The Fermi gerbe of Weyl semimetals

TL;DR

This paper introduces the Fermi gerbe, a topological structure that encodes spectral data in Weyl semimetals, revealing how topological invariants protect their Fermi surfaces.

Contribution

It explicitly constructs the generator of the third homotopy group for gap topology using Dirac operators related to Weyl semimetals, linking abstract topology with physical spectral data.

Findings

The Fermi gerbe encodes spectral interpolation between gaps.

The Dixmier-Douady invariant protects the Fermi surface.

Topological invariants ensure the stability of Weyl semimetals.

Abstract

In the gap topology, the unbounded self-adjoint Fredholm operators on a Hilbert space have third homotopy group the integers. We realise the generator explicitly, using a family of Dirac operators on the half-line, which arises naturally in Weyl semimetals in solid-state physics. A "Fermi gerbe" geometrically encodes how discrete spectral data of the family interpolate between essential spectral gaps. Its non-vanishing Dixmier-Douady invariant protects the integrity of the interpolation, thereby providing topological protection of the Weyl semimetal's Fermi surface.