# Chern numbers as half-signature of the spectral localizer

## TL;DR

This paper offers a new proof linking Chern numbers to the spectral localizer's half-signature, and demonstrates its effectiveness as a topological phase indicator in disordered insulators through numerical analysis.

## Contribution

It provides a novel proof for even index pairings using spectral flow and explores the spectral localizer's practical use in disordered topological insulators.

## Key findings

- Half-signature indicates topological phase in disordered systems
- Spectral gap and half-signature are numerically studied in a 2D disordered insulator
- The approach works even in regimes without a bulk gap

## Abstract

Two recent papers proved that complex index pairings can be calculated as the half-signature of a finite dimensional matrix, called the spectral localizer. This paper contains a new proof of this connection for even index pairings based on a spectral flow argument. It also provides a numerical study of the spectral gap and the half-signature of the spectral localizer for a typical two-dimensional disordered topological insulator in the regime of a mobility gap at the Fermi energy. This regime is not covered by the above mathematical results (which suppose a bulk gap), but nevertheless the half-signature of the spectral localizer is a clear indicator of a topological phase.