Spectral localization for semimetals and Callias operators

TL;DR

This paper uses semiclassical analysis and localization techniques to connect the spectrum of a spectral localizer with the count of Dirac or Weyl points in semimetals, and extends these ideas to Callias operators.

Contribution

It introduces a novel spectral localization approach that links the low-lying spectrum to topological features in semimetals and applies similar methods to Callias operators.

Findings

Spectral localizer determines the number of Dirac or Weyl points.

Explicit computation for Dirac and Weyl models supports the spectral localization.

Techniques relate spectral flow to topological invariants in operator analysis.

Abstract

A semiclassical argument is used to show that the low-lying spectrum of a selfadjoint operator, the so-called spectral localizer, determines the number of Dirac or Weyl points of an ideal semimetal. Apart from the IMS localization procedure, an explicit computation for the local toy models given by a Dirac or Weyl point is the key element of proof. The argument has numerous similarities to Witten's reasoning leading to the strong Morse inequalities. The same techniques allow to prove a spectral localization for the Callias operator in terms of a multi-parameter spectral flow of selfadjoint Fredholm operators.