Discovery of Higher-Order Topological Insulators using the Spin Hall Conductivity as a Topology Signature

The discovery and realization of topological insulators, a phase of matter which hosts metallic boundary states when the $d$-dimension insulating bulk is confined to ($d-1$)-dimensions, led to several potential applications. Recently, it was shown that protected topological states can manifest in ($d-2$)-dimensions, such as hinge and corner states for three- and two-dimensional systems, respectively. These nontrivial materials are named higher-order topological insulators (HOTIs). Here we show a connection between spin Hall effect and HOTIs using a combination of {\it ab initio} calculations and tight-binding modeling. The model demonstrates how a non-zero bulk midgap spin Hall conductivity (SHC) emerges within the HOTI phase. Following this, we performed high-throughput density functional theory calculations to find unknown HOTIs, using the SHC as a criterion. We calculated the SHC of 693 insulators resulting in seven stable two-dimensional HOTIs. Our work guides novel experimental and theoretical advances towards higher-order topological insulators realization and applications.