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Physical review. B, Condensed matter

Dynamical structure factor of an electron liquid. I

K. Awa, Hiroshi Yasuhara, T. Asahi

Published Mar 15, 1982• 48 citations

Abstract

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.

DOI: 10.1103/physrevb.25.3670

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ELECTRON CORRELATION IN METALS. Annual Summary Report No. 2, November 1, 1960-October 31, 1961

N. H. March, S. Sampanthar

Results for the diagonal element of the second-order density matrix (pair function) and for the momentum distribution of an interacting electron gas are presented. In the low density limit, it is shown that, to a similar degree of approximation, the first-order density matrix and hence the momentum distribution may be obtained. In the highdensity problem, the pair function has also been calculated using the Green's function approach. From the information available in these high and low density limits, the range of usefulness of the concept of the Fermi surface in an interacting electron gas is discussed. Also, results of varia. tional calculations on the stability of Cverhauser spindensity waves in metals are reported. It is concluded that such waves reduce the energy below the Hartree-Fock value only when the interaction becomes very strong, or equivalently when the density is very low. Some preliminary work on the correlation problem in a non-uniform electron gas is also briefly discussed. (auth)

•Oct 31, 1963•0 citations

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 dynamic density response function, structure factor, and spectral function of a Luttinger liquid with Coulomb interaction are studied with the emphasis on the short-range electron correlations. The Coulomb interaction changes dramatically the density response function as compared to the case of the short-ranged interaction. The coordinate dependence of the density response is smoothing with time, and the oscillatory structure appears. However, the spectral functions remain qualitatively the same. The dynamic structure factor contains the $\ensuremath{\delta}$ peak in the long-wave region, corresponding to one-boson excitations. In addition, the multi-boson-excitations band exists in the wave-number region near to

{2k}_{F},

where

{k}_{F}

is the Fermi wave number. We develop a method to analyze the asymptotic behavior of the spectral functions near to the edges of the multi-boson-excitations band at zero temperature. The dynamic structure factor diverges at the edges of this band, while the dielectric function goes to zero there. Hence, a new collective charge mode appears in the one-dimensional electron liquid as a consequence of short-range dynamic correlations. The mode energy goes to zero at

{2k}_{F},

which means that the

{2k}_{F}

mode is soft.

We use the Ceperley and Alder Green's-function Monte Carlo data for the static form factor,

S(q)

, of an electron liquid to calculate the static local-field correction,

G(q)

, to the dielectric function which takes into account short-range electron correlations. The resulting local-field correction in the long-wavelength limit is obtained by smoothly fitting to the results of Iwamoto and Pines, which are exact in this limit. We use the

G(q)

thus obtained to calculate the static and dynamic form factors, the contribution to the correlation energy from different momentum transfers, plasmon dispersion, and the plasmon-dispersion coefficient. We compare these theoretical quantities with the available experimental data, and use the ${\ensuremath{\omega}}^{3}$-sum rule (which we also calculate from the Monte Carlo data) to discuss the limitations arising from the use of the static local-field correction.

Multipair excitations in the interacting-electron gas at metallic densities are quantitatively investigated at large momentum transfers with use of the microscopic Green-Neilson-Szyma\ifmmode \acute{n}\else \'{n}\fi{}ski (GNS) theory. The effect of single-pair and multipair excitations on the dynamic response function \ensuremath{\chi}(q,\ensuremath{\omega}) are contrasted, and the relative strengths of these excitations are compared. It is shown that the multiple-peak structures in the dynamic structure factor S(q,\ensuremath{\omega}) as calculated in the GNS theory are associated with pure multipair effects. The essential role multipair effects play in determining the high-energy-transfer tail of S(q,\ensuremath{\omega}) is stressed, and the way this affects the frequency moment sum rules is examined. Other properties of these sum rules, when they are generated within approximate but conserving theories, are discussed.

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.

Results for the diagonal element of the second-order density matrix (pair function) and for the momentum distribution of an interacting electron gas are presented. In the low density limit, it is shown that, to a similar degree of approximation, the first-order density matrix and hence the momentum distribution may be obtained. In the highdensity problem, the pair function has also been calculated using the Green's function approach. From the information available in these high and low density limits, the range of usefulness of the concept of the Fermi surface in an interacting electron gas is discussed. Also, results of varia. tional calculations on the stability of Cverhauser spindensity waves in metals are reported. It is concluded that such waves reduce the energy below the Hartree-Fock value only when the interaction becomes very strong, or equivalently when the density is very low. Some preliminary work on the correlation problem in a non-uniform electron gas is also briefly discussed. (auth)