A detailed investigation is made of the influence of particle self-energies on the dynamic response of the metallic electron gas. Two quite different approaches, the Green-Neilson-Szyma\ifmmode \acute{n}\else \'{n}\fi{}ski (GNS) theory and the quasiparticle-lifetime theory, are examined and contrasted. It is shown that the inclusion of off-shell effects in the quasiparticle-lifetime theory drastically alters its central conclusions. Moreover, that theory's neglect of cancellations between self-energy terms and other classes of terms seriously violates the major sum rules. These cancellations are closely connected with conservation principles. The GNS theory, being conserving, reproduces these cancellations in detail.
Electron correlations of an electron liquid at metallic densities are studied by means of a diagrammatic method, the primary aim being the interpretation of the spectral structure observed for several metals in recent inelastic electron and x-ray scattering experiments. Several sum rules and requirements such as the positivity of the pair distribution function, the compressibility sum rule, the frequency moment sum rules, and the conservation law of the local electron number which have hitherto been regarded as guiding principles in theoretical formulation are first examined and discussed in a critical manner. Detailed examinations are made of the proper polarization function within the Hartree-Fock approximation. The lowest-order perturbation expressions giving two-pair excitations and one-pair---one-plasmon excitations are next examined. It is pointed out that these expressions are indeed divergent in the usual one-pair excitation region; the renormalization of the one-particle Green's function is indispensable there. From the above considerations and further discussions, the quasi-one-pair excitation approximation is presented for the purpose of describing the spectral shape in the dynamical structure factor $S(q,\ensuremath{\omega})$ in the intermediate wave-number and intermediate frequency regions. In the quasi-one-pair excitation approximation the proper polarization function is constructed from a coherent part of the renormalized one-particle Green's function and a static part of the irreducible particle-hole interaction including strong short-range correlations.
The effects of short-range correlations on the properties of the electron gas are studied using an approach phrased in the polarization potential language of Pines and Nozi\`eres. It was argued by Lowy and Brown that two-body terms make up the dominant contribution to the effective interaction for large momentum transfers. In the region of small-momentum transfers it is known that the random-phase approximation provides an adequate picture of the electron gas. With these two regions of large- and small-momentum transfers in mind, a polarization propagator is constructed to interpolate between them. A number of sum rules, among them the
f-sum rule, involving the imaginary part of the polarization propagator, were found to be satisfied. The failure of our theory to satisfy the compressibility sum rule is discussed in detail. We present here the results for the local
t-matrix, the dynamic and static form factors, and the pair correlation function. That our pair-correlation function remains positive and is essentially the same as in the paper of Lowy and Brown indicates that the physics and not the formalism is responsible for the results we obtain.
The quadratic density response function and the quadratic dynamical structure function (the Fourier transform of the equilibrium three-point density correlations) contain important information about a many body system; they are also ingredients for an improved dynamical mean field theory for strongly coupled Fermi systems. We examine the analytic properties of the density response function and establish new single frequency and double frequency moment sum rules. We relate the sum rule coefficients to the high frequency expansion of the response function. Next we invoke the quadratic fluctuation-dissipation theorem to relate these frequency moments to weighted frequency moments of the dynamical structure function. These latter reduce to straight frequency moments in the high temperature classical and zero temperature degenerate limits. \textcopyright{} 1996 The American Physical Society.
We present an analytical study of the dynamical local-field factors associated with the response of a homogeneous two-dimensional interacting electron liquid as functions of momentum, frequency, and density. We derive sum rules that constrain their asymptotic forms (in momentum and frequency) for both the spin-symmetric and spin-antisymmetric cases. Parametrized expressions for the local-field factors are proposed, based on all available sum rules and on many-body perturbation theory, and these are found to be in good agreement with quantum Monte Carlo calculations. Finally, these expressions are used to evaluate the effective electron-electron interaction in a local approximation for two-dimensional systems. It is shown that both the quantitative and qualitative behaviors of the interaction are sensitive to the inclusion of dynamical correlations.
This paper is concerned with the application of sum rules satisfied by the density correlation function to the calculation of properties of the interacting Bose system, particularly liquid
He4 and the hard-core gas. In addition to the two well-known sum rules which have been extensively discussed in the literature, we derive a third useful relation which directly involves the two-body interaction. The assumption that a single excitation exhausts all the sum rules (which is known to lead to the Feynman theory when only the first two sum rules are used) now provides, with the help of the additional sum rule, an explicit integral equation for the liquid structure factor
S(k). This equation can be solved quite easily for the hard-core gas, and we find the low-lying excitation spectrum to have the form ${{\ensuremath{\omega}}_{\mathrm{hc}}}^{2}(k)={k}^{2}{{v}_{T}}^{2}+{\left(\frac{{k}^{2}}{2m}\right)}^{2}+4\left(\frac{{k}^{2}}{2m}\right)\left[\left(\frac{P}{\ensuremath{\rho}}\ensuremath{-}\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)({\mathcal{P}}_{1}(\mathrm{ka})\ensuremath{-}{\mathcal{P}}_{1}(0))+\left(m{{v}_{T}}^{2}\ensuremath{-}\frac{2P}{\ensuremath{\rho}}\ensuremath{-}\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)({\mathcal{P}}_{2}(\mathrm{ka})\ensuremath{-}{\mathcal{P}}_{2}(0))\right],$ where
vT,
P,
〈N〉〈KE〉 and
a are, respectively, the sound velocity, pressure, kinetic energy per particle, and core diameter.
P1 and
P2 are sinusoidal functions given in the text. Although the sum rules themselves are exact, there are several difficulties associated with the use, even in the low-
k limit, of the sharp resonance assumption in all such integral relations. In spite of these difficulties, which are discussed briefly, we have pursued the consequences of the sharp-resonance approximation to obtain an integral equation for the liquid structure factor
S(k) together with the above closed-form solution for the hard-core gas excitation spectrum. The speculation that the approximation becomes essentially exact in the low-
k limit provides, through the third sum rule, an expression for the sound velocity in terms of the two-body potential
v(r) and the radial distribution function
S(r). We apply this relation to the case of liquid
He4 and discuss extensions of the single-resonance approximation which can be made with the help of the third sum rule.