Is Fermi liquid topologically protected?

TL;DR

This paper examines whether the topological protection of Fermi liquids, as suggested by topological arguments, holds true given the potential for Cooper instability, questioning the stability of Fermi liquid poles.

Contribution

It critically analyzes the topological argument for Fermi liquid stability and discusses the apparent contradiction with Cooper instability.

Findings

Topological argument suggests Fermi liquid poles are stable.

Contradiction with known Cooper instability challenges this view.

Discussion highlights the origin of the controversy.

Abstract

The book by Volovik [1] contains the argument, which can be considered as the topological proof of the Luttinger theorem. The Green function of the ideal Fermi gas has a pole in the (E,|p|) plane (where E and p are energy and momentum). This pole is considered to be analogous to a vortex in liquid helium. Since a vortex is topologically stable against restricted perturbations, one can include interaction adiabatically (as a succession of small perturbations), and observe transformation of the Fermi gas pole to the Fermi liquid pole. In this argument, the topological stability arises already on the level of the ideal Fermi gas, which is in conflict with its Cooper instability. We discuss the origin of this controversy.