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Progress of Theoretical Physics

Stability Condition of the Hartree-Fock States against Four-Body Correlations

K. Suzuki

Published May 1, 1968• 0 citations
Publisher

Abstract

The stability condition of the Hartree-Fock states against the four-body correlations is discussed and formulated in the framework of the new Tamm-Dancoff approximation extended to the four-particle scattering eigenmode. Important contributions of the pair-scattering interactions to the stability problem are asserted. Finally, the conclusion of the instability of the Hartree-Fock state is deduced.

DOI
10.1143/ptp.39.1180

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The stability condition of the Hartree-Fock states against the four-body correlations is discussed and formulated in the framework of the new Tamm-Dancoff approximation extended to the four-particle scattering eigenmode. Important contributions of the pair-scattering interactions to the stability problem are asserted. Finally, the conclusion of the instability of the Hartree-Fock state is deduced.
A necessary condition for the stability of the Hartree-Fock solution in many-body problem is presented. If this condition is not satisfied, then the solution becomes no longer stable as was discussed by Overhauser in connection with the one-dimensional spin wave model. A variational method is proposed in this case to construct a stable solution which has definitely a lower energy than the Hartree-Fock value. The workability of our method is tested in some realistic examples like the B.C.S.-Bogolyubov theory of superconductivity. Then the method is applied to field theory which seems to be inconsistent in view of the presence of “ghost states”. A canonical transformation leads to a new vacuum state with lower energy, but the high momentum part of the coupling is not damped.
The Hartree-Fock (HF) approximation is generalized to the study of n -particle states, taking correlation effects in interacting quantum-mechanical many-particle systems systematically into account. The case of n = 2 is developed explicitly. It is shown that the form of the HF equations remains essentially the same as in the usual formulation, but with appropriate interpretations of various parameters in terms of n -particle, rather than single-particle, states. We show results of calculations for model systems that illustrate the power of the method when compared to ordinary versions of the HF treatment.
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.
A new theory of treating the ground-state correlations and the correspond-ing renormalization effects is proposed. In the theory, two-particle-two-hole and four-particle-four-hole excitations are taken up as the important parts in the ground-state correlations. The correlated true ground state is given through multiplication of the unitary operator to the Hartree-Fock ground state. The unitary operator is determined from the condition of minimizing the energy of the correlated true ground state. New single-particle operators are defined through the unitary transformation of the Hartree-Fock single-particle operators in the second-quantization formalism. The transformed Hamiltonian represented by the new single-particle operators is established to display the renormalization effects in the single-particle energies and the two-body matrix elements, and further it is proved that in the Hamiltonian the two-body interactions inducing the ground-state correlati6ns are reduced to small and negligible values. With this Hamiltonian, we are led to a simple