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Physical review. B, Condensed matter

Exact limits of the many-body local fields in a two-dimensional electron gas

Giuseppe E. Santoro, Gabriele F. Giuliani

Published Mar 15, 1988• 43 citations
Publisher

Abstract

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 ${G}_{+}$(\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.

DOI
10.1103/physrevb.37.4813

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We have derived the exact limiting behavior of the many-body local fields G\ifmmode±\else\textpm\fi{G}_{\ifmmode\pm\else\textpm\fi{}}G\ifmmode±\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\ensuremath−{G}_{\mathrm{\ensuremath{-}}}G\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.
We analyze the spin and charge susceptibilities of a spin-polarized electron gas subject to a weak space- and time-dependent field coupled to the electronic spins, with the main attention given to the case of space dimensionality D=2.D=2.D=2. Exchange and correlations enter the dynamic susceptibilities through spin-dependent local-field factors ${G}_{\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}(\mathbf{q},\ensuremath{\omega}).$ For an arbitrary degree of polarization, we determine the exact analytic expressions of ${G}_{\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}(\mathbf{q},\ensuremath{\omega})$ in two limiting cases: (i) the limit of large wave number q at finite frequency $\ensuremath{\omega},$ already considered in D=3D=3D=3 by D. C. Marinescu and J. J. Quinn [Phys. Rev. B 56,56,56, 1114 (1997)]; and (ii) the static limit at small wave number. In the latter case, we obtain thermodynamic sum rules of general validity in both dimensionalities. Our work gives insight into many-body vertex corrections and basic information for calculations of the effects of the electron-electron interactions on physical properties.
We discuss interaction effects for the one-dimensional electron gas with a repulsive \ensuremath{\delta}-function interaction potential by using the random-phase approximation and a local-field correction. Analytical results for the local-field correction of charge-density fluctuations and spin-density fluctuations are obtained. The ground-state energy is found to be in better agreement with the exact result than the ground-state energy calculated within the random-phase approximation. We calculate the pair-correlation functions for electrons with parallel and antiparallel spins and the paramagnetic susceptibility. The energies of the collective density modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q) and the collective spin modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q) are calculated and compared with the energy of the electron-hole excitations ${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q): ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q\ensuremath{\rightarrow}0). The critical exponents for the long-distance behavior of correlation functions are found to be described by local-field corrections. We compare our results for the critical exponents with recent results obtained by bosonization techniques and the conformal field theory.
We calculate the pair distribution function, g ( r ), in a two-dimensional electron gas and derive a simple analytical expression for its value at the origin as a function of r s . Our approach is based on solving the Schrödinger equation for the two-electron wavefunction in an appropriate effective potential, leading to results that are in good agreement with quantum Monte Carlo data and with the most recent numerical calculations of g (0) (Bulutay and Tanatar 2002 Phys. Rev. B 65 195116). We also show that the spin-up spin-down correlation function at the origin, , is mainly independent of the degree of spin polarization of the electronic system.
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.
We extend Niklasson's method in order to study the spin response of an electron liquid in a weak sinusoidal magnetic field. The many-body correction $G\ensuremath{-}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ caused by exchange and correlation is introduced to describe the correct wave-vector- and frequency-dependent spin susceptibility. The exact behavior of $G\ensuremath{-}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ in the large-qqq limit is related to the pair distribution function $g(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ at r=0r=0r=0. ${G}_{\ensuremath{-}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})\ensuremath{\rightarrow}\frac{[4g(0)\ensuremath{-}1]}{3}$, as $q\ensuremath{\rightarrow}\ensuremath{\infty}$. At metallic densities this value is negative, opposite in sign to the limit at small wave vector. Thus the spin susceptibility for large wave vectors is suppressed, rather than enhanced, by many-body effects.
The quasi-particle spectral weight at the Fermi level (Z(F)) in the two-dimensional (2D) electron gas is determined as a function of density. A model local field correction factor (G(q)) possessing the correct features both in the small and large q limits is introduced and used to calculate the Z(F), by evaluating the electron self-energy within the GW approximation. It is found that the Z(F) calculated using the present G(q) is smaller than the one calculated using the Hubbard G(q) implying that the correlation interaction makes the system more localized. Our result also indicates that high T(c) superconductors, if described by a 2D electron gas system, will exhibit a small but nonzero (approximately 0.2) quasi-particle spectral weight at the Fermi level.