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Physical review. B, Condensed matter

Electron and hole self-energy contributions to the dynamic structure factor in interacting-electron systems

F. Green, David Neilson, J. Szymański

Published Jan 1, 1987• 14 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.35.124

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arXiv:cond-mat/9811356•Physical review. B, Condensed matter•Aug 15, 1999•0 citations
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.
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.
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.
There is increasing interest in many-body perturbation theory as a practical tool for the calculation of ground-state properties. As a consequence, unambiguous sum rules such as the conservation of particle number under the influence of the Coulomb interaction have acquired an importance that did not exist for calculations of excited-state properties. In this paper we obtain a rigorous, simple relation whose fulfilment guarantees particle-number conservation in a given diagrammatic self-energy approximation. Hedin's G0W0{G}_{0}{W}_{0}G0​W0​ approximation does not satisfy this relation and hence violates the particle-number sum rule. Very precise calculations for the homogeneous electron gas and a model inhomogeneous electron system allow the extent of the nonconservation to be estimated.
The one-particle Green function of a many-electron system is traditionally formulated within the self-energy picture. A different formalism was recently proposed, in which the self-energy is replaced by a dynamical exchange-correlation field, which acts on the Green function locally in both space and time. It was found that there exists a fundamental quantity, referred to as the dynamical exchange-correlation hole, which can be interpreted as effective density fluctuations induced in a many-electron system when a hole or an electron is introduced into the system, as in photoemission and inverse photoemission experiments. The dynamical exchange-correlation potential is simply the Coulomb potential of this exchange-correlation hole, which fulfils a sum rule and an exact constraint, identical to those satisfied by the static exchange-correlation hole in density-functional theory. The proposed formalism has been applied to a number of model systems such as the half-filled one-dimensional Hubbard model, the one-dimensional antiferromagnetic Heisenberg model, and the single-impurity Anderson model. The dynamical exchange-correlation hole and field of the homogeneous electron gas have also been studied with the view of constructing a density-functional approximation such as the local-density approximation. The availability of simple but accurate approximations for the exchange-correlation potential would circumvent costly computations of the traditional self-energy. The formalism may also provide new perspectives and insights into the many-body problem.
We evaluate the one-particle and double-occupied Green functions for the Hubbard model at half-filling using the moment approach of Nolting [Z. Phys. 255, 25 (1972); Grund Kurs: Theoretische Physik. 7 Viel-Teilchen-Theorie (Verlag Zimmermann-Neufang, Ulmen, 1992)]. Our starting point is a self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ which has a single pole, $\ensuremath{\Omega}(\stackrel{\ensuremath{\rightarrow}}{k}),$ with spectral weight, $\ensuremath{\alpha}(\stackrel{\ensuremath{\rightarrow}}{k}),$ and quasiparticle lifetime, $\ensuremath{\gamma}(\stackrel{\ensuremath{\rightarrow}}{k})$ [J. J. Rodr\'{\i}guez-N\'u\~nez and S. Schafroth, J. Phys. Condens. Matter 10, L391 (1998); J. J. Rodr\'{\i}guez-N\'u\~nez, S. Schafroth, and H. Beck, Physica B (to be published); (unpublished)]. In our approach, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ becomes the central feature of the many-body problem and due to three unknown $\stackrel{\ensuremath{\rightarrow}}{k}$ parameters we have to satisfy only the first three sum rules instead of four as in the canonical formulation of Nolting [Z. Phys. 255, 25 (1972); Grund Kurs: Theoretische Physik. 7 Viel-Teilchen-Theorie (Verlag Zimmermann-Neufang, Ulmen, 1992)]. This self-energy choice forces our system to be a non-Fermi liquid for any value of the interaction, since it does not vanish at zero frequency. The one-particle Green function, $G(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ shows the fingerprint of a strongly correlated system, i.e., a double peak structure in the one-particle spectral density, $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ vs $\ensuremath{\omega}$ for intermediate values of the interaction. Close to the Mott insulator-transition, $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ becomes a wide single peak, signaling the absence of quasiparticles. Similar behavior is observed for the real and imaginary parts of the self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}).$ The double-occupied Green function, ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega}),$ has been obtained from $G(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ by means of the equation of motion. The relation between ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega})$ and the self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ is formally established and numerical results for the spectral function of ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ ${\ensuremath{\chi}}^{(2)}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})\ensuremath{\equiv}\ensuremath{-}(1/\ensuremath{\pi}){limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\delta}}{0}^{+}}\mathrm{Im}[{G}_{2}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})],$ are given. Our approach represents the simplest way to include (1) lifetime effects in the moment approach of Nolting, as shown in the paper, and (2) Fermi or/and marginal Fermi liquid features as we discuss in the conclusions.