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An Exact Dynamics of an Electron Gas at r_s≈3.5 and a Short-wavelength Limit(New Developments in Statistical Physics Similarities in Diversities,YITP Workshop)

Hong, Jongbae

Published Jul 20, 1993• 0 citations
OpenAlex

Abstract

We show the exact dynamic structure factor S(κ, ω) of the many-electron system by calculating the relaxation function. Our S(κ, ω) is asymptotically exact as κ→∞ and at a particular metallic density r^*_s≈3.5. From the exact solution, we derive two static relations, x^2=3/5y, and x^2=3/16ω^2_ρ[1-g(0)] valid at r^*_s, where x and y are averages of the one-particle kinetic energy and its square, respectively. Our S(κ, ω) is compared with an experimental measurement for Li at κ=2.08κ_F.

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We show the exact dynamic structure factor S(κ, ω) of the many-electron system by calculating the relaxation function. Our S(κ, ω) is asymptotically exact as κ→∞ and at a particular metallic density r^_s≈3.5. From the exact solution, we derive two static relations, x^2=3/5y, and x^2=3/16ω^2_ρ[1-g(0)] valid at r^_s, where x and y are averages of the one-particle kinetic energy and its square, respectively. Our S(κ, ω) is compared with an experimental measurement for Li at κ=2.08κ_F.
The exact nonequilibrium time evolution of the momentum distribution for a finite many-particle system in one dimension with a linear energy dispersion coupled to optical phonons is presented. For distinguishable particles the influence function of the phonon bath can be evaluated also for a finite particle density in the thermodynamic limit. In the case of fermions the exact fulfillment of the Pauli principle involves a sum over permutations of the electrons and the numerical evaluation is restricted to a finite number of electrons. In the dynamics the antisymmetry of the wave function shows up in the obvious Pauli blocking of momentum states as well as more subtle interference effects. The model shows the expected physical features known from approximate treatments of more realistic models for the relaxation in the energy regime far from the bottom of the conduction band and provides an excellent testing ground for quantum kinetic equations.
The dynamic structure function S(k,\ensuremath{\omega}) and the dielectric function \ensuremath{\epsilon}(k,\ensuremath{\omega}) of the two-dimensional electron gas is calculated phenomenologically at rs{\mathit{r}}_{\mathit{s}}rs​=1 and 5 in the memory-function formalism. The density-density response function \ensuremath{\chi}(k,\ensuremath{\omega}) is expressed in terms of that of a noninteracting system and an effective potential. The interaction part of the second-order memory function that determines the effective potential is approximated by a Gaussian in time domain. We use a consistent set of highly accurate ground-state static properties as input, coming from a Green's-function Monte Carlo calculation. The calculated S(k,\ensuremath{\omega}) shows a single-broad-peak structure at intermediate k, which is from the case in three dimensions.
The exact nonequilibrium time evolution of the momentum distribution for a finite many particle system in one dimension with a linear energy dispersion coupled to optical phonons is presented. For distinguishable particles the influence function of the phonon bath can be evaluated also for a finite particle density in the thermodynamic limit. In the case of fermions the exact fulfillment of the Pauli principle involves a sum over permutations of the electrons and the numerical evaluation is restricted to a finite number of electrons. In the dynamics the antisymmetry of the wavefunction shows up in the obvious Pauli blocking of momentum states as well as more subtle interference effects. The model shows the expected physical features known from approximate treatments of more realistic models for the relaxation in the energy regime far from the bottom of the conduction band and provides an excellent testing ground for quantum kinetic equations.
We use the Ceperley and Alder Green's-function Monte Carlo data for the static form factor, S(q)S(q)S(q), of an electron liquid to calculate the static local-field correction, G(q)G(q)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)G(q)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.
We have derived the exact limiting behavior of the many-body local fields ${G}_{\ifmmode\pm\else\textpm\fi{}}$(q,\ensuremath{\omega}) in a two-dimensional electron gas. At large wave vectors and finite frequencies these functions can be expressed in terms of g(0), the value at the origin of the pair correlation function of the electron gas. We find in particular that in this limit G+{G}_{+}G+​(\ensuremath{\infty},\ensuremath{\omega})=1-g(0), and ${G}_{\mathrm{\ensuremath{-}}}$(\ensuremath{\infty},\ensuremath{\omega})=g(0). Our results provide new insight into the problem of the many-body vertex corrections in this interacting system.
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.