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CRC Press eBooks

Response and Correlation in Homogeneous Electron Systems

Philippe Nozières, David Pines

Published Mar 5, 2018• 0 citations
Publisher

Abstract

In the previous chapter we have used Poisson equations to define the frequency–and wave vector-dependent dielectric response function, ϵ( q , ω). Not surprisingly, there is an intimate connection between this measure of the longitudinal response and the density-density response function introduced in Chapter 2 . We now proceed to develop and exploit this connection.

DOI
10.1201/9780429495717-5

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In the previous chapter we have used Poisson equations to define the frequency–and wave vector-dependent dielectric response function, ϵ( q , ω). Not surprisingly, there is an intimate connection between this measure of the longitudinal response and the density-density response function introduced in Chapter 2 . We now proceed to develop and exploit this connection.
Introduction There are countless situations in physics when one is interested in calculating the response of a system to a small time-dependent perturbation acting on it. With some luck the response can be expanded in a power series of the strength of the perturbation, so that, to first order, it is a linear function of the latter. To compute this function is the objective of the linear response theory (LRT). Linear response theory has many important applications to the study of electronic matter. Virtually all interactions of electrons with experimental probes (electromagnetic fields, beams of particles) can be regarded as small perturbations to the system: if they were not, one would not be probing the system, but the system modified by the probe. Consequently, the results of these experiments can be expressed in terms of linear response functions, which are properties of the unperturbed system. In particular it will turn out that the analytic structure of these functions is entirely determined by the eigenvalues and eigenfunctions of the unperturbed system. Conversely, a measure of the linear response as a function of the frequency of the perturbation enables us to determine the excitation energies of the system. Beside being a cornerstone for the theory of single-particle properties to be developed in Chapter 8, the linear response functions also provide invaluable information in their own right. For example, as we will discuss in Chapter 5, the extent to which an external electrostatic potential is reduced by screening is controlled by the dynamical dielectric function which, in turn, is determined by the density–density response function.
The chapter begins with a brief survey of concepts and techniques of experimental condensed matter physics. It will be shown how correlation functions provide a bridge between concrete experimental data and the theoretical formalism developed in previous chapters. Specifically we discuss – an example of outstanding practical importance – how the response of many-body systems to various types of electromagnetic perturbation can be described in terms of correlation functions and how these functions can be computed by field theoretical means. In the previous chapters we have introduced important elements of the theory of quantum many-body systems. Perhaps most importantly, we have learned how to map the basic microscopic representations of many-body systems onto effective low-energy models. However, to actually test the power of these theories, we need to understand how they can be related to experiment. This will be the principal subject of the present chapter. Modern condensed matter physics benefits from a plethora of sophisticated and highly refined techniques of experimental analysis including the following: electric and thermal transport; neutron, electron, Raman, and X-ray scattering; calorimetric measurements; induction experiments; and many more (for a short glossary of prominent experimental techniques, see Section 7.1.2 below). While a comprehensive discussion of modern experimental condensed matter would reach well beyond the scope of the present text, it is certainly profitable to attempt an identification of some structures common to most experimental work in many-body physics.
The chapter begins with a brief survey of concepts and techniques of experimental condensed matter physics. It will be shown how correlation functions provide a bridge between concrete experimental data and the theoretical formalism developed in previous chapters. Specifically we will discuss – an example of outstanding practical importance – how the response of many-body systems to various types of electromagnetic perturbations can be described in terms of correlation functions and how these functions can be computed by field theoretical means. In the previous chapters we have introduced important elements of the theory of quantum many-body systems. Perhaps most importantly, we have learned how to map the basic microscopic representations of many-body systems onto effective low-energy models. However, to actually test the power of these theories, we need to understand how they can be related to experiment. This will be the principal subject of the present chapter. Modern condensed matter physics benefits from a plethora of sophisticated and highly refined techniques of experimental analysis including the following: electric and thermal transport; neutron, electron, Raman, and X-ray scattering; calorimetric measurements; induction experiments; and many more (for a short glossary of prominent experimental techniques, see Section 7.1.2 below). While a comprehensive discussion of modern experimental condensed matter would reach well beyond the scope of the present text, it is certainly profitable to attempt an identification of some structures common to most experimental work in many-body physics.
Motivated by recent experiments, we append long ranged Coulomb interactions to dominant strong local correlations and study the resulting t-J-VC model for the 2-dimensional cuprate materials. This model includes the effect of short ranged Hubbard–Gutzwiller–Kanamori type correlations and long ranged Coulomb interactions on tight binding electrons. We calculate the {q→,ω} dependent charge density fluctuations in this model using the extremely correlated Fermi liquid theory characterized by quasiparticles with very small weight Z. We develop a novel set of formulae to represent the dynamical charge susceptibility and the dielectric function, using a version of the charge-current continuity equation for a band system, valid for arbitrary q→. Combining these ingredients, we present results for the irreducible dynamical charge susceptibility χ˜ρρ(q→,ω), (longitudinal) dielectric function ɛ(q→,ω), current susceptibility χ˜JJ(q→,ω), conductivity σ(q→,ω), and the plasma frequency for any q→. We also present calculations for the first moment of the structure function and discuss a characteristic energy scale Ωp(q→) which locates a peak in Imχ˜ρρ(q→,ω).
The dielectric screening function epsilon (k, omega ) of the fluid of charged bosom at zero temperature is evaluated in a range of low-to-intermediate coupling strength, in view of recent data on static screening and fluid structure from quantum Monte Carlo methods. Correlations beyond the random phase approximation are included within a class of approximations which are well known for the electron fluid, i.e. by introducing a frequency-independent local field factor to be determined through self-consistency requirements connecting various aspects of the physics contained in the dielectric function. The static dielectric function epsilon (k,0) is negative over a range of wavenumbers at all values of the density, leading to oscillatory screening of a foreign charge and to an effective long-range attraction between equi-charged impurities. Quantitative agreement with the Monte Carlo data on static screening is achieved by imposing self-consistency on the compressibility of the fluid in addition to self-consistency on the pair distribution function. Good agreement is also obtained on the pair distribution function and the correlation energy. Within the present class of approximations, the dispersion relation of longitudinal excitations takes the Feynman form, starting at the plasma frequency with a negative dispersion coefficient and going through a minimum before ending at the single-particle recoil frequency.