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Electron Liquid Beyond the Random Phase Approximation

Mehmet Tomak, Ceyhun Bulutay

Published Jan 1, 1999• 1 citation
OpenAlex

Abstract

A deeper understanding of low-dimensional and small-scale electronic systems requires an accurate characterization of the many-electron effects. Based on the electron liquid (EL) model, these many-body effects in the three-dimensional (3D) and two-dimensional (2D) electronic systems are dealt going beyond the random phase approximation within the zero-temperature framework. The dielectric formulation of the many-body problem is recapitulated and the longitudinal dielectric function of the EL is obtained using the self-consistent local-field correction scheme, known as STLS, under a general spin-valley degeneracy. Extensive explicit expressions are provided for the use of other researchers. The performance of this formalism is compared in both 3D and 2D by the quantum Monte Carlo data and the pseudopotential approach introduced by Pines and co-workers. STLS is observed to be a highly satisfactory technique, with some reservations on the long-wavelength behaviour. Ground-state energy, correlation energy and the isothermal compressibility of the 2D EL are computed using a variety of approaches. Negative compressibility and overscreening are discussed in the light of the recent experiments.

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A deeper understanding of low-dimensional and small-scale electronic systems requires an accurate characterization of the many-electron effects. Based on the electron liquid (EL) model, these many-body effects in the three-dimensional (3D) and two-dimensional (2D) electronic systems are dealt going beyond the random phase approximation within the zero-temperature framework. The dielectric formulation of the many-body problem is recapitulated and the longitudinal dielectric function of the EL is obtained using the self-consistent local-field correction scheme, known as STLS, under a general spin-valley degeneracy. Extensive explicit expressions are provided for the use of other researchers. The performance of this formalism is compared in both 3D and 2D by the quantum Monte Carlo data and the pseudopotential approach introduced by Pines and co-workers. STLS is observed to be a highly satisfactory technique, with some reservations on the long-wavelength behaviour. Ground-state energy, correlation energy and the isothermal compressibility of the 2D EL are computed using a variety of approaches. Negative compressibility and overscreening are discussed in the light of the recent experiments.
The Thomas-Fermi (TF) approximation for the static dielectric function of a three-dimensional (3D) electron liquid can be derived by minimizing the TF local-density approximation for the kinetic-energy functional. Here we show that this connection between energy functionals and model dielectric functions is not an artifact, but a general paradigm. Four examples are worked out in detail, by calculating the model dielectric functions that follow, respectively, from (i) exchange corrections to TF theory in 3D, i.e., TF-Dirac theory, (ii) further correlation corrections to TF-Dirac theory in 3D, (iii) TF theory in two dimensions (2D), and (iv) exchange corrections to TF theory in 2D. Each of these cases has certain interesting features, revealing connections between independent many-body methods, that are discussed in some detail. As a by-product of these investigations we also find that a common textbook statement about the long-wavelength $(k\ensuremath{\rightarrow}0)$ limit of the random-phase approximation is not fully correct.
We present a quantitative study of the quasiparticle properties of a three-dimensional electron Fermi liquid. Our approach is based on the theory of the many-body local field factors which we use to include vertex corrections associated with charge and spin fluctuations. Extensive use is made of the results of recent quantum Monte Carlo calculations. Several models for the wave-vector dependence of the spin-antisymmetric many-body local field factor ${G}_{\ensuremath{-}}$, for which no numerical results are currently available, are discussed and compared. Both the real and imaginary parts of the self-energy as well as the quasiparticle renormalization constant and the enhancement of the effective mass are calculated in the experimentally accessible range of electron densities. The results obtained by means of the on-shell approximation are critically compared with those given by a self-consistent solution of the Dyson equation. An ultraviolet catastrophe in the Coulomb-hole part of the self-energy is identified and a satisfactory resolution of this impasse is presented. The same many-body local field factors are also used to obtain the Landau interaction function. This allows us to obtain both the spin susceptibility and the proper compressibility of the system.
The parametrized form of local-field corrections is introduced to construct the dielectric function and the nonlocal spin susceptibility that describe the dielectric properties of a paramagnetic degenerate electron liquid in two dimensions beyond the random-phase approximation (RPA). The parameters that take into account the spin-parallel and spin-antiparallel electron-electron interactions at short distances are determined in such a way that the response functions correctly give the Monte Carlo values for the compressibility and the spin susceptibility, respectively, in the long-wavelength limit. The local-field correction thus obtained is compared with that in the Hubbard approximation; the static dielectric function and the spin susceptibility are also compared with those in the RPA. It is found that the sign of the static dielectric function depends on both the dimensionless coupling constant and the wave number, which is a feature markedly different from the RPA results.
The problem of the Friedel oscillations in two- and three-dimensional electron Fermi liquids is examined by means of the many-body local fields theory as aided by the most recent results of accurate numerical studies based on the quantum Monte Carlo method. Within linear response, an exact answer is obtained for the amplitude of the electron density distortion due to both short- and long-ranged impurity potentials. It is discussed how a measurement of the local environment of an impurity embedded in an otherwise homogeneous electron liquid can be used to characterize the systems as a Fermi liquid as well as to extract the magnitude of the static many-body corrections beyond the random phase approximation for wave vector equal to 2kF2{k}_{F}2kF​.
Motivated by a number of recent experimental studies we have revisited the problem of the microscopic calculation of the quasiparticle self-energy and many-body effective mass enhancement in an unpolarized two-dimensional electron liquid. Our systematic study is based on the many-body local field theory and takes advantage of the results of the most recent diffusion Monte Carlo calculations of the static charge and spin response of the electron liquid. We report extensive calculations of both the real and imaginary parts of the quasiparticle self-energy. We also present results for the many-body effective mass enhancement and the renormalization constant over a broad range of electron densities. In this respect we critically examine the relative merits of the on-shell approximation, commonly used in weak coupling situations versus the actual self-consistent solution of the Dyson equation. We show that already for ${r}_{s}\ensuremath{\simeq}3$ and higher, a solution of the Dyson equation proves necessary in order to obtain a well-behaved effective mass. Finally we find confirmation that the inclusion of both charge- and spin-density fluctuations beyond the random phase approximation is indeed crucial to get reasonable agreement with recent measurements.
Two-dimensional electron liquid (2D EL) at full Fermi degeneracy is revisited, giving special attention to the spin-polarization effects. First, we extend the recently proposed classical-map hypernetted-chain (CHNC) technique to the 2D EL, while preserving the simplicity of the original proposal. An efficient implementation of CHNC is given utilizing Lado's quadrature expressions for the isotropic Fourier transforms. Our results indicate that the paramagnetic phase stays to be the ground state until the Wigner crystallization density, even though the energy separation with the ferromagnetic and other partially polarized states become minute. We analyze compressibility and spin stiffness variations with respect to density and spin polarization, the latter being overlooked until now. Spin-dependent static structure factor and pair-distribution functions are computed; agreement with the available quantum Monte Carlo data persists even in the strong-coupling regime of the 2D EL.