Punctured-Chern topological invariants for semi-metallic bandstructures

TL;DR

This paper introduces a new wavefunction-based 'punctured-Chern' invariant to classify semi-metallic band structures with gapless points, extending topological methods beyond gapped insulators.

Contribution

It develops a general topological invariant applicable to semi-metallic systems with gap-closing points, demonstrated on models with fragile topology and triple-point nodal defects.

Findings

The invariant captures various band-topological transitions.

It characterizes semi-metallic topology with half-integers affecting transport.

Classifies Nexus triple-points with symmetry restrictions.

Abstract

Topological insulator-based methods underpin the topological classification of gapped bands, including those surrounding semi-metallic nodal defects. However, multiple bands with gap-closing points can also possess non-trivial topology. We construct a general wavefunction-based ``punctured-Chern" invariant to capture such topology. To show its general applicability, we analyze two systems with disparate gapless topology: 1) a recent two-dimensional fragile topological model to capture the various band-topological transitions and 2) a three-dimensional model with a triple-point nodal defect to characterize its semi-metallic topology with \emph{half-integers} that govern physical observables such as anomalous transport. This invariant also gives the classification for Nexus triple-points ($\mathbb{Z}\times\mathbb{Z}$) with certain symmetry restrictions, which is re-confirmed by abstract algebra.