Differential topology of semimetals

TL;DR

This paper explores the topological and geometric properties of semimetals, revealing new classes with subtle invariants and predicting torsion Fermi arcs through advanced mathematical frameworks.

Contribution

It introduces a novel topological classification of semimetals using cohomological invariants, Euler structures, and geometric sequences, extending beyond Dirac Hamiltonians.

Findings

Identification of new semimetal classes with Atiyah-Dupont-Thomas invariants

Prediction of torsion Fermi arcs in certain semimetals

Establishment of links between topological invariants and surface states

Abstract

The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the torsion of manifolds. Dually, a topological semimetal can be represented by Euler chains from which its surface Fermi arc connectivity can be deduced. These dual pictures, and the link to topological invariants of insulators, are organised using geometric exact sequences. We go beyond Dirac-type Hamiltonians and introduce new classes of semimetals whose local charges are subtle Atiyah-Dupont-Thomas invariants globally constrained by the Kervaire semicharacteristic, leading to the prediction of torsion Fermi arcs.