Higher-Order Topological Phases

• Higher-order topological phases are quantum states with robust, lower-dimensional boundary modes localized at corners or hinges through discrete symmetry-breaking mass terms.
• They are characterized by quantized multipole moments and nontrivial bulk invariants that extend conventional bulk-boundary correspondence.
• Experimental realizations in crystalline, fractal, and Floquet systems validate the theory with observable corner and hinge states.

Higher-order topological phases (HOTPs) are states of quantum matter exhibiting robust boundary modes localized at lower-dimensional boundaries, such as corners or hinges, fundamentally extending the conventional (first-order) bulk-boundary correspondence. In these phases, a nontrivial bulk invariant protects boundary states of codimension $n>1$ (i.e., on boundaries of dimension $d-n$ in a $d$-dimensional system). The general construction involves adding discrete symmetry-breaking mass terms to Dirac-type models, leading to gapping out successively higher-dimensional boundaries and localizing zero modes to junctions of mass domain walls—a process characterized by the Jackiw–Rebbi mechanism and systematically classified via Clifford algebra structures and topological invariants such as quantized multipole moments. Recent theoretical and experimental advances reveal HOTPs in both crystalline lattices and a wide range of noncrystalline geometries, including fractals, amorphous networks, and Floquet systems, many of which support boundary phenomena beyond the reach of traditional first-order symmetry-protected topological phases.

1. General Construction and Classification Principles

A HOTP is constructed from a Dirac-type Hamiltonian $$H_0(\mathbf{k})=\sum_{i=1}^{m} d_i(\mathbf{k})\,\Gamma_i$$ and $n-1$ mutually anticommuting Wilson–Dirac mass terms $\{M_j\}$, each gapping out boundary modes in distinct real-space directions, resulting in boundary zero modes of codimension $n$ (Calugaru et al., 2018). The hierarchy of topological orders is dictated by the interplay of Clifford algebra ($\{\Gamma_i, \Gamma_k\}=2\delta_{ik}$; $\{M_j, M_\ell\}=2\delta_{j\ell}$; $\{\Gamma_i, M_j\}=0$), with the maximum attainable order given by the Clifford module structure per Altland–Zirnbauer symmetry class.

For a second-order phase in 2D, the addition of a mass term which changes sign under $C_4$ rotation generates domain walls on the edges, whose intersections at the corners host robust zero-energy modes. This principle generalizes to higher orders and dimensions; e.g., a third-order HOTP in 3D supports zero modes on corners due to two independent mass terms (Calugaru et al., 2018, Roy, 2019).

The codimension $d_c$ of the boundary zero modes is simply the number of mass terms added, $d_c=n$, and the boundary hosts modes of dimension $d-n$.

2. Archetypal Models and Bulk Invariants

Prototypical lattice models include the Benalcazar–Bernevig–Hughes (BBH) quadrupole insulator: $$H_{\mathrm{BBH}}(\mathbf{k}) = (γ_x + λ_x \cos k_x)\,\Gamma_4 + λ_x \sin k_x\,\Gamma_3 + (γ_y + λ_y \cos k_y)\,\Gamma_2 + λ_y \sin k_y\,\Gamma_1$$ where $\{\Gamma_i\}$ are four anticommuting Dirac matrices, and the bulk topological invariant is the quantized electric quadrupole moment $q_{xy}$, defined in real space as

$$q_{xy} = (1/2\pi)\, \mathrm{Im}\, \log\, \langle \Psi_G | e^{i\,2\pi\,Q_{xy}} | \Psi_G \rangle \;\mathrm{mod}\; 1$$

with $$Q_{xy} = \sum_{r=(x,y)} x\,y/(L_x\,L_y) (n_r - \bar{n})$$ (Yang et al., 2023).

In 3D, chiral hinge insulators are constructed analogously, with surface mass terms changing sign under a point-group symmetry (e.g., $C_4$), and characterized by either Chern–Simons, Wannier Chern, or quadrupole winding invariants (Yang et al., 2023).

Second-order topology persists away from strictly crystalline geometries: models such as the Sierpinski carpet or triangle fractals, with real-space hopping generated to accommodate arbitrary site connectivity, exhibit quantized quadrupole moments $Q_{xy}=1/2$ and sharply localized corner modes, as confirmed both numerically and analytically (Manna et al., 2021).

