The $Z_2$ Index of Disordered Topological Insulators with Time Reversal Symmetry

TL;DR

This paper defines a noncommutative $Z_2$ index for disordered topological insulators with time reversal symmetry, proving its robustness against perturbations that preserve the spectral gap.

Contribution

It introduces a noncommutative $Z_2$ index based on the noncommutative index theorem, extending topological classification to disordered systems with time reversal symmetry.

Findings

The $Z_2$ index is well-defined for disordered topological insulators.

The index remains invariant under time-reversal symmetric perturbations.

The spectral gap's preservation ensures the index's robustness.

Abstract

We study disordered topological insulators with time reversal symmetry. Relying on the noncommutative index theorem which relates the Chern number to the projection onto the Fermi sea and the magnetic flux operator, we give a precise definition of the $Z_2$ index which is a noncommutative analogue of the Atiyah-Singer $Z_2$ index. We prove that the noncommutative $Z_2$ index is robust against any time-reversal symmetric perturbation including disorder potentials as long as the spectral gap at the Fermi level does not close.