An introduction to universality and renormalization group techniques

TL;DR

This paper introduces the concepts of universality and renormalization group techniques, illustrating their applications in dynamical systems and quantum field theories, including fixed points and critical phenomena.

Contribution

It provides an accessible overview of renormalization group methods and their application to universality in both classical and quantum systems, with explicit examples and derivations.

Findings

Feigenbaum constants computed via renormalization

Derivation of Wetterich equation for scalar fields

Calculation of Wilson-Fisher fixed point and critical exponents

Abstract

These lecture notes have been written for a short introductory course on universality and renormalization group techniques given at the VIII Modave School in Mathematical Physics by the author, intended for PhD students and researchers new to these topics. First the basic ideas of dynamical systems (fixed points, stability, etc.) are recalled, and an example of universality is discussed in this context: this is Feigenbaum's universality of the period doubling cascade for iterated maps on the interval. It is shown how renormalization ideas can be applied to explain universality and compute Feigenbaum's constants. Then, universality is presented in the scenario of quantum field theories, and studied by means of functional renormalization group equations, which allow for a close comparison with the case of dynamical systems. In particular, Wetterich equation for a scalar field is derived and discussed, and then applied to the computation of the Wilson-Fisher fixed point and critical exponent for the Ising universality class. References to more advanced topics and applications are provided.