Discovery of higher-order topological insulators using the spin Hall conductivity as a topology signature

The emergence of symmetry-protected topological phases established that d-dimensional topological insulators (TIs) host robust gapless boundary states in (d−1) dimensions, mediated by symmetries such as time-reversal, mirror, non-symmorphic and particle-hole. A newer class, higher-order topological insulators (HOTIs), apparently violates conventional bulk–boundary correspondence by exhibiting gapped bulk and (d−1)-dimensional boundaries but protected gapless states in (d−2) dimensions (e.g., hinges or corners). While first-order TIs are abundant, HOTIs are comparatively rare, motivating new discovery strategies. Historically, topological protection has been linked to quantized conductivities (e.g., TKNN relation for the quantum Hall effect). Murakami et al. predicted finite spin Hall conductivity (SHC) in narrow-gap insulators later identified as TIs, suggesting a link between SHC and nontrivial topology: in ideal TIs, SHC is quantized by the spin Chern number, whereas ordinary insulators show zero midgap SHC. The relation between SHC and HOTIs, however, remained unexplored due to the scarcity of known HOTIs. This work proposes and demonstrates that a finite bulk midgap SHC is a signature of HOTI phases, then uses this connection to identify new 2D HOTIs by high-throughput screening of candidate materials solely assuming time-reversal symmetry. The study reports seven stable 2D HOTIs and validates their higher-order topology via nanoflake calculations and topological invariants.

Prior work established bulk–boundary correspondence for first-order TIs protected by various symmetries (Kane & Mele; Hasan & Kane; Qi & Zhang). The TKNN invariant relates quantized Hall conductance to topology (Thouless et al.). Murakami, Nagaosa, and Zhang predicted finite SHC in narrow-gap insulators, later understood as TIs, linking SHC and topology via the spin Chern number. Higher-order topology was theoretically introduced and explored in various platforms (Benalcazar–Bernevig–Hughes; Schindler et al.; Ezawa; Park et al.; Sheng et al.; Kempkes et al.; Wang et al.). High-throughput topology classifications using symmetry indicators and cataloging of topological materials (Vergniory et al.; Tang et al.; Zhang et al.) provided broad context but are not tailored for HOTIs. Symmetry-indicator approaches can misclassify 2D HOTIs as trivial when using certain invariants, necessitating alternative signatures. Recent works also examined local SHC contributions and the role of spin-rotation symmetry breaking on SHC quantization. This paper builds on these foundations by proposing bulk midgap SHC as a practical signature for HOTIs.

- Tight-binding model: An eight-band tight-binding Hamiltonian adapted from Schindler et al. for 2D systems, with two spinful orbitals per site and Pauli matrices acting on orbital and spin spaces, is used to realize a time-reversal- and C4-rotation-symmetric 2D HOTI. Model parameters include mass M, hopping t, spin–orbit coupling strength, and a mixing term A that drives a TI into a HOTI. Example parameters yield a gapped bulk band structure (~2 eV) with corner-localized modes in finite flakes. The SHC is computed as a function of Fermi energy and mapped over parameter space by varying M, t, SOC strength, and A, showing how the HOTI phase hosts a finite but non-quantized midgap SHC that is suppressed as A increases. - High-throughput DFT screening: Starting from the C2DB database (~4000 2D compounds), a screening for thermodynamically and dynamically stable, non-magnetic insulators produced 693 candidates. Fully relativistic DFT (PBE-GGA with PAW pseudopotentials from pslibrary) was performed using Quantum ESPRESSO. Lattice parameters from C2DB were used; full structural relaxations proceeded until forces < 0.01 eV/Å. Plane-wave and charge-density cutoffs were set 40% above recommendations. k-point densities were 6.0/Å for optimization and 12.0/Å for self-consistent runs; vacuum spacing of 15 Å was used to avoid spurious interactions. - SHC calculations: Local effective Hamiltonians were generated by projecting plane-wave Kohn–Sham states onto pseudo-atomic orbitals (PAO) using PAOFLOW. The SHC was computed via linear-response Kubo formalism with degenerate perturbation theory to avoid unphysical midgap SHC in trivial insulators. The spin current operator was defined via the anticommutator of the spin Pauli matrix and the velocity operator. Dense Brillouin-zone sampling (72.0/Å) was employed. Consistency with C2DB topological classifications was checked to exclude known TIs/TCIs when searching for HOTIs with large SHC. - Nanoflake calculations: For predicted HOTIs, triangular nanoflakes preserving C3 symmetry were constructed and computed with VASP to probe (d−2)-dimensional corner states. Real-space projections of midgap wavefunctions confirmed corner localization and the expected near-degeneracy of corner modes. - Topological invariant (representative case BiSe): An adaptation of the inversion- and C3-protected HOTI invariant was evaluated by tracking inversion parity products of occupied bands at time-reversal-invariant momenta, organized by C3 eigenvalues. The global index ν was obtained from contributions at Γ and the three symmetry-equivalent M points (ν = ν_Γ ν_M^(1/3)); ν = −1 indicates a 2D HOTI.

