Renormalization group methods have a long history in quantum field theory and statistical physics. In the context of solid state physics they have been applied since the 1970s, initially mainly in impurity problems and critical phenomena. There RG schemes were used to sum the leading divergencies in perturbation theory and typically only a small number of coupling constants had to be considered. With the advent of high-temperature superconductivity new interest in non-Fermi liquid behavior in interacting Fermi systems arose, and renormalization group methods have proved valuable in the analysis of both the stability and also the instability of the Fermi liquid state. More recently it has been realized that the so-called functional renormalization group (fRG) is a source of new powerful computation tools for interacting Fermi systems, especially for low-dimensional systems with competing instabilities and entangled infrared singularities. Instead of renormalizing only a small number of coupling constants, the fRG captures in principle the complete vertex or correlation functions of the system, which leads to a much more flexible framework.
At the heart of the fRG methods is an exact hierarchy of differential flow equations for the Green or vertex functions of the system, which is obtained by taking derivatives with respect to an energy scale Λ. Usually the scale parameter Λ is taken to be an infrared cutoff in momentum or frequency space, but other choices such as the temperature are possible as well. Approximations are then constructed by truncating the hierarchy and parametrizing the vertex functions with a manageable set of variables or functions. The truncation procedure is guided by small parameters and physical intuition.
