The Hartree-Fock theory for many fermions with plane-wave orbitals but with abnormal occupation is studied. A one-dimensional problem with an interparticle interaction of finite range which saturates the $N$ body system at a fixed density and finite binding energy is solved explicitly. A direct variation of the resulting Hartree-Fock energy with respect to the additional variational parameters introduced is carried out numerically and results reminiscent of a gas-to-liquid phase transition are found.NUCLEAR STRUCTURE plane-wave Hartree-Fock; model calculation; nuclear matter.The Hartree-Fock theory for many fermions with plane-wave orbitals but with abnormal occupation is studied. A one-dimensional problem with an interparticle interaction of finite range which saturates the $N$ body system at a fixed density and finite binding energy is solved explicitly. A direct variation of the resulting Hartree-Fock energy with respect to the additional variational parameters introduced is carried out numerically and results reminiscent of a gas-to-liquid phase transition are found.NUCLEAR STRUCTURE plane-wave Hartree-Fock; model calculation; nuclear matter.

Physical Review CMar 1, 197937 citations

Many-Particle Theory,

Eberhard K. U. Gross, E. Runge, Olle Heinonen

Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.

Jan 1, 1991137 citations

Perturbation theory of a many-fermion system

Robert William Jones, Franz Mohling

Nuclear Physics AAug 1, 197021 citations

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

Nobuyuki Fukuda, T. Ogawa, T. Soda

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 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.

Progress of Theoretical Physics SupplementMay 16, 20130 citations

N

body system at a fixed density and finite binding energy is solved explicitly. A direct variation of the resulting Hartree-Fock energy with respect to the additional variational parameters introduced is carried out numerically and results reminiscent of a gas-to-liquid phase transition are found.NUCLEAR STRUCTURE plane-wave Hartree-Fock; model calculation; nuclear matter.