Single-point spin Chern number in a supercell framework

TL;DR

This paper introduces a single-point formula for calculating the spin Chern number in supercell frameworks, enabling topological invariant analysis in disordered and non-crystalline quantum spin Hall systems.

Contribution

The authors develop a novel single-point method for computing the spin Chern number applicable to disordered systems and supercells, even with non-commuting spin operators, extending topological analysis capabilities.

Findings

Validated on Kane-Mele model with and without disorder

Successfully identified disorder-driven topological phase transition

Applicable to systems with Rashba spin-orbit coupling

Abstract

We present an approach for the calculation of the $\mathbb{Z}_2$ topological invariant in non-crystalline two-dimensional quantum spin Hall insulators. While topological invariants were originally mathematically introduced for crystalline periodic systems, and crucially hinge on tracking the evolution of occupied states through the Brillouin zone, the introduction of disorder or dynamical effects can break the translational symmetry and imply the use of larger simulation cells, where the $\bf{k}$-point sampling is typically reduced to the single $\Gamma$-point. Here, we introduce a single-point formula for the spin Chern number that enables to adopt the supercell framework, where a single Hamiltonian diagonalisation is performed. Inspired by the work of E. Prodan [Phys. Rev. B, $\textbf{80}$, 12 (2009)], our single-point approach allows to calculate the spin Chern number even when the spin operator $\hat{s}_z$ does not commute with the Hamiltonian, as in the presence of Rashba spin-orbit coupling. We validate our method on the Kane-Mele model, both pristine and in the presence of Anderson disorder. Finally, we investigate the disorder-driven transition from the trivial phase to the topological state known as topological Anderson insulator. Beyond disordered systems, our approach is particularly useful to investigate the role of defects, to study topological alloys and in the context of ab-initio molecular dynamics simulations at finite temperature.