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Physical Review

Self-Consistent Pair Interaction for Many-Fermion System

K. Sawada, Toshio Soda

Published Aug 15, 1961• 1 citation
Publisher

Abstract

The equations which determine the one-particle energy and effective two-body interaction in an interacting Fermi gas are constructed within the approximation which sums up all pair creation-annihilation processes. The equation corresponds to the familiar equation for the $K$ matrix which represents the interaction between particles (or holes) and sums up the particle-particle (or hole-hole) scattering processes. The method of the equation of motion is used in this paper. Our result for the one-particle energy is shown to lead to the result previously obtained by Quinn and Ferrell, and by Rockmore for the case of the electron gas with Coulomb interactions, when we replace screened potentials by bare potential in the self-consistent energy equation. For nuclear matter, it is shown that the presence of an attractive interaction in the equation of motion for number density causes an "enhancement" of exchange forces, whereas in the electron gas repulsive Coulomb interactions lead to "screening" of the exchange force. The strength of the isospin density interaction pseudopotential is enhanced by a factor of two when one solves the self-consistent equation; and a simple estimate shows that the Goldhaber-Teller mode lies about 15% higher than the value $\frac{{p}_{F}q}{m}$ previously estimated by Glassgold et al. ($q$: momentum of the oscillation, ${p}_{F}$: Fermi momentum).

DOI
10.1103/physrev.123.1087

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The equations which determine the one-particle energy and effective two-body interaction in an interacting Fermi gas are constructed within the approximation which sums up all pair creation-annihilation processes. The equation corresponds to the familiar equation for the KKK matrix which represents the interaction between particles (or holes) and sums up the particle-particle (or hole-hole) scattering processes. The method of the equation of motion is used in this paper. Our result for the one-particle energy is shown to lead to the result previously obtained by Quinn and Ferrell, and by Rockmore for the case of the electron gas with Coulomb interactions, when we replace screened potentials by bare potential in the self-consistent energy equation. For nuclear matter, it is shown that the presence of an attractive interaction in the equation of motion for number density causes an "enhancement" of exchange forces, whereas in the electron gas repulsive Coulomb interactions lead to "screening" of the exchange force. The strength of the isospin density interaction pseudopotential is enhanced by a factor of two when one solves the self-consistent equation; and a simple estimate shows that the Goldhaber-Teller mode lies about 15% higher than the value pFqm\frac{{p}_{F}q}{m}mpF​q​ previously estimated by Glassgold et al. (qqq: momentum of the oscillation, pF{p}_{F}pF​: Fermi momentum).
In pair density-functional theory, the only unknown piece of the energy is the kinetic energy TTT as a functional of the pair density P(x1,x2)P({x}_{1},{x}_{2})P(x1​,x2​). Although T[P]T\phantom{\rule{0.1em}{0ex}}[P]T[P] has a simpler structure than the Hohenberg-Kohn functional of conventional density-functional theory, computational requirements are still moderate. In the present work, a particularly convenient model system to represent many-electron pair densities is introduced. This ``boson pair model'' (BPM) approximately treats electron pairs as noninteracting bosons. The resulting explicit model for the kinetic energy T2[P]{T}_{2}[P]T2​[P] is shown to be exact for two-electron systems and a lower bound to T[P]T\phantom{\rule{0.1em}{0ex}}[P]T[P] for more than two electrons. The one- and two-particle density matrices obtained from the BPM yield upper bounds for the corresponding many-electron quantities. This suggests a partitioning T[P]=T2[P]+Teff[P]T\phantom{\rule{0.1em}{0ex}}[P]={T}_{2}[P]+{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]T[P]=T2​[P]+Teff​[P], where only the remainder ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]\ensuremath{\geqslant}0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schr\"odinger equation with an effective local two-particle potential results; the latter is identified as a sum of the bare Coulomb interaction and the functional derivative of Teff[P]{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]Teff​[P]. This self-consistent scheme to minimize the energy with respect to PPP is more efficient than previous procedures. Further information on the functional derivative of Teff[P]{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]Teff​[P] is derived from a contracted Schr\"odinger equation. Since Teff[P]{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]Teff​[P] is explicitly known in the two-electron and noninteracting (Hartree-Fock) limits, the present method provides an alternative to density-matrix functional theories, which can be exact in the same limits and are similar in computational cost.
On the basis of a microscopic model of self-consistent field, the thermodynamics of the many-particle Fermi system at finite temperatures with account of three-body interactions is built and the quasiparticle equations of motion are obtained.It is shown that the delta-like three-body interaction gives no contribution into the selfconsistent field, and the description of three-body forces requires their nonlocality to be taken into account.The spatially uniform system is considered in detail, and on the basis of the developed microscopic approach general formulas are derived for the fermion's effective mass and the system's equation of state with account of contribution from three-body forces.The effective mass and pressure are numerically calculated for the potential of "semi-transparent sphere" type at zero temperature.Expansions of the effective mass and pressure in powers of density are obtained.It is shown that, with account of only pair forces, the interaction of repulsive character reduces the quasiparticle effective mass relative to the mass of a free particle, and the attractive interaction raises the effective mass.The question of thermodynamic stability of the Fermi system is considered and the three-body repulsive interaction is shown to extend the region of stability of the system with the interparticle pair attraction.The quasiparticle energy spectrum is calculated with account of three-body forces.
We study the dynamics of a small number of impurity particles coupled to the ideal Fermi gas in a ddd-dimensional box. The impurities interact with the fermions via a two-body potential λv(x)λv(x)λv(x) where λλλ is a coupling constant and v(x)v(x)v(x) for instance a screened Coulomb potential. After taking the large-volume limit at positive Fermi momentum kFk_FkF​ we consider the regime of high density of the fermions, that is, kFk_FkF​ large compared to one. For coupling constants that scale like λ2∼kF(2−d)λ^2 \sim k_F^{(2-d)}λ2∼kF(2−d)​ we show that the impurity particles effectively decouple from the fermions but evolve with an attractive pair interaction among each other which is induced by fluctuations in the Fermi gas.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
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