The Renormalization Group and the Superconducting Susceptibility of a Fermi Liquid

TL;DR

This paper investigates how the superconducting susceptibility behaves in a Fermi liquid, finding that it does not diverge logarithmically as in a free Fermi gas, due to the marginal irrelevance of repulsive interactions.

Contribution

It derives a Callan-Symanzik equation for a repulsive Fermi liquid and analyzes the superconducting susceptibility using renormalization group techniques.

Findings

Superconducting susceptibility does not diverge logarithmically in a Fermi liquid.

Two logarithms are not more significant than one for the susceptibility.

Develops a renormalization group approach to study low-energy properties of Fermi liquids.

Abstract

A free Fermi gas has, famously, a superconducting susceptibility that diverges logarithmically at zero temperature. In this paper we ask whether this is still true for a Fermi liquid and find that the answer is that it does {\it not}. From the perspective of the renormalization group for interacting fermions, the question arises because a repulsive interaction in the Cooper channel is a marginally irrelevant operator at the Fermi liquid fixed point and thus is also expected to infect various physical quantities with logarithms. Somewhat surprisingly, at least from the renormalization group viewpoint, the result for the superconducting susceptibility is that two logarithms are not better than one. In the course of this investigation we derive a Callan-Symanzik equation for the repulsive Fermi liquid using the momentum-shell renormalization group, and use it to compute the long-wavelength behavior of the superconducting correlation function in the emergent low-energy theory. We expect this technique to be of broader interest.