Unification of topological invariants in Dirac models

TL;DR

This paper introduces a unified wrapping number approach for calculating topological invariants in Dirac models across various dimensions and symmetry classes, simplifying and unifying the characterization of topological phases.

Contribution

It proposes a gauge-invariant wrapping number method that unifies the calculation of topological invariants like Chern and winding numbers in Dirac models.

Findings

The wrapping number describes the degree of the map from the Brillouin zone to the Dirac sphere.

The method unifies different topological invariants under a single framework.

It is gauge-invariant and based on intrinsic features of Dirac models.

Abstract

Topological phases of materials are characterized by topological invariants that are conventionally calculated by different means according to the dimension and symmetry class of the system. For topological materials described by Dirac models, we introduce a wrapping number as a unified approach to obtain the topological invariants in arbitrary dimensions and symmetry classes. Given a unit vector that parametrizes the momentum-dependence of the Dirac model, the wrapping number describes the degree of the map from the Brillouin zone torus to the sphere formed by the unit vector that we call Dirac sphere. This method is gauge-invariant and originates from the intrinsic features of the Dirac model, and moreover places all known topological invariants, such as Chern number, winding number, Pfaffian, etc, on equal footing.