The Renormalization Group

Abstract

Before presenting a specific model, we would like to revisit in more depth and from a more general perspective the renormalization group (RG) formalism previously introduced in Chap. 2. The RG analysis, introduced in field theory and in critical phase transitions, is a very general mathematical (and conceptual) tool, which allows one to decompose the problem of finding the “macroscopic” behavior of a large number of interacting parts into a succession of simpler problems with a decreasing number of interacting parts, whose effective properties vary with the scale of observation. Technically, this is done as we will see by defining a mapping between the observational scale and the distance |T − T c| from the critical point. The term “observational scale” usually refers to the physical scale of an observation. In the spin context, the observational scale rather refers to the size of the block of spins that one analyzes within the system. The usefulness of the RG approach is based on the existence of scale invariance and self-similarity of the observables at the critical point. The purpose of the RG is to translate in mathematical language the concept that a critical point results from the aggregate response of an ensemble of elements. In addition, the RG formalism can be used as a tool of model construction [803]. In this presentation, we follow [629, 706].