Introduction

In the last chapter we introduced some new techniques of calculation which, though approximate, gave asymptotically exact results in the limit N → ∞. This is a chapter of advanced topics in which we develop these techniques further, particularly with a view to calculating dynamic quantities and response functions. We first of all extend the perturbational scheme of section 7.2 to a self-consistent scheme which takes into account all non-crossing diagrams, and known generally as the Non-Crossing Approximation (NCA). This is used to calculate dynamic response functions at finite temperatures. It has been used extensively for making predictions to compare with experiment for anomalous rare earth systems. Following this we also reconsider the mean field slave boson approach. The aim here is to go beyond the mean field theory and take some account of fluctuations. It is possible to generalize the approach so that the corrections to the mean field can be systematically treated in a 1/N expansion. We shall also develop approximations so that the full spectrum of the one particle Green's function can be calculated, and not just the quasi-particle contribution. Finally we discuss briefly yet another formulation of a 1/N expansion, the variational approach. This has been developed for the calculation of the spectral density of Green's functions and response functions for comparison with several types of photoemission experiments, and also for one electron absorption spectra. It has been used extensively in the interpretation of data on Ce and other rare earth compounds.