Stability of Z$_2$ topological order in the presence of vacancy-induced impurity band

TL;DR

This study examines how vacancies affect the stability of Z₂ topological order in topological insulators, showing that topological properties and edge states persist despite impurity-induced midgap states and impurity bands until a phase transition occurs.

Contribution

It demonstrates that Z₂ topological order remains stable in the presence of vacancies, with impurity bands and midgap states not destroying topological properties until a critical phase transition.

Findings

Vacancies induce midgap states associated with edge states.

Impurity bands form and mix with edge states but do not destroy topological order.

Topological order persists until Dirac cones are depleted, causing a phase transition.

Abstract

Although topological insulators (TIs) are known to be robust against non-magnetic perturbations and exhibit edge or surface states as their distinct feature, experimentally it is known that vacancies often occur in these materials and impose strong perturbations. Here we investigate effects of vacancies on the stability of Z$_{2}$ topological order using the Kane-Mele (KM) model as a prototype of topological insulator. It is shown that even though a vacancy is not classified as a topological defect in KM model, it generally induces a pair of degenerate midgap states only in the TI phase. We show that these midgap states results from edge states that fit into vacancies and are characterized by the same Z$_{2}$ topological order. Furthermore, in the presence of many vacancies, an impurity band that is degenerate with edge states in energy is induced and mixes directly with edge states. However, the Z$_{2}$ topological order persists and edge states exist between the impurity band and perturbed bulk bands until a phase transition occurs when Dirac cones near Dirac points are depleted. Our analyses indicate that the same scenario holds for point vacancies or line of vacancies in 3D TIs as well.