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.
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.
There exists a regime of overlap between the equilibrium and non-equilibrium behavior of a system, the non-equilibrium behavior of weakly perturbed states. When a system is perturbed ever so slightly, its response will be linear in the perturbation, say the current of the conduction electrons in a metal will be proportional to the strength of the applied electric field. This regime is called the linear response regime, and though the system is in a non-equilibrium state all its characteristics can be inferred from the properties of its equilibrium state. In the next chapter we shall go beyond the linear regime by showing how to obtain quantum kinetic equations. The kinetic-equation approach to transport is a general method, and allows in principle nonlinear effects to be considered. However, in many practical situations one is interested only in the linear response of a system to an external force. The linear response limit is a tremendous simplification in comparison with general non-equilibrium conditions, and is the subject matter of this chapter. In particular the linear response of the density and current of an electron gas are discussed. The symmetry properties of response functions, and the fluctuation–dissipation theorem are established. Lastly we demonstrate how correlation functions can be measured in scattering experiments, as illustrated by considering neutron scattering from matter. Needless to say, in measurements of (say) the current in a macroscopic body, far less information in the current correlation function is probed. Linear response In this section we consider the response of an arbitrary property of a system to a general perturbation.
Introduction The calculation of the linear response functions of an electron liquid is a very important, but obviously extremely difficult task. Even after many years of study, and in spite of much progress, a complete solution of the problem is still lacking. In preparation to the study of this difficult problem, we consider in this chapter the main response functions of the non-interacting electron gas, namely, the density–density, spin–spin, and current–current response functions, all of which can be calculated analytically. It turns out that understanding the response of the non-interacting electron gas is an essential prerequisite for understanding the richer and more complex response of the interacting liquid. Indeed, one of the most fruitful ideas in many-body physics is that the response of an interacting system can be pictured as the response of a non-interacting system to an effective self-consistent field, which depends on global properties such as the particle density, the density matrix, the current–density etc. ‥ This is true, in particular, for every dynamical mean field theory , which can be derived from the corresponding static mean field theory (such as the HF theory) with the help of the techniques introduced in Section 4.7. In the first part of this chapter we present a detailed study of the response functions of the homogeneous non-interacting electron gas in three, two, and one dimension. Electron–impurity scattering is included only at the most elementary level, and only to demonstrate its main effect, namely, the emergence of diffusion in the dynamics of density fluctuations. In the next chapter we shall present approximations for the effective self-consistent field.
Many properties of chemical or materials systems are characterized by how they respond to external perturbations. When the external perturbation is sufficiently small that only the leading-order perturbation is important, this is known as the linear response regime. It is in some sense analogous to the linear stability analysis of a dynamical system or that in fluid mechanics. This chapter will give an in-depth discussion of the linear response theory for both static and dynamic perturbations and their applications. The analysis will be from the perspective of the perturbation theory of Green's functions. Towards the end of the chapter, the perturbation theory of a many-body quantum system and its connection to DFT will also be briefly addressed.
Introduction and guide to the chapter Linear response functions contain a wealth of information about the physical properties of a many-body system. In the case of the electron liquid, for example, the density–density response function provides a unified framework for the understanding of different phenomena such as static screening, effective interaction, collective modes, electron energy loss spectra (inelastic scattering of electrons), and Raman spectra (inelastic scattering of photons). The spin–spin response function provides the corresponding information for the spin density fluctuations as probed, for example, by cross-polarized Raman scattering experiments, in which the incident and scattered photon have perpendicular polarizations, or by spin-flip electron energy loss spectroscopy, in which the incoming and outgoing electrons have opposite spin orientations. As we have seen, the Hartree–Fock approximation describes each electron as an independent particle moving in a self-consistent field generated by all the other electrons. There is no correlation between this self-consistent field and the instantaneous position of the electron. Reality is however quite different: whenever an electron moves, it acts on the surrounding electrons, causing a collective disturbance which eventually feeds back on its own motion. In order to study these effects we need, first of all, to learn how the electron liquid as a whole responds to disturbances caused by the charge, the spin, and the current of a single electron. The linear response theory of Chapter 3 provides the natural framework for this description. Strictly speaking, the linear response functions describe the readjustments of the electronic density, spin, or current, in response to externally controlled fields.
The general expressions for the number density and current are first considered under the action of an external time-dependent perturbing potential of arbitrary strength and cast in terms of the single-particle lesser Green’s function. The expansion of the number density up to linear order in the perturbing potential is then considered, yielding the density–density correlation function of linear-response theory. A connection is also considered with the temperature correlation function of the Matsubara formalism via an analytic continuation in frequency space.
Linear response theorySuppose that a solid is hit with a hammer.Sound waves will propagate outwards from the point of contact.How is the frequency of the sound wave related to its wave number?How does a light wave propagate through plasma?What happens when a charge impurity is embedded in an electrically neutral medium?Is it screened, and if so how is that screening described quantitatively?If a medium is disturbed by a small amount one might expect its response also to be small.The quantitative formalism for dealing with small disturbances is called linear response theory.The beauty of the theory is that the response of the system can be expressed as a folding of the external source causing the disturbance with a response function that is computed using equilibrium correlation functions not dependent on the strength of the external source.Therefore, details of the internal dynamics of the thermodynamic system can be studied using weak external probes.Other areas of science where linear response theory has proven to be extremely useful are quite extensive, and include x-ray scattering from crystals and molecules, electron scattering from protons and nuclei, and sound waves generated by earthquakes propagating through the earth's interior. Linear response to an external fieldSuppose we apply some external field to our system, which is initially in equilibrium.The goal of linear response theory is to calculate the change in the ensemble average value of any operator Y (x, t) caused by the external field, to first order in that external field.Let