9 - Electrons with Coulomb interaction

Summary

Using the techniques described in the previous chapter, we can in a unified way discuss properties of the electron gas with Coulomb interaction and consider such effects as optical response, screening, plasmons, etc. In many textbooks these properties are obtained using a variety of methods. The virtue of the Green function method is its universality and, I would say, not much simpler, but standardized form. This method permits one to obtain all the properties mentioned above in the form of one general expression, and it also gives the possibility to generalize the results quite easily to the cases of low-dimensional (1d, 2d) systems, or to take into account the details of the band structure of the material, etc. But more important is the fact that it leads naturally to a number of special interesting consequences which would be rather difficult to obtain with the usual classical methods. In this and in the next two chapters I will demonstrate how to reproduce, using this method, the familiar results such as Debye screening or the plasmon energy, but I will mostly concentrate on less frequently discussed effects which are quite naturally obtained using this technique.

Dielectric function, screening: random phase approximation

We start by studying the form of the effective electron–electron interaction in metals. The ordinary Coulomb interaction V(q) = 4πe²/q² is modified by the reaction of the electronic system. The first, well-known effect is just the screening of the electric charge. But there are other interesting effects as well.