Aspects of Renormalization in Finite Density Field Theory

TL;DR

This paper develops a Wilsonian RG framework for finite density field theory, addressing previous renormalization challenges and analyzing Fermi surface interactions near three dimensions.

Contribution

It introduces a novel RG approach with independent boson and fermion decimation, resolving divergences and multilogarithms in finite density field theory.

Findings

Resolved logarithmic divergences in non-local interactions

Identified UV-IR mixing responsible for Fermi velocity renormalization

Applied RG to 2k_F singularities

Abstract

We study the renormalization of the Fermi surface coupled to a massless boson near three spatial dimensions. For this, we set up a Wilsonian RG with independent decimation procedures for bosons and fermions, where the four-fermion interaction "Landau parameters" run already at tree level. Our explicit one loop analysis resolves previously found obstacles in the renormalization of finite density field theory, including logarithmic divergences in non-local interactions and the appearance of multilogarithms. The key aspects of the RG are the above tree level running, and a UV-IR mixing between virtual bosons and fermions at the quantum level, which is responsible for the renormalization of the Fermi velocity. We apply this approach to the renormalization of $2 k_F$ singularities, and a companion paper considers the RG for Fermi surface instabilities. We end with some comments on the renormalization of finite density field theory with the inclusion of Landau damping of the boson.