Mean-Field Theory and the Gaussian Approximation

Abstract

The Wilsonian renormalization group (RG) was invented in order to study the effect of strong fluctuations and the mutual coupling between different degrees of freedom in the vicinity of continuous phase transitions. Before embarking on the theory of the RG, let us in this chapter describe two less sophisticated methods of dealing with this problem, namely the mean-field approximation and the Gaussian approximation. Within the mean-field approximation, fluctuations of the order parameter are completely neglected and the interactions between different degrees of freedom are taken into account in some simple average way. The Gaussian approximation is in some sense the leading fluctuation correction to the mean-field approximation. Although these methods are very general and can also be used to study quantum mechanical many-body systems¹, for our purpose it is sufficient to introduce these methods using the nearest-neighbor Ising model in D dimensions as an example.