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arXiv:cond-mat/0311292 · Journal of Physics Condensed Matter

Hartree–Fock energy of a density wave in a spin-polarized two-dimensional electron gas

Juana Moreno, D. C. Marinescu

Published Mar 19, 2004• 0 citations
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Abstract

We calculate the Hartree-Fock energy of a density-wave in a spin polarized two-dimensional electron gas using a short-range repulsive interaction. We find that the stable ground state for a short-range potential is always either the paramagnetic state or the uniform ferromagnetic state. The energy of a density-wave state is, however, reduced by a factor proportional to (1 - zeta^2), where zeta is the polarization of the electron gas. Since this situation occurs in the most unfavorable conditions (short-range repulsive interaction) it is therefore conceivable that by including higher order many-body corrections to the interaction a density-wave ground state is indeed found to be stable.

DOI
10.1088/0953-8984/16/13/011

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We calculate the Hartree-Fock energy of a density-wave in a spin polarized two-dimensional electron gas using a short-range repulsive interaction. We find that the stable ground state for a short-range potential is always either the paramagnetic state or the uniform ferromagnetic state. The energy of a density-wave state is, however, reduced by a factor proportional to (1 - zeta^2), where zeta is the polarization of the electron gas. Since this situation occurs in the most unfavorable conditions (short-range repulsive interaction) it is therefore conceivable that by including higher order many-body corrections to the interaction a density-wave ground state is indeed found to be stable.
A detailed investigation has been made of the application of the Hartree-Fock approximation to the stability and thermodynamic behaviour of spin-density waves in a model with a single electronic energy band and a delta-function-type interaction between electrons. It is shown that for a half-filled band the ground state at zero temperature is always a spin-density wave, provided only that the width of the band is sufficiently small. For other bands the conditions for the relative stability of the spin-density wave, ferromagnetic and paramagnetic states are more complicated, and are studied in detail for a special band form. For a half-filled band the energy per atom and specific heat have been calculated as functions of temperature for the spin-density wave, ferromagnetic and paramagnetic states. The temperature dependence of the amplitude for the spin-density wave and ferromagnetic states, and of the entropy for the spin-density wave and paramagnetic states have also been calculated. The formulation of the problem has been made using a Green function method in order to allow the development in a later paper of a higher-order approximation to the correlation problem.
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 condition that the plane wave state is the lowest Hartree-Fock state of a large fermion system with repulsive interactions, is generally investigated and applied to the systems with $\ensuremath{\delta}$-type interactions, finite-range interactions, and long-range Coulomb interactions. It is found that for finite-range interactions the plane wave state is stable in both the high-density and low-density regions, but may become unstable in the intermediate-density region. The high-density stable region disappears for $\ensuremath{\delta}$-type interactions, while the low-density stable region vanishes for long-range Coulomb interactions. Especially for an electron gas, the critical value rs{r}_{s}rs​ is smaller than 4.5. A speculation concerning the transition of the true ground state is discussed.
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.