The cohomology invariant for class DIII topological insulators

TL;DR

This paper introduces a new cohomology invariant for class DIII topological insulators that generalizes existing invariants and provides a comprehensive classification of phases in multiple dimensions.

Contribution

It defines a novel cohomology class for class DIII insulators that extends the $ ext{Z}_2$-invariant and relates to KR-theory and gerbes, advancing topological phase classification.

Findings

Defines a cohomology class generalizing the $ ext{Z}_2$-invariant.

Enables complete description of strong and weak phases in 2D.

Connects the invariant with KR-theory and gerbes.

Abstract

This work concerns with the description of the topological phases of band insulators of class DIII by using the equivariant cohomology. The main result is the definition of a cohomology class for general systems of class DIII which generalizes the well known $\mathbb{Z}_2$-invariant given by the Teo-Kane formula in the one-dimension case. In the two-dimensional case this cohomology invariant allows a complete description of the strong and weak phases. The relation with the KR-theory, the Noether-Fredholm index and the classification of "Real" gerbes are also discussed.