Higher-Order Topological Structures

Updated 9 April 2026

Higher-order topological structures are systems showing robust boundary states on higher codimension surfaces such as hinges and corners, enabled by symmetry-protected topological invariants.
They are realized across domains—including electronic, photonic, mechanical, and acoustic systems—using techniques like nested Wilson loops and Dirac mass perturbations.
They offer promising applications in robust signal routing, quantum computing, and sensing through multifunctional devices that leverage mixed boundary phenomena.

A higher-order topological structure refers to a physical or mathematical system in which robust boundary states manifest not at the usual boundaries (of codimension one), but rather at boundaries of higher codimension—such as hinges or corners—enabled by topological invariants protected by global or crystalline symmetries. This concept generalizes the conventional bulk–boundary correspondence of topological band theory, yielding an expanded hierarchy of topological phases with mixed boundary phenomenology. Higher-order topology has been realized and classified in electronic, photonic, mechanical, acoustic, and bosonic systems, as well as engineered in quasicrystals, fractal lattices, superconductors, spin systems, and machine-learning architectures.

1. General Formulation and Order Hierarchy

In a $d$ -dimensional system, a conventional (first-order) topological insulator (TI) or superconductor (TSC) supports gapless states on $(d-1)$ -dimensional boundaries (edges or surfaces), protected by bulk topological indices and relevant symmetries. An $n$ th-order topological phase in $d$ dimensions admits robust boundary states on $(d-n)$ -dimensional submanifolds (e.g., hinges for $n=2$ , corners for $n=3$ in $d=3$ systems), while all lower-codimension boundaries are gapped. The relation $d_c = d - n$ specifies the dimension $d_c$ of topologically protected boundary modes in an $(d-1)$ 0th-order phase.

For example:

In 2D, a second-order phase (n=2) supports 0D corner modes but gapped bulk and edges.
In 3D, a second-order phase (n=2) localizes 1D hinge states; a third-order phase (n=3) localizes 0D corner modes at the intersection of three gapped surfaces (Schindler, 2020, Schindler et al., 2017).
In 3D Dirac semimetals, explicit construction shows that up to third-order (codimension-3) modes (corners) can arise by adding mutually anticommuting “Wilsonian mass” perturbations, each successively raising the localization codimension by one (Calugaru et al., 2018).

This sequence is realized by recursively adding symmetry-allowed, mutually anticommuting mass terms in Dirac-type Hamiltonians, each term gapping the boundary of the previous order except at domain walls where the mass switches sign—localizing states at the intersection.

2. Topological Invariants and Symmetry Constraints

The quantization, classification, and stability of higher-order phases derive from symmetry-constrained topological indices that generalize the Chern number or bulk $(d-1)$ 1 index:

Multipole Moments: Quantized multipole moments (e.g., quadrupole in 2D, octupole in 3D) captured by nested Wilson loops or real-space formulas serve as bulk invariants. For the Benalcazar–Bernevig–Hughes (BBH) model, the bulk quadrupole $(d-1)$ 2 is directly related to the number and nature of corner states (Schindler, 2020, Kang et al., 19 Dec 2025, Zheng et al., 2023).
Symmetry Indicators: Rotational, inversion, and mirror symmetries quantize indicators such as parity products or rotation eigenvalues at high-symmetry Brillouin zone points. For example, in the cubic-symmetric 3D HOTI, the quantized corner charge $(d-1)$ 3 is protected by $(d-1)$ 4 symmetry and bulk filling anomaly (Kachin et al., 2021).
Polarized Topological Charges: In chiral-symmetric systems, momentum-space polarized topological charges $(d-1)$ 5 characterize $(d-1)$ 6th-order phases with a universal relationship $(d-1)$ 7, linking momentum-space winding and boundary polarization (Jia et al., 2024).
Dirac Mass Representation: Real representations $(d-1)$ 8 of the point-group symmetries acting on mass terms of the boundary Dirac Hamiltonian determine the possible codimension and symmetry protection of the domain walls, leading to $(d-1)$ 9-theoretic classification (Trifunovic et al., 2020).

Robust higher-order phases rely on preserved crystalline symmetries (rotation $n$ 0, inversion $n$ 1, mirror $n$ 2), antiunitary symmetries (time reversal $n$ 3, particle-hole $n$ 4), and (in superconducting cases) combinations thereof. The codimension and localization of protected modes are determined by how these symmetries constrain the allowed mass domain-wall configurations at the boundaries.

3. Minimal Models and Prototypical Realizations

Explicit lattice and continuum Hamiltonians have been constructed across physical platforms:

Lattice Constructions

BBH Model: $n$ 5-classified, four-band model with quantized bulk quadrupole and four zero-energy corner modes; supports sub-symmetry-protected boundary states under partial symmetry breaking (Kang et al., 19 Dec 2025).
3D SSH-Type Models: Cubic lattices with alternating strong and weak links implement 3D octupole insulators with corner charges at the intersection of three surfaces (Kachin et al., 2021, Zhang et al., 2019).
Creutz Ladder Generalization: Direct product of two 1D chiral Creutz ladders with flux yields a 2D model supporting both first-order (edge) and second-order (corner) modes, with analytical phase boundaries dictated by winding numbers (Lahiri et al., 2022).
Higher-Order Topological Dirac Superconductors (HOTDSCs): 8-band BdG models yield coexisting 3D Dirac nodes, gapped surfaces, and symmetry-protected helical Majorana hinge modes under $n$ 6, $n$ 7, and $n$ 8 symmetry (Zhang et al., 2019).
Magnetic-Flux Induced HOTSCs: Staggered flux, Zeeman field, and antiferromagnetism generate higher-order topology in $n$ 9-wave superconductors without spin-orbit interaction, with corner, hinge, and corner states tunable via flux and stacking (Xiao et al., 16 Oct 2025).

