Noncommutative topological $\mathbb{Z}_2$ invariant

TL;DR

This paper extends the concept of the $bZ_2$ topological invariant for insulators into noncommutative geometry, establishing new models and proving their equivalence, thus broadening the mathematical framework for topological phases.

Contribution

It introduces two novel noncommutative models of the $bZ_2$ invariant and proves their equivalence, connecting analytic K-homology and the Pfaffian formalism.

Findings

Defined a noncommutative $bZ_2$ index as a topological K-homology index.

Formulated a noncommutative Kane--Mele invariant on the noncommutative 2-torus.

Proved the equivalence between the noncommutative $bZ_2$ index and the noncommutative Kane--Mele invariant.

Abstract

We generalize the $\mathbb{Z}_2$ invariant of topological insulators using noncommutative differential geometry in two different ways. First, we model Majorana zero modes by KQ-cycles in the framework of analytic K-homology, and we define the noncommutative $\mathbb{Z}_2$ invariant as a topological index in noncommutative topology. Second, we look at the geometric picture of the Pfaffian formalism of the $\mathbb{Z}_2$ invariant, i.e., the Kane--Mele invariant, and we define the noncommutative Kane--Mele invariant over the fixed point algebra of the time reversal symmetry in the noncommutative 2-torus. Finally, we are able to prove the equivalence between the noncommutative topological $\mathbb{Z}_2$ index and the noncommutative Kane--Mele invariant.