Topological charge of fermions and Landau theory of Fermi liquid

TL;DR

This paper explores the topological stability of Fermi surfaces in fermionic liquids and extends the Landau theory of Fermi liquids to include topological invariants, with implications for non-Fermi liquids and insulators.

Contribution

It introduces a topological perspective on fermionic charge and applies it to the Landau theory of Fermi liquids and non-Fermi liquids.

Findings

Fermi surface topological invariants relate to fermionic charge.

Conservation of fermionic charge corresponds to topological charge conservation.

Application of topological charge to non-Fermi liquids and insulators.

Abstract

In the fermionic liquids, the Fermi surface is topologically stable,\cite{Volovik2003} which is at the origin of the applicability of the Landau theory of Fermi liquid (LFL). The LFL exists under special condition, when the Green's function has a pole with nonzero residue $Z$. Otherwise one has non-Landau Fermi liquid (NLFL), such as Luttinger liquid, which is described by the same topological invariant. It appears that in general this topological invariant is the property of the fermionic particle, i.e. the particle charge (or the electric charge of electron) is equivalent to the topological charge of the fermion. The conservation of the fermionic charge is equivalent to the conservation of the topological charge. We consider the application of this topological charge to the Landau theory of Fermi liquids. We also consider the application to non-Fermi liquids and crystalline insulators in relation to the Luttinger theorem.