Boundary Mode Analysis

Domain-Wall Mass Theory: In all dimensions, the Jackiw–Rebbi mechanism applies: domain walls in Dirac mass profiles, enforced by symmetry constraints, localize midgap states at discrete boundary submanifolds (Schindler, 2020, Schindler et al., 2017, Saha et al., 2021).
Nested Wilson Loops: Successive Wilson loop constructions extract multipole invariants in $d$ 0th order phases, predicting the location, multiplicity, and robustness of boundary states (Trifunovic et al., 2020, Kachin et al., 2021).

Fractal and Quasiperiodic Structures

Fractal Metamaterials: Sierpinski- and rhombic-fractal elastic plates with generalized SSH coupling exhibit a proliferation of inner and outer edge/corner states, with quantized real-space quadrupole moments and heightened robustness in high-symmetry geometries (Ma et al., 2023).
Quasicrystals: Aperiodic Fibonacci chains and squares inherit higher-order topology from periodic SSH models via cut-and-project methods, yielding corner and interface modes despite the absence of conventional Brillouin zone or Wannier center definitions (Ouyang et al., 2024).

Correlated and Interacting Systems

Higher-Order Quantum Paramagnets: Frustrated spin-1/2 Heisenberg models demonstrate coexistence of long-range plaquette order and symmetry-protected corner-like states, with quantized local Berry phase as a diagnostic (González-Cuadra, 2021).

Machine Learning and Network Science

Graph Neural Networks: Higher-order topological structures, defined as clique complexes, power multi-scale Personalized PageRank propagation in GNNs, yielding improved accuracy and robustness on heterophilic graphs (Wang et al., 22 Jul 2025).

4. Experimental Realizations and Probes

Higher-order topological structures have been realized in:

Phononic/Acoustic Metamaterials: 3D-printed lattices of resonant cavities/spheres coupled by tuning tube widths implement SSH and HOTI models. Direct field mapping via scanning microphones verifies dimensional hierarchy—coexisting surface, hinge, and corner states in a single sample (Zhang et al., 2019, Zheng et al., 2023).
Photonic Crystals: Inverse-designed dielectric metastructures (e.g., silicon-on-insulator) employ topology optimization to engineer second-order photonic topological insulators (SPTIs) with spectrally tuned edge and corner states for robust routing (Chen et al., 2020, Kachin et al., 2021).
Quantum Simulators: Cold atoms in optical lattices realize the 2D extended BBH and Aubry-André–Harper models, permitting direct pseudospin measurements of momentum-space topological invariants via Rabi oscillations (Jia et al., 2024).
Electronic Materials: IV–VI semiconductors (SnTe under strain) and van der Waals materials (Bi $d$ 1Br $d$ 2 chains, multilayer WTe $d$ 3) exhibit helical/hinge modes consistent with HOTI/HOTSC predictions, detectable via ARPES and STM (Schindler et al., 2017, Saha et al., 2021).

5. Extensions: Fractals, Semimetals, Semimetal-Superconductor Hybrids

The notion of higher-order topology extends beyond conventional insulators and superconductors:

Fractals: Non-integer Hausdorff dimension lattices break translation invariance but, via real-space quadrupole formulas, support a superabundance of edge/corner states, with experimental demonstrations showing up to fivefold increases in boundary state diversity compared to periodic structures. Robustness is enhanced in highly symmetric (rhombic) configurations due to fractional corner charge matching (Ma et al., 2023).
Semimetals: The construction principle applies to gapless systems (Dirac, Weyl, nodal-loop). Addition of Wilsonian mass terms sequentially gaps lower-dimensional boundary states, producing higher-order Fermi arcs or corner flat bands while preserving bulk nodal features (Calugaru et al., 2018, Wang et al., 2020).
Higher-Order Weyl Points: Chiral tetragonal crystals exhibit HOWPs as topological transitions between Chern-insulator (with surface arcs) and quadrupole-insulator (with hinge modes) $d$ 4-slices, resulting in spatially and momentum-dependent fractional hinge charges (Wang et al., 2020).
Superconductor Hybrids: HOTDSCs and higher-order Weyl superconductors feature coexisting 3D Dirac/Weyl nodes, gapped surfaces, and Majorana hinge or corner states, realized via symmetry-tuned BdG Hamiltonians (Zhang et al., 2019).

6. Applications and Outlook

Higher-order topological structures underlie a broad array of prospective devices and functionalities:

Robust Signal Routing: Frequency-selective edge/corner routing in photonic and phononic chips (Chen et al., 2020).
Quantum Computing: Hinge or corner-confined Majorana modes for fault-tolerant qubits in HOTSC architectures (Zhang et al., 2019, Xiao et al., 16 Oct 2025).
Sensing and Energy Concentration: Multi-frequency, multi-scale topological states enhance sensitivity and spatial concentration in mechanical and elastic metamaterials (Ma et al., 2023).
Topology-Driven Information Processing: Graph neural networks exploiting higher-order topology for noise-robust learning in networks with complex interaction patterns (Wang et al., 22 Jul 2025).

Unification of higher-order topology via K-theory, real-space multipole invariants, and domain-wall mass network models suggests further generalizations to correlated, non-Hermitian, or amorphous systems (Trifunovic et al., 2020, Schindler, 2020). The field is poised to expand toward programmable multifunctional quantum and classical systems—leveraging the full hierarchy of bulk, edge, hinge, and corner states, and beyond.