Topological insulators (TIs), materials exhibiting metallic boundary states despite an insulating bulk, have garnered significant attention due to their potential applications. The bulk-boundary correspondence principle governs first-order TIs, where protected gapless states appear at the (d-1)-dimensional boundaries of a d-dimensional system. However, higher-order topological insulators (HOTIs) present a more complex scenario, exhibiting gapless states at (d-2)-dimensional boundaries (e.g., hinge states in 3D, corner states in 2D) while remaining gapped at the (d-1)-dimensional boundaries. The discovery and characterization of HOTIs have been challenging due to their less common occurrence compared to first-order TIs. The quantized Hall conductivity has historically provided a connection between topology and conductivity in quantum Hall effects and first-order TIs, where a quantized spin Hall conductivity (SHC) has been observed. This paper investigates whether SHC can serve as a signature for HOTIs, a connection not previously explored in depth. The paucity of known HOTIs highlights the need for novel strategies to discover and design these materials. The authors aim to establish this connection between SHC and HOTI phases, providing a new pathway for the discovery of these materials.

The paper reviews the existing literature on topological insulators, focusing on the bulk-boundary correspondence principle in first-order TIs and the recent emergence of higher-order topological insulators (HOTIs). It highlights the connection between quantized conductivity and topological protection, citing the TKNN number for the quantum Hall effect and the work of Murakami et al. on the spin Hall conductivity (SHC) in narrow-gap insulators that were later identified as first-order TIs. The authors note the lack of exploration into the relationship between SHC and HOTIs, emphasizing the limited number of known HOTI materials as a significant obstacle to understanding this connection. The literature review sets the stage for the research presented in the paper, emphasizing the novelty of using SHC as a predictive tool for discovering new HOTIs.

The research employs a two-pronged approach: tight-binding modeling and high-throughput density functional theory (DFT) calculations. First, an eight-band tight-binding Hamiltonian, adapted from Schindler et al., is used to model a 2D HOTI with time-reversal and four-fold rotational symmetry. The model allows for the investigation of the relationship between the SHC and model parameters, demonstrating how a non-zero midgap SHC arises within the HOTI phase and how the SHC value is inversely related to a model parameter (A) that influences the transition from a TI to a HOTI phase. Subsequently, a high-throughput computational approach is employed. The authors leverage the 2D materials database C2DB, which contains DFT-calculated properties for approximately 4000 2D compounds. They screen this database, selecting thermodynamically and dynamically stable, non-magnetic insulators, resulting in a set of 693 candidates. Fully relativistic DFT calculations using the QUANTUM ESPRESSO package are performed to compute the SHC of these materials. The SHC is calculated using local effective Hamiltonians constructed via the pseudo-atomic orbital (PAO) projection method implemented in the PAOFLOW code, employing linear response theory and the Kubo formula. This method helps to avoid issues with non-degenerate perturbation theory that can lead to unphysical midgap SHC in trivial insulators. For HOTI candidates identified through this process, the electronic structure of triangular nanoflakes is calculated using VASP to investigate the localization of corner states. Finally, a topological invariant is calculated for a representative candidate (BiSe) to confirm the presence of higher-order topology.

The tight-binding model successfully demonstrates a non-zero midgap SHC within the HOTI phase, establishing a crucial link between the SHC and HOTI topology. The high-throughput DFT calculations, utilizing SHC as a selection criterion, identify seven stable 2D HOTI candidates: BiSe, BiTe (in two structures), PbF, PbBr, PbCl, and HgTe. These materials display metallic states localized at the corners (d-2 dimensional boundaries) in open boundary conditions, confirming the presence of HOTI states protected by C3 symmetry. The topological invariant calculation for BiSe validates the identification of BiSe as a HOTI. The calculated SHC values for the discovered HOTIs are generally a constant fraction of e²/h, demonstrating a clear relationship between the bulk SHC and the topologically protected states. The magnitude of the SHC, however, shows dependency on the material’s structure and composition.

The findings confirm the existence of a previously unexplored connection between the spin Hall effect and higher-order topology. The SHC serves as a reliable indicator of the HOTI phase, providing a valuable tool for the prediction and discovery of novel HOTI materials. The success in identifying seven new 2D HOTI candidates validates the effectiveness of this method. The materials identified all exhibit characteristics consistent with HOTI behavior, including the presence of corner states and a non-zero midgap SHC. While the current method has been shown to be successful, the authors acknowledge the possibility of discovering additional HOTIs among the materials investigated, particularly those with lighter elements and smaller SHC. The study highlights the importance of SHC as a topological signature for HOTIs and opens up avenues for developing analytical expressions for SHC in HOTI models and implementing more efficient high-throughput invariants.

This research demonstrates a clear link between the spin Hall effect and higher-order topological insulators, establishing the spin Hall conductivity as a valuable indicator for identifying HOTIs. The high-throughput computational approach successfully identified seven new 2D HOTI candidates, confirming the predictive power of the method. Further research could focus on developing analytical expressions for SHC in HOTI models and implementing HOTI invariants suitable for high-throughput calculations, expanding the search to 3D materials and other material classes.

The study primarily focuses on 2D materials and materials with strong spin-orbit coupling. The method might not be equally effective for identifying HOTIs composed of lighter elements with weaker spin-orbit coupling, where the SHC may be too small to be reliably detected. The computational cost associated with high-throughput DFT calculations limits the number of materials that can be screened.