Universal and non-universal renormalizations in Fermi liquids

TL;DR

This paper examines the relationship between Fermi-liquid theory and diagrammatic perturbation methods, highlighting how mass renormalization involves contributions from both Fermi surface and away from Fermi surface fermions, with implications for renormalization group approaches.

Contribution

It demonstrates the equivalence of Fermi-liquid and perturbative approaches for mass renormalization and emphasizes the importance of high-energy contributions in renormalization group analysis.

Findings

Mass renormalization $m^*/m$ involves fermions both at and away from the Fermi surface.

The equivalence of FL and perturbative approaches relies on a specific relation between high- and low-energy self-energy contributions.

Derived an expression for the anisotropic Fermi-liquid vertex in large-$N$ SU(N) systems.

Abstract

We discuss an interplay between the Fermi-liquid (FL) theory and diagrammatic perturbative approach to interacting Fermi systems. In the FL theory for Galilean-invariant systems, mass renormalization $m^*/m$ comes exclusively from fermions at the Fermi surface. We show that in a diagrammatic perturbation theory the same result for $m^*/m$ comes from fermions both at and away from the Fermi surface. The equivalence of the FL and pertubative approaches is based on a particular relation between self-energy contributions from high- and low-energy fermions. We argue that care has to be exercised in the renormalization group approach to a FL in order not to miss the high-energy contribution to $m^*/m$. As particular examples, we discuss $m^*/m$ and the quasiparticle residue $Z$ for 2D and 3D systems with both SU(2) and SU(N) symmetries, and with a short-range interaction. We derive an expression for the anisotropic part of the Fermi-liquid vertex in the large-$N$ limit of the SU(N) case.