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arXiv:0910.0534 · Physical Review B

Noncollinear spin-spiral phase for the uniform electron gas within reduced-density-matrix-functional theory

F. G. Eich, Stefan Kurth, César R. Proetto, S. Sharma, Eberhard K. U. Gross

Published Jan 29, 2010• 11 citations
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Abstract

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.

DOI
10.1103/physrevb.81.024430

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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 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 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 stability condition of the Hartree-Fock states, which are constructed out of the plane waves times spin-up or -down eigenfunctions. i.e. the paramagnetic (P), ferromagnetic (F) and unsaturated ferromagnetic (UF) states, is examined in some detail in the case of repulsive δ-function. Yukawa and Coulomb interactions. The spherical Fermi surface leads of course to the minimum energy in each case; this energy is then minimized as the function of polarization n=(n+-n-)/(n++n-,n± being the total number of spin-up and -down electrons. We thus can determine the domains of density where each of the three states is most stable. Then the stability of these states against small deformations of one-particle wave functions is examined in their respective domains. In the case of δ-function potential no instability occurs, while for an electron gas the P state (rs ≪ 5.45) is always unstable and the F state (rs > 5.45) is unstable near the critical density. The variational method in the form of canonical transformation is presented to obtain a new Hartree-Fock solution having helical spin structure in case instability occurs. The case of δ-function interaction is worked out for the P state and it is actually shown that such a solution can exist if and only if the P state becomes unstable. This new solution is of non-perturbative nature in the sense that it can never be obtained by perturbation theory starting from the free ground state.
The Hartree-Fock ground state phase diagram of the one-dimensional Hubbard model is calculated, constrained to uniform phases, which have no charge density modulation. The allowed solutions are saturated ferromagnetism (FM), a spiral spin density wave (SSDW) and a double spin density wave} (DSDW). The DSDW phase comprises two canted interpenetrating antiferromagnetic sublattices. FM occurs for small filling, SSDW in most of the remainder of the phase diagram, and DSDW in a narrow tongue near quarter (and three-quarter) filling. Itinerant electrons lift the degeneracy with respect to canting angle in the DSDW. The Hartree-Fock states are metallic except at multiples of a quarter filling. Near half filling the uniform SSDW phase is unstable against phase separation into a half-filled antiferromagnetic phase and a hole-rich SSDW phase. The dependence of the ground state wave number on chemical potential is conjectured to be a staircase. Comparison is made with higher dimensional Hubbard models and the J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg model.