Punctured-Chern topological invariants for semi-metallic bandstructures

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.

Significant revision compared to the previous version with additional results on a fragile topological model. Previous results are unaffected. Codes are available at https://www.dropbox.com/s/wgy3ugj7nfk0c5w/Nexus_2d_3d.nb?dl=0 on how to calculate the punctured-Chern numbers in both the 2D and 3D applications shown in the paper. 6 pages of main text with 3 figures, 4 pages of supplementary text

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