3. Symmetry Classes, Protection Mechanisms, and Disorder Effects

HOTPs are protected by a variety of symmetry mechanisms:

• Crystalline symmetry: Point-group operations (rotation, mirror, inversion) can pin mass domain walls and enforce multipole quantization or symmetry-indicator formulas (Yang et al., 2023, Liu et al., 2020, Wang et al., 2021).
• Antiunitary spectral symmetry: Operators (e.g., complex conjugation $K$ in suitable gamma-matrix representations) can enforce $\{A, H\} = 0$ and thus spectral pairing, pinning unpaired zero modes at $E=0$ (Roy, 2019).
• Chiral symmetry: In class AIII, higher-order topology is protected by multipole chiral numbers $\nu_{xy}$ defined via projected multipole operators and singular value decomposition, quantifying zero modes per boundary (Benalcazar et al., 2021).
• Subsystem symmetry and non-invertible symmetries: Subsystem-protected and fractonic HOTPs leverage symmetries acting on rigid submanifolds, pinning projective representations and enforcing boundary anomalies via group-cohomology or mixed anomaly inflow (You, 2019, Gliozzi et al., 2022, Mana et al., 23 May 2025, Zhang et al., 2022).

Quenched disorder preserves higher-order topology under symmetry-respecting perturbations, with real-space invariants such as $q_{xy}$ and $\nu_{xy}$ remaining quantized up to critical disorder strengths. Disorder-induced HOTIs (HOTAI) exhibit transitions marked by gap closings and plateaus in localization diagnostics or conductance as tracked by e.g. the inverse participation ratio or edge Green functions (Yang et al., 2023).

4. Topological Invariants: Multipole, Quaternion, and Polarized Charge Descriptions

The hierarchy of HOTPs is characterized by bulk invariants beyond conventional winding or Chern numbers:

• Quadrupole and higher multipole moments: Electric $q_{xy}$ and magnetic $\mathcal M_{ij}$ quadrupole moments, computed via real-space or layer-resolved methods, are signatures of second- and third-order topology in insulators or magnetic systems, with derivatives $\partial \mathcal M_{ij}/\partial \mu$ acting as quantized invariants counting hinge channels (Gliozzi et al., 2022).
• Multipole chiral numbers (MCNs): Integer invariants $\nu_{xy}$, $\nu_{xyz}$ defined from projected sublattice multipole operators in chiral-symmetric systems specify the degeneracy and sublattice occupancy of corner or hinge states (Benalcazar et al., 2021).
• Non-Abelian quaternion charges: In certain coupled-wire HOTPs, a non-Abelian quaternion invariant $q \in Q_8$ (arising from the Wilson loop over band frames) unites with an Abelian winding number to specify edge and corner degeneracies. True higher-order corner modes emerge only when both quaternion and winding invariants are nontrivial, yielding rich bulk–edge–corner correspondence (Pan et al., 24 Dec 2025).
• Polarized topological charges: HOTPs in chiral-symmetric Bloch Hamiltonians are characterized by local "polarized topological charges"—products of point-winding and mass polarization—whose sum (with prefactor, e.g. $1/4$ in 2D, $–1/8$ in 3D) yields the higher-order invariant. Phase transitions correspond to creation/annihilation of these charges (Jia et al., 2024).

5. HOTPs in Noncrystalline, Amorphous, and Fractal Geometries

HOTPs extend beyond crystals to nonperiodic and even fractional-dimensional systems:

• Amorphous HOTIs: Real-space quadrupole invariants remain quantized in random networks, with protected corner states insensitive to spatial symmetry (Yang et al., 2023).
• Quasicrystals and hyperbolic lattices: HOTPs are stabilized in Penrose, Ammann–Beenker, or $\{p,3\}$ hyperbolic tessellations, with boundary phenomena controlled by effective mass profiles and symmetry indicators.
• Fractal HOTPs: Sierpinski carpet and triangle lattices, at arbitrary generation $f$, host corner modes at "outer" corners, with bulk invariants (quadrupole, etc.) quantifying fractionalization. This demonstrates HOTPs in fractional Hausdorff dimensions $d_\mathrm{frac}<2$ (Manna et al., 2021).
• Floquet HOTPs: Periodically driven (Floquet) systems support gapless higher-order phases, with zero and $\pi$ corner states appearing robustly at bulk critical points, classified by generalized winding-number invariants and preserved chiral symmetry. Bulk–corner correspondence applies uniformly across gapped and gapless transitions (Zhou et al., 14 Jan 2025).

