Stability and Instability of Plane-Wave Hartree-Fock Solutions and the Helical Spin Arrangement

概要

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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 (γ_s<5.45) is always unstable and the F state (γ_s<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 nonperturbative nature in the sense that it can never be obtained by perturbation theory starting from the free ground state.