Introduction to a renormalisation group method

TL;DR

This book introduces a Wilson-style renormalisation group method, analyzing the 4D hierarchical $| ext{varphi}|^4$ model, and discusses extensions to Euclidean and supersymmetric settings with pedagogical clarity.

Contribution

It provides a detailed, pedagogical analysis of the renormalisation group for the 4D hierarchical $| ext{varphi}|^4$ model, including stability and perturbation theory, with extensions outlined for Euclidean and supersymmetric cases.

Findings

Analysis of the hierarchical $| ext{varphi}|^4$ model using renormalisation group techniques

Development of norms and stability analysis for the RG map

Outline of extension to Euclidean nearest-neighbour and supersymmetric models

Abstract

This book provides an introduction to a renormalisation group method in the spirit of that of Wilson. It starts with a concise overview of the theory of critical phenomena and the introduction of several tools required in the renormalisation group approach, including Gaussian integration and finite range decomposition. The bulk of the book consists of an analysis of the 4-dimensional hierarchical $|\varphi|^4$ model. This includes definition of the renormalisation group map, perturbation theory, development of a general family of norms that we use to analysis the renormalisation group map, and of a stability analysis of the dynamical system that the renormalisation group map defines. The analysis of the hierarchical $|\varphi|^4$ model is presented in a pedagodical way and set up in such that the extension to the more complicated Euclidean nearest-neighbour setting is very much parallel. This extension is outlined in an appendix with precise pointers to the literature where it is carried out in detail. Finally, we provide a short introduction to supersymmetry and the supersymmetric representation of self-avoiding walks. The renormalisation group method can be extended to this supersymmetric setting and has been used to study the 4-dimensional weakly self-avoiding walk.