6. Experimental Realizations and Physical Signatures

HOTPs have been engineered and detected in a variety of platforms:

• Designer electronic fractals: STM-assembled atomic Sierpinski carpets and molecular fractals on solid-state substrates realize predicted HOTP spectra and boundary modes (Manna et al., 2021).
• Photonic and acoustic metamaterials: Fabrication of waveguide arrays and resonator networks allows for measurement of corner states, hinge arcs, and nontrivial bulk invariants. Photonic crystals with tunable geometry (e.g., $C_3$-symmetric configurations) demonstrate phase transitions controlled by symmetry-indicator indices and fractional corner charges (Wang et al., 2021).
• Topolectric circuits and superconducting qubits: Impedance/resonance and quantum-state tomography provide direct detection of HOTP invariants and phase transitions, via bulk pseudo-spin or corner population measurements (Niu et al., 2020, Yang et al., 2023).
• Cold atom quantum simulators: Implementation of chiral-symmetric Hamiltonians in multilevel atomic setups allows for momentum-space tomography of polarized topological charges, verifying corner modes and phase transitions (Jia et al., 2024).
• Multilayer TI/SC stacks under Zeeman field: In-plane Zeeman-coupled topological insulator/superconductor multilayers manifest Majorana hinge flat bands, hinge arcs, and nodal-line HOTSCs, with phase diagrams explicitly calculated in terms of critical pairing and field strengths (Nakai et al., 2023).

Measurable physical signatures include:

• Local density of states peaks at corners or hinges.
• Impedance anomalies in electric circuits.
• Quantized magnetization/hyperfine response in atomic systems.
• Chiral conductance/hinge current anomalies in 3D HOTIs (Gliozzi et al., 2022).

7. Outlook: HOTPs Beyond Conventional Paradigms

The HOTP landscape continues to expand:

• HOTPs and HOTSCs are realizable in fractal geometries, Cantor dust, Apollonian gaskets, and multi-fractal composites, suggesting new directions for topology in fractional dimensions (Manna et al., 2021).
• Subsystem and non-invertible symmetries underlie anomaly-protected HOTPs with fractonic dynamics, generalizing group cohomology to include fusion and duality symmetries (Zhang et al., 2022, Mana et al., 23 May 2025).
• Ongoing research explores HOTPs in non-Hermitian, Floquet, and strongly correlated models, seeking universal classification schemes for higher-order topology in arbitrary (possibly fractional) dimensions.

Open questions include the stability of fractionalized corner modes under strong correlations, the fate of HOTPs in Floquet or non-Hermitian settings, the role of interaction-driven topology in fractal boundary localization, and the full classification of higher-order topological and symmetry-protected phases beyond spatial symmetry constraints.

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References:

• (Calugaru et al., 2018) Higher Order Topological Phases: A General Principle of Construction
• (Yang et al., 2023) Higher-order topological phases in crystalline and non-crystalline systems: a review
• (Roy, 2019) Antiunitary symmetry protected higher order topological phases
• (Manna et al., 2021) Higher-order topological phases on fractal lattices
• (Wang et al., 2021) Higher-order topological phases in tunable $C_3$-symmetric photonic crystals
• (Gliozzi et al., 2022) Orbital magnetic quadrupole moment in higher order topological phases
• (Pan et al., 24 Dec 2025) Coupled-wire construction of non-Abelian higher-order topological phases
• (Zhou et al., 14 Jan 2025) Gapless higher-order topology and corner states in Floquet systems
• (Benalcazar et al., 2021) Chiral-Symmetric Higher-Order Topological Phases of Matter
• (Jia et al., 2024) Unveiling Higher-Order Topology via Polarized Topological Charges
• (Zhang et al., 2022) Classification and construction of interacting fractonic higher-order topological phases
• (Mana et al., 23 May 2025) Higher-order topological phases protected by non-invertible and subsystem symmetries
• (Nakai et al., 2023) Higher-order topological superconductor phases in a multilayer system