approximation taking into account electron loop-type diagrams. Within the framework of this approximation, both the dynamic magnetic and dielectric susceptibilities are calculated for a wide interval of electron concentrations, 0<n<1. The magnetic susceptibility shows a crossover in the magnetic behaviour of the system from pure itinerant magnetism to magnetism with localized magnetic moments. The crossover occurs at a critical point, nc, when the chemical potential for electrons of the lower Hubbard band changes sign. Magnetic phase diagrams are constructed on the (t/U, n) plane at T=0 for different types of crystal lattices. The behaviour of magnetic phases with temperature is also studied. The GRPA leads to a too drastic crossover, especially at low temperatures, because of insufficient allowance for the charge and spin fluctuations on a site. Summation of special diagram series shows that the Gaussian fluctuations of the effective electric and magnetic fields lead to a smoother picture of the crossover.