Introduction to the Renormalization Group with Applications to Non-Relativistic Quantum Electron Gases

TL;DR

This paper reviews rigorous mathematical work on many-fermion models, highlighting the construction of interacting Fermi liquids in two dimensions and the influence of Fermi surface shape, with extensions to three dimensions.

Contribution

It introduces a constructive renormalization approach around the Fermi surface, summing all orders of perturbation theory rigorously for the first time.

Findings

Construction of interacting Fermi liquids in 2D

Identification of different scaling regimes based on Fermi surface shape

Progress on rigorous results in 3D fermion models

Abstract

We review the rigorous work on many Fermions models which lead to the first constructions of interacting Fermi liquids in two dimensions, and allowed to prove that there are different scaling regimes in two dimensions, depending on the shape of the Fermi surface. We also review progress on the three dimensional case. We start with a pedagogical introduction on quantum field theory and perturbative renormalization. Emphasis is then put on using renormalization around the Fermi surface in a constructive way, in which all orders of perturbation theory are summed rigorously.