Hinge TBICs in Higher-Order Topology

• Hinge TBICs are one-dimensional topological channels localized at crystal hinges that form when adjacent gapped facets exhibit incompatible topological masses.
• They emerge in three-dimensional second-order topological insulators where gapped surfaces coexist with gapless hinge modes, as demonstrated in bismuth and bismuth–antimony systems.
• Their distribution is governed by symmetry, surface termination, and external perturbations like magnetism and superconductivity, enabling precise control over transport in both electronic and classical-wave platforms.

Hinge TBICs are one-dimensional topological channels localized at crystal hinges, or more generally at codimension-2 boundaries where adjacent gapped facets carry incompatible topological masses. In the higher-order-topology literature, they are the characteristic boundary modes of three-dimensional second-order topological insulators and related magnetic topological phases: the bulk is insulating, the two-dimensional surfaces can be fully gapped, and gapless conduction reappears only on selected hinges as helical Kramers pairs or chiral modes (Schindler et al., 2017, Aggarwal et al., 2021, Perez-Piskunow et al., 2021). In finite crystals these channels need not occur on every geometrically available hinge; their distribution is fixed by symmetry, surface termination, and the sign structure of facet-dependent masses, and in some bismuth crystals they form a continuous closed loop encircling the sample (Zhao et al., 11 Feb 2025).

1. Concept and classification

Higher-order topological insulators are bulk-gapped phases whose protected gapless modes live on boundaries that are more than one dimension lower than the bulk. In three dimensions, a second-order phase has one-dimensional hinge states while all surfaces may be gapped. This distinguishes hinge TBICs from the gapless two-dimensional surfaces of ordinary three-dimensional topological insulators and from the one-dimensional edges of two-dimensional topological insulators (Schindler et al., 2017).

Two principal symmetry classes recur in the literature. In helical HOTIs, time-reversal symmetry and crystalline symmetries protect Kramers pairs of counterpropagating hinge channels. In chiral HOTIs, time reversal is broken, while a spatio-temporal symmetry such as $\mathsf{C}_{2n}\mathcal T$ or $\mathsf{C}_4\mathcal T$ protects unidirectional hinge transport. The noninteracting classifications quoted in the cited works are correspondingly different: the helical class is described by a $\mathbb{Z}$ classification associated with mirror Chern numbers, whereas the chiral class discussed in the $\mathsf{C}_4\mathcal T$ setting is $\mathbb{Z}_2$-classified (Schindler et al., 2017, Tiwari et al., 2019).

Magnetic topological insulators provide a related but distinct realization. There, exchange-gapped surfaces with opposite Dirac mass signs can localize chiral channels at hinges, and the local spin of those modes is strongly in-plane, flips sign between top and bottom surfaces, and reverses with propagation direction at the opposite edge (Perez-Piskunow et al., 2021). This produces hinge TBICs that are topological boundary currents confined to the intersections of insulating facets rather than to the facets themselves.

A recurrent misconception is that higher-order topology requires every hinge to be gapless. The bismuth-based experiments and the quasi-one-dimensional bismuth-halide theory show the opposite: symmetry can require hinge modes only on a subset of edges, and the exact hinge pattern can depend on termination even when the bulk phase is topological (Aggarwal et al., 2021, Yoon et al., 2020).

2. Surface-mass mechanism and bulk–boundary correspondence

A standard low-energy description treats each facet as a gapped surface Dirac cone with a facet-dependent mass. In the Bi and BiSb analysis, the minimal surface Hamiltonian is written as

$$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$

where $m(r)$ is the Dirac mass. When $m(r)$ changes sign across the boundary between adjacent facets, the hinge is a mass domain wall. The resulting Jackiw–Rebbi bound state is a one-dimensional helical mode with

$E(k_h)=\pm v_h k_h,$

localized at the hinge and protected by time-reversal symmetry (Aggarwal et al., 2021).

The same logic appears in several guises. In magnetic topological insulators, the exchange field generates opposite masses on top and bottom surfaces, so the sidewall hinges become domain walls hosting chiral modes (Perez-Piskunow et al., 2021). In the quasi-one-dimensional $\alpha$ phases of $\mathsf{C}_4\mathcal T$0, side surfaces are described as chains of coupled helical edges with an SSH-type dimerization, and a domain wall in the dimerization binds a one-dimensional hinge channel (Yoon et al., 2020). In mirror-protected HOTIs, the bulk mirror Chern number controls the number of hinge Kramers pairs through the relation $\mathsf{C}_4\mathcal T$1, where $\mathsf{C}_4\mathcal T$2 counts protected helical pairs on a mirror-related hinge pair (Schindler et al., 2017).

