Weakly interacting electrons and the renormalization group

TL;DR

This paper introduces a controlled renormalization group method to analyze weak-coupling instabilities in electron models, revealing phenomena like superconductivity and competing phases, with a focus on the Hubbard model.

Contribution

It develops a unified, unbiased renormalization group framework for studying weakly interacting electrons, connecting parquet summation and Wilson's effective action, and clarifies the validity of one-loop approximations.

Findings

Electron gas is unstable towards superconductivity without phonons.

Demonstrates d-wave pairing in the 2D Hubbard model.

Shows equivalence of three RG approaches within their regime of validity.

Abstract

We present a general method to study weak-coupling instabilities of a large class of interacting electron models in a controlled and unbiased way. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons, since high-energy spin fluctuations create an effective attraction between the quasi-particles. As an example, we show the occurrence of d-wave pairing in the repulsive Hubbard model in two dimensions. In one dimension or if the Fermi surface is nested, there are several competing instabilities. The required renormalization group formalism for this case is presented to lowest (one-loop) order on a most elementary level, connecting the idea of the ``parquet summation'' to the more modern concept of Wilson's effective action. The validity and restrictions of the one-loop approximation are discussed in detail. As a result, three different renormalization group approaches known in the literature are shown to be equivalent within the regime of applicability. We also briefly discuss the open problem of a two-dimensional Fermi system at Van Hove filling without nesting.