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Physical review. B, Condensed matter

Exact behavior of the density and spin susceptibilities of a Fermi liquid for large wave vectors: Derivation from diagrammatic many-body theory

Giovanni Vignale

Published Oct 1, 1988• 23 citations
Publisher

Abstract

The exact asymptotic behavior of the density and spin response functions of a uniform Fermi system for large wave vectors is derived from diagrammatic many-body theory. Various exact relationships between the self-energy, the particle-hole interaction, and the vertex corrections are derived for the first time and applied to this problem. The final results agree with those provided by the kinetic equation approach. Different behavior of the polarization potentials and local-field factors for interactions that decrease faster or slower than 1/${q}^{2}$ for large wave vector q, is pointed out.

DOI
10.1103/physrevb.38.6445

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The exact asymptotic behavior of the density and spin response functions of a uniform Fermi system for large wave vectors is derived from diagrammatic many-body theory. Various exact relationships between the self-energy, the particle-hole interaction, and the vertex corrections are derived for the first time and applied to this problem. The final results agree with those provided by the kinetic equation approach. Different behavior of the polarization potentials and local-field factors for interactions that decrease faster or slower than 1/q2{q}^{2}q2 for large wave vector q, is pointed out.
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.
Dynamical density and spin responses of a two-dimensional fermion system are studied with the self-consistent second-order perturbation theory using a model Hamiltonian. Special attention is paid to the density response at a large wave vector, where it was reported that a sharp mode develops below the particle-hole continuum in two-dimensional liquid 3{}^{3}3He. We find that at a large wave vector, a peak in the dynamical density structure factor shifts to lower energy as interaction sets in, but it is not sharp enough to represent a well-defined mode because of the strong damping effect of the quasiparticles. We point out that it is necessary to treat the large damping effect properly in explaining the propagating mode found in two-dimensional liquid 3{}^{3}3He.
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 main topic of the present thesis is represented by the many-body effects which characterize the physical behavior of an electron liquid in various realizations. We begin by studying the problem of the response of an otherwise homogeneous electron liquid to the potential of an impurity embedded in its bulk. The most dramatic consequence of this perturbation is the existence of so called Friedel density oscillations. We present calculations of their amplitude valid in two as well as in three dimensions. The second problem we will discuss is that of the correlation effects in a three dimensional electron liquid in the metallic density regime. A number of quasiparticle properties are evaluated: the electron self-energy, the quasiparticle effective mass and the renormalization constant. We also present an analysis of the effective Landé g-factor as well as the compressibility. The effects of the Coulomb interactions beyond the random phase approximation have been treated by means of an approach based on the many-body local field factors theory and by utilizing the latest numerical results of Quantum Monte Carlo numerical simulations. The final chapter includes the results of our extensive work on various aspects regarding the two dimensional Fermi liquid in the presence of linear Rashba spin-orbit coupling. By using a number of many-body techniques, we have studied the interplay between spin-orbit coupling and electron-electron interaction. After proving an extension to the famous Overhauser Hartree-Fock instability theorem, a considerable amount of work will be presented on the problem of the density and spin response functions. For the study of the spin response, we will present the results of extensive numerical calculations based on the time dependent mean field theory approach.
The calculation of the induced spin density of a normal fermion system, such as the electron gas, in the limit of small wave numbers, is carried beyond the random phase approximation (R.P.A.) by the method of canonical transformation so as to include the first non-R.P.A. corrections. An expression for the induced spin density in the same limit, exact to all orders of particle-particle coupling, is then derived by more general methods of many-body perturbation theory. It depends only on the knowledge, to all orders of interparticle coupling, of the effective mass and the forward scattering of quasi-particles at the Fermi surface, and also yields the results of the extended R.P.A. on appropriate expansion. In the limit of zero wave number, the resulting expression for the magnetic susceptibility is found identical with that deduced by Landau from a phenomenological basis and may be regarded as an additional confirmation of the microscopic validity of the theory of the Fermi liquid.