This mass-domain-wall picture also explains why hinge patterns are facet selective. In the Bi(110) island geometry, bulk $\mathsf{C}_4\mathcal T$3 symmetry about $\mathsf{C}_4\mathcal T$4 and time-reversal symmetry constrain the relative signs of adjacent surface masses, but inversion is naturally broken by the finite termination, so multiple allowed mass-sign configurations remain. The experimentally observed three-of-four edge pattern is therefore interpreted as a symmetry-constrained but termination-sensitive realization of higher-order bulk–boundary correspondence (Aggarwal et al., 2021).

Boundary-sensitive diagnostics are often necessary because bulk symmetry indicators can be incomplete. The most explicit example is $\mathsf{C}_4\mathcal T$5-$\mathsf{C}_4\mathcal T$6: its symmetry indicators are trivial, yet its hinge TBICs are established by boundary winding numbers and surface dimerization/domain-wall analysis. This shows that higher-order topology can exist beyond the scope of bulk symmetry indicators (Yoon et al., 2020).

3. Bismuth and bismuth–antimony as benchmark hinge-TBIC systems

Elemental Bi and $\mathsf{C}_4\mathcal T$7 occupy a central place in the experimental literature on hinge TBICs. In the 2021 scanning-tunneling study, highly crystalline Bi and $\mathsf{C}_4\mathcal T$8 films were grown by molecular beam epitaxy on $\mathsf{C}_4\mathcal T$9-doped Si(111) wafers, with thicknesses of $\mathbb{Z}$0–$\mathbb{Z}$1 bilayers chosen to represent bulk-like behavior. Measurements were performed at $\mathbb{Z}$2, and large-area STM on the (110) films revealed rectangular islands whose long edges run parallel to the $\mathbb{Z}$3 direction. Atomic-scale STM and FFTs showed the pseudo-cubic Bi(110) lattice with measured periodicities $\mathbb{Z}$4 and $\mathbb{Z}$5 along orthogonal in-plane directions (Aggarwal et al., 2021).

The spectroscopic signature of the hinge channel is a sharp, edge-localized peak in the local density of states around $\mathbb{Z}$6 above the Fermi level. In both Bi(110) and $\mathbb{Z}$7, line cuts across the edge show that this feature is confined within a few nanometers from the hinge into the adjacent facets, while the terrace spectra are comparatively featureless apart from a broad DOS increase above $\mathbb{Z}$8 associated with Rashba-split surface states. The decisive observation is the “three-of-four” pattern: on rectangular islands, three edges show the sharp hinge-localized DOS peak, while one edge does not, and the missing hinge is consistently the short edge perpendicular to $\mathbb{Z}$9 (Aggarwal et al., 2021).

That pattern was interpreted as inconsistent with simple chemical or bonding asymmetry and as naturally explained by HOTI physics in Bi/BiSb. The paper further argued that the observation places $\mathsf{C}_4\mathcal T$0 in a HOTI regime with trivial strong $\mathsf{C}_4\mathcal T$1 and higher-order crystalline topology, thereby challenging the standard classification of $\mathsf{C}_4\mathcal T$2 near $\mathsf{C}_4\mathcal T$3 (Aggarwal et al., 2021).

A later STM study on mesoscopic Bi crystals grown on superconducting $\mathsf{C}_4\mathcal T$4 extended the phenomenology from local hinge detection to a global hinge-state loop. The crystals were approximately $\mathsf{C}_4\mathcal T$5 laterally and $\mathsf{C}_4\mathcal T$6 high, exposed $\mathsf{C}_4\mathcal T$7, $\mathsf{C}_4\mathcal T$8, and $\mathsf{C}_4\mathcal T$9 facets, and generated five hinge types. Pronounced one-dimensional van Hove peaks at $\mathbb{Z}_2$0–$\mathbb{Z}_2$1, for example $\mathbb{Z}_2$2 on type-4 hinges, were observed in hinge spectra but not on facets; the typical localization width was $\mathbb{Z}_2$3–$\mathbb{Z}_2$4. Quasiparticle-interference line scans showed hole-like dispersions with $\mathbb{Z}_2$5 up to $\mathbb{Z}_2$6–$\mathbb{Z}_2$7, and Fe-cluster deposition produced an additional $\mathbb{Z}_2$8 branch on type-1 and type-4 hinges, consistent with spin-flip backscattering in spin-helical channels. Combined with first-principles calculations and a global symmetry analysis for space group $\mathbb{Z}_2$9 and point group $$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$0, the work identified hinge types $$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$1, $$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$2, and $$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$3 as topological and concluded that they connect into a single closed loop of spin-helical hinge states around the crystal. The same work reported proximity-induced superconductivity with substrate gap $$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$4 and induced gap $$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$5–$$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$6 on Bi hinges and facets, with enhanced coherence peaks at the hinge location (Zhao et al., 11 Feb 2025).

