Theory of local $\mathbb{Z}_{2}$ topological markers for finite and periodic two-dimensional systems

TL;DR

This paper introduces two space-resolved $ ext{Z}_2$ topological markers that can determine local topological properties in both finite and inhomogeneous two-dimensional systems, extending the usual reciprocal space methods.

Contribution

The paper presents two novel local $ ext{Z}_2$ topological markers applicable to finite and inhomogeneous systems, broadening the scope of topological invariant calculations beyond periodic models.

Findings

Markers successfully identify topological phases in the Kane-Mele model

Markers work in the presence of disorder and heterojunctions

Markers are valid for both periodic and open boundary conditions

Abstract

The topological phases of two-dimensional time-reversal symmetric insulators are classified by a $\mathbb{Z}_{2}$ topological invariant. Usually, the invariant is introduced and calculated by exploiting the way time-reversal symmetry acts in reciprocal space, hence implicitly assuming periodicity and homogeneity. Here, we introduce two space-resolved $\mathbb{Z}_{2}$ topological markers that are able to probe the local topology of the ground-state electronic structure also in the case of inhomogeneous and finite systems. The first approach leads to a generalized local spin-Chern marker, that usually remains well-defined also when the perpendicular component of the spin, $S_{z}$, is not conserved. The second marker is solely based on time-reversal symmetry, hence being more general. We validate our markers on the Kane-Mele model both in periodic and open boundary conditions, also in presence of disorder and including topological/trivial heterojunctions.