Fermi liquid theory: a renormalization group point of view

TL;DR

This paper demonstrates how Fermi liquid theory can be derived using a renormalization group approach, providing a systematic and order-by-order framework for understanding collective modes, response functions, and quasiparticle properties.

Contribution

It introduces a renormalization group method to recover Fermi liquid results, extending calculations to all orders and comparing different RG formulations.

Findings

RG equations reproduce Fermi liquid properties at one-loop order.

Two-loop calculations match results from bosonization and Ward identities.

Discussion of zero-temperature limit and importance of multi-body interactions.

Abstract

We show how Fermi liquid theory results can be systematically recovered using a renormalization group (RG) approach. Considering a two-dimensional system with a circular Fermi surface, we derive RG equations at one-loop order for the two-particle vertex function $\Gamma $ in the limit of small momentum (${\bf Q}$) and energy ($\Omega $) transfer and obtain the equation which determines the collective modes of a Fermi liquid. The density-density response function is also calculated. The Landau function (or, equivalently, the Landau parameters $F_l^s$ and $F_l^a$) is determined by the fixed point value of the $\Omega $-limit of the two-particle vertex function (${\Gamma ^\Omega }^*$). We show how the results obtained at one-loop order can be extended to all orders in a loop expansion. Calculating the quasi-particle life-time and renormalization factor at two-loop order, we reproduce the results obtained from two-dimensional bosonization or Ward Identities. We discuss the zero-temperature limit of the RG equations and the difference between the Field Theory and the Kadanoff-Wilson formulations of the RG. We point out the importance of $n$-body ($n\geq 3$) interactions in the latter.