4. Other material platforms and boundary-sensitive topology

The quasi-one-dimensional bismuth halides $$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$7-$$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$8 and $$H_s = v_x k_x \sigma_y - v_y k_y \sigma_x + m(r)\sigma_z,$$9-$m(r)$0 provide an analytically transparent setting for hinge TBICs. Each (001) monolayer is a two-dimensional $m(r)$1 topological insulator with a helical edge along the chain direction $m(r)$2. In the $m(r)$3 phases, stacking converts the side surfaces into chains of coupled helical edges, and weak interlayer edge tunneling produces an SSH-type dimerization that gaps the side surfaces while leaving domain-wall hinge channels running precisely along the quasi-one-dimensional $m(r)$4 axis (Yoon et al., 2020).

The distinction between the two compounds is set by the inversion-center location. In $m(r)$5-$m(r)$6, the inversion center lies within a (001) layer, the two side cleavage surfaces dimerize oppositely, and the hinge channels are intrinsic: they are robust to adding or removing a topological-insulator layer at the surface. In $m(r)$7-$m(r)$8, the inversion center lies between adjacent layers, the two side surfaces dimerize in the same way, and the hinge channels are extrinsic and termination dependent. The cited analysis therefore classified $m(r)$9-$m(r)$0 as an intrinsic HOTI with indicator $m(r)$1 and $m(r)$2-$m(r)$3 as an extrinsic HOTI with trivial indicators $m(r)$4 but nontrivial boundary invariants. Reported hinge properties include $m(r)$5–$m(r)$6 and localization length smaller than $m(r)$7 (Yoon et al., 2020).

This family also sharpens the difference between weak and higher-order topology. The $m(r)$8 phases are quasi-one-dimensional weak topological insulators with gapless side surfaces, whereas the $m(r)$9 phases have gapped side surfaces and hinge TBICs. Because $E(k_h)=\pm v_h k_h,$0 undergoes a $E(k_h)=\pm v_h k_h,$1 structural transition around $E(k_h)=\pm v_h k_h,$2, the literature identifies a route for switching between side-surface transport and hinge transport by thermal cycling or local strain (Yoon et al., 2020).

Mirror-protected helical hinge states were placed in a broader materials context by the foundational HOTI work on SnTe and “dual” weak topological insulators such as $E(k_h)=\pm v_h k_h,$3, BiSe, and BiTe. In that framework, SnTe is a topological crystalline insulator with mirror Chern number $E(k_h)=\pm v_h k_h,$4, and realistic distortions such as (111) rhombohedral ferroelectric displacement or uniaxial strain along (110) gap the relevant surfaces while leaving a single Kramers pair on each hinge. For $E(k_h)=\pm v_h k_h,$5, BiSe, and BiTe, the cited proposal was to gap weak-TI surface Dirac cones without breaking time reversal or the relevant mirror symmetry, for example by a commensurate charge density wave, thereby converting those materials into helical HOTIs with one Kramers pair per mirror-invariant hinge (Schindler et al., 2017).

5. Transport control, magnetism, superconductivity, and interactions

Magnetic topological insulators supply a transport-oriented realization of hinge TBICs. Using a Fu–Kane–Mele lattice Hamiltonian on a diamond cubic lattice with spin–orbit coupling and a layer-dependent exchange term, the cited work considered ferromagnetic, antiferromagnetic, and canted-antiferromagnetic phases. For a slab perpendicular to $E(k_h)=\pm v_h k_h,$6, a uniform exchange $E(k_h)=\pm v_h k_h,$7 gaps the top and bottom Dirac surfaces and yields a quantum anomalous Hall phase whose boundary modes are concentrated at hinges. Their local spin polarization is dominated by $E(k_h)=\pm v_h k_h,$8, flips sign between top and bottom hinges, and reverses at the opposite edge. In a four-terminal device with spin-filtering ferromagnetic contacts near the top hinge, the matching configuration gives $E(k_h)=\pm v_h k_h,$9 inside the gap, while the mismatching configuration increases $\alpha$0 by more than one order of magnitude; the switch survives Anderson disorder up to $\alpha$1 with $\alpha$2 and structural edge vacancy probability up to $\alpha$3 (Perez-Piskunow et al., 2021). The transport formalism is the standard Landauer–Büttiker relation

