Linear Response Theory

Abstract

A general formalism for describing the response of a system to weak external perturbations can be developed in the linear approximation. In this chapter we consider the response of electrons to weak electromagnetic fields, while the case of nonmechanical perturbations (for example, temperature gradient) is discussed in Chapter 5. Using a solution of the linearized equation for the density matrix, one can write exact expressions for the kinetic coefficients connecting induced currents to the external fields. Such coefficients are expressed through the equilibrium characteristics of the system so that the linear-response problem is reduced to a statistical averaging (or to simple integrations in the case of non-interacting quasiparticles). The most effective and unified approach to such averaging is based upon diagrammatic expansion of the Green’s function. The simplest variant of this method, developed for electronimpurity systems, is described in this chapter. Another approach assumes expression of the Green’s function through the path integral, when the quantum-mechanical and statistical averaging can be done separately. Both these methods allow one to describe the case of strong scattering, when the quantum kinetic equation with the collision integral (7.17) is not valid.