- Theory-model link: The 2D HOTI tight-binding model exhibits a finite bulk midgap SHC (e.g., ~0.75 e^2/h in a representative parameter set), despite gapped bulk and edge spectra, and shows corner-localized states in finite flakes. Mapping SHC over parameters demonstrates that the mixing term A, which drives TI → HOTI, reduces SHC away from quantized TI values; larger A yields smaller fractional SHC. - High-throughput results: Of 693 screened stable non-magnetic insulators, seven 2D HOTIs were identified based on large midgap SHC and subsequent validations: BiSe; BiTe in GaSe and CH prototypes; PbF; PbBr; PbCl; HgTe. - Representative material properties: - BiSe (GaSe prototype): Inversion-, time-reversal-, and C3-symmetric. Direct bulk gap ~0.46 eV at Γ. SHC ~0.5 e^2/h constant across the gap. Triangular ~50 Å flakes exhibit midgap corner modes localized at the three corners. The invariant ν = ν_Γ ν_M^(1/3) = −1 confirms HOTI character. - PbBr (CH prototype): Large midgap SHC ≈ −0.5 e^2/h; triangular flakes show corner-localized midgap states. - Across predicted HOTIs, small bulk gaps consistent with strong SOC elements were found: 260, 406, 58, 398, 57, and 94 meV for BiTe (GaSe), PbBr, BiTe (CH), PbCl, PbF, and HgTe, respectively. For BiTe, SHC depends on structure: ~0.75 e^2/h (GaSe) vs ~0.5 e^2/h (CH). - General observation: Although not quantum spin Hall insulators, these HOTIs display a constant fractional midgap SHC (a robust fraction of e^2/h) correlated with higher-order topology and the presence of corner states in (d−2)-dimensional flakes.

The results establish that a finite bulk midgap SHC serves as a practical and robust signature of 2D HOTI phases. The tight-binding analysis clarifies why HOTIs exhibit non-quantized but finite midgap SHC, in contrast to the quantized SHC in first-order TIs, and how this signature evolves with parameters that tune the system deeper into the HOTI phase. High-throughput DFT calculations leveraging SHC as a screening metric enabled the discovery of seven stable 2D HOTIs, which were corroborated by nanoflake calculations showing corner-localized midgap states and by evaluation of a symmetry-based topological invariant. This connects bulk transport-like signatures (SHC) with protected (d−2)-dimensional boundary phenomena, offering a design principle for identifying HOTIs in materials databases without requiring explicit symmetry-indicator calculations for higher-order topology.

This work uncovers and validates a direct connection between spin Hall conductivity and higher-order topology in 2D materials. A finite bulk midgap SHC indicates the HOTI phase, a principle demonstrated via a tight-binding model and used for materials discovery. Screening 693 stable non-magnetic insulators from C2DB yielded seven 2D HOTIs (BiSe; BiTe in two structures; PbF; PbBr; PbCl; HgTe) with strong SOC, small gaps, fractional midgap SHC, and corner states in symmetry-preserving flakes. Future directions include deriving analytical expressions for SHC in HOTI models, developing and implementing HOTI invariants suitable for high-throughput workflows (e.g., addressing fractional corner anomaly invariants that require d−2 and d−1 density-of-states), and extending the SHC-based criterion to 3D HOTIs, as preliminarily suggested by a 3D semiconducting HOTI (α-BiBr4).

- Sensitivity to SOC: The SHC-based criterion primarily identifies HOTIs with significant spin–orbit coupling; systems with negligible SOC (e.g., graphdiyne/graphyne) may not be captured despite HOTI topology. - Symmetry requirements: Corner states and the higher-order invariant rely on preservation of protecting symmetries (e.g., C3 and time-reversal) in both bulk and finite geometries; symmetry breaking in flakes can obscure signatures. - SHC quantization: Spin-rotation symmetry breaking reduces midgap SHC from quantized values; while not inducing spurious midgap SHC in trivial insulators (with degenerate perturbation theory), it complicates direct comparison across materials. - Computational aspects: Accurate SHC evaluation requires dense k-point sampling and careful treatment of band degeneracies; without degenerate perturbation theory, unphysical midgap SHC can appear in trivial systems. Some HOTI invariants (e.g., fractional corner anomaly) need DOS information of (d−2) and (d−1) systems, challenging to integrate into high-throughput pipelines. - Potential false negatives: The screening may miss HOTIs formed by lighter elements (weak SOC) or those whose SHC is too small to pass thresholds, meaning additional HOTIs could exist among the surveyed materials.