$\alpha$4

The same channels can be reshaped by local perturbations. In the surface theory of inversion-broken HOTIs, an out-of-plane Zeeman field $\alpha$5 does not simply gap the helical hinge mode; instead it shifts the mass domain wall differently in the two valleys and can split a single helical hinge into two spatially separated chiral electron channels. An $\alpha$6-wave superconducting proximity term can analogously split the hinge into helical Majorana channels around a superconducting patch, and the combination of Zeeman field and superconductivity can leave a single chiral Majorana mode around a superconducting island (Queiroz et al., 2018).

At the device level, extrinsic second-order topological insulators with hinges carrying one or two co-propagating chiral modes can realize interferometric circuits. In the lattice model of a first-order TI in a magnetic field, surface gating and orbital field produce hinges with either one or two chiral channels, and two-mode hinges act as beam splitters. Ohmic contacts on four single-mode hinges then define a hinge-based Mach–Zehnder interferometer whose conductance is

$\alpha$7

with interference phase controlled by the enclosed magnetic flux (Chaou et al., 2022).

Interactions can also remove hinge conduction without breaking the protecting symmetry globally. For $\alpha$8-protected chiral HOTIs and HOTSCs, non-Abelian surface topological orders placed in a $\alpha$9-preserving alternating pattern on side faces, together with appropriate top and bottom surface orders, allow all hinge channels to be gapped by symmetry-preserving anyon condensation. In that setting the hinge TBICs are “unhinged”: the anomalous surface termination exists only on the boundary of the three-dimensional bulk phase and not as a standalone two-dimensional system (Tiwari et al., 2019).

6. Classical-wave realizations and terminological breadth

Hinge TBIC physics is not confined to electronic materials. In a three-dimensional acoustic higher-order topological insulator built from a bilayer hexagonal phononic crystal, hinge states were predicted and observed simultaneously along three independent directions of a single sample. The acoustic fields obey

$\mathsf{C}_4\mathcal T$00

and the hinge modes are understood as Jackiw–Rebbi states generated by sign inversions of surface or interface Dirac masses. Experimentally, the platform used lattice constants $\mathsf{C}_4\mathcal T$01 and $\mathsf{C}_4\mathcal T$02, hinge transport was strongest around $\mathsf{C}_4\mathcal T$03–$\mathsf{C}_4\mathcal T$04, and measured pressure maps showed hinge-to-hinge routing in a $\mathsf{C}_4\mathcal T$05 sample with negligible degradation under $\mathsf{C}_4\mathcal T$06 rotation disorder of selected scatterers (Wei et al., 2021).

A different phononic-crystal platform realized a dimensional hierarchy of topological bound states in the continuum. There, exact separability block-diagonalized the Bloch Hamiltonian into orthogonal subspaces, so surface and hinge modes of one subspace were embedded in the bulk continuum of the other without hybridization. The system hosted coexisting two-dimensional surface TBICs and one-dimensional hinge TBICs protected by valley Chern numbers, with operating gap $\mathsf{C}_4\mathcal T$07–$\mathsf{C}_4\mathcal T$08, measured interface TBICs near $\mathsf{C}_4\mathcal T$09, and hinge TBICs near $\mathsf{C}_4\mathcal T$10 (Yin et al., 30 Aug 2025). In that literature, “TBIC” denotes topological bound states in the continuum rather than conducting channels, but the hinge localization and topological protection are directly analogous.

The acronym is not fully standardized across arXiv subfields. In the interferometer literature, TBIC is explicitly expanded as “topological beam-interference circuits” and refers to hinge-mode device architectures rather than to a distinct quasiparticle type (Chaou et al., 2022). In computational mechanics, “hinge TBICs” denotes thin bending constraints on triangular meshes, with edge-based, FVM, and smoothed hinge models formulated in a corotational frame and characterized by constant bending-energy Hessians under small strain and small curvature assumptions (Liang, 15 Feb 2025). This suggests that contextual disambiguation is necessary: in condensed-matter and wave-topology papers, hinge TBICs usually denote hinge-localized topological channels or hinge-localized topological BICs, whereas outside that literature the same acronym can describe mathematically unrelated hinge-based constructions.