Non-Fermi liquid Superconductivity: Eliashberg versus the Renormalization Group

TL;DR

This paper compares the renormalization group and Eliashberg methods for analyzing non-Fermi liquid superconductivity, demonstrating their consistency and exploring quantum critical points and unconventional metals.

Contribution

It establishes a connection between RG and Eliashberg approaches for non-Fermi liquids and applies this to quantum critical points and unconventional metallic states.

Findings

RG beta function derived from Eliashberg equations

Consistency shown for phonon-mediated superconductivity

Identified transition near quantum critical points

Abstract

We address the problem of superconductivity for non-Fermi liquids using two commonly adopted, yet apparently distinct methods: 1) the renormalization group (RG) and 2) Eliashberg theory. The extent to which both methods yield consistent solutions for the low energy behavior of quantum metals has remained unclear. We show that the perturbative RG beta function for the 4-Fermi coupling can be explicitly derived from the linearized Eliashberg equations, under the assumption that quantum corrections are approximately local across energy scales. We apply our analysis to the test case of phonon mediated superconductivity and show the consistency of both the Eliashberg and RG treatments. We next study superconductivity near a class of quantum critical points and find a transition between superconductivity and a "naked" metallic quantum critical point with finite, critical BCS couplings. We speculate on the applications of our theory to the phenomenology of unconventional metals.