The medium: correlation and response functions

Abstract

In the previous chapters the response of the medium to the electromagnetic waves was described in a phenomenological manner in terms of the frequency and wavevector dependent complex dielectric constant and conductivity. Our task at hand now is to relate these parameters to the changes in the electronic states of solids, brought about by the electromagnetic fields or by external potentials. Several routes can be chosen to achieve this goal. First we derive the celebrated Kubo formula: the conductivity given in terms of current–current correlation functions. The expression is general and not limited to electrical transport; it can be used in the context of different correlation functions, and has been useful in a variety of transport problems in condensed matter. We use it in the subsequent chapters to discuss the complex, frequency dependent conductivity. This is followed by the description of the response to a scalar field given in terms of the density–density correlations. Although this formalism has few limitations, in the following discussion we restrict ourselves to electronic states which have well defined momenta. In Section 4.2 formulas for the so-called semiclassical approximation are given; it is utilized in later chapters when the electrodynamics of the various broken symmetry states is discussed. Next, the response to longitudinal and transverse electromagnetic fields is treated in terms of the Bloch wavefunctions, and we derive the well known Lindhard dielectric function: the expression is used for longitudinal excitations of the electron gas; the response to transverse electromagnetic fields is accounted for in terms of the conductivity.