Higher-Order Topological Phases

• Higher-order topological phases are defined by robust gapless or midgap boundary states localized on codimension greater than one (e.g., corners or hinges).
• They are characterized by quantized multipole moments, nested Wilson loops, and Dirac-mass constructions that systematically capture their topological invariants.
• These phases are realized in diverse platforms including twisted bilayer graphene, photonic crystals, and Floquet-driven systems, enabling innovative applications in quantum materials.

Higher-order topological phases (HOTPs) generalize the conventional notion of bulk–boundary correspondence by supporting robust gapless or midgap states at boundaries of codimension greater than one, such as hinges or corners, in contrast to conventional (first-order) topological insulators which host boundary states at faces of codimension one. Advances in HOTPs encompass broad developments in classification schemes, theoretical models, quantitative topological invariants, symmetry requirements, dynamical and Floquet phenomena, and applications in both electronic and synthetic quantum materials.

1. Fundamental Definition and Hierarchy of Higher-Order Topological Phases

A HOTP in $d$ spatial dimensions and $n$th order exhibits protected boundary states on $(d-n)$-dimensional manifolds, leaving all boundaries of lower codimension fully gapped. The codimension-$n$ property implies, for instance, zero-dimensional (0D) corner modes for second-order topology in two dimensions, or one-dimensional (1D) hinge states for second-order topology in three dimensions. This contrasts with the conventional “first-order” paradigm where $d$-dimensional systems host $(d-1)$-dimensional boundary modes (e.g., edge states in 2D quantum spin Hall insulators) (Saha et al., 2021, Hua et al., 2024, Pan et al., 24 Dec 2025). The higher-order bulk–boundary correspondence asserts that the presence of HOTP-induced boundary states is enforced by quantized topological invariants and protected by symmetry, not by fine-tuning or accidental degeneracy.

Formally, the boundary dimension for an $n$th-order topological phase in $m$ dimensions is $d_{c} = m-n$ (Saha et al., 2021). Table 1 summarizes prototypical boundary phenomena:

HOTP Order ($n$)	Spatial Dim. ($m$)	Mode Dim. ($d_c$)	Example
1 (First)	2	1	Edge states in 2D TIs
2 (Second)	2	0	Corner states in 2D SOTIs
2 (Second)	3	1	Hinge states in 3D SOTIs
3 (Third)	3	0	Corner states in 3D TIs

2. Symmetry, Classification, and Theoretical Frameworks

The existence and robustness of HOTP boundary modes fundamentally depend on the presence of certain symmetries:

• Internal symmetries (Altland–Zirnbauer classes): time-reversal $\mathcal{T}$, particle–hole $\mathcal{C}$, chiral $\mathcal{S}$, classifying strong TIs/TSCs.
• Crystalline symmetries: rotational ($C_n$), mirror ($M$), inversion ($I$), glide, or screw, essential for HOTP protection, e.g., quantized $C_4 \mathcal{T}$ or two-mirror symmetries (Hua et al., 2024).
• Subsystem symmetries: HOTPs may exist in the absence of global crystalline symmetries if protected by subsystem (e.g., planar) symmetries (You, 2019).

The minimal theoretical description of HOTPs is via Clifford-algebra Hamiltonians containing several mutually anticommuting Dirac-like matrices. For a $D$-dimensional first-order phase with $m$ Dirac matrices, the addition of $p-1$ further anticommuting “mass” matrices $M_a$ allows the systematic construction of $n=p$th-order HOTPs, with the locality and sign structure of each “Wilsonian mass” term engineering domain walls of progressively higher codimension (Calugaru et al., 2018, Lei et al., 2022). The domain wall construction combined with the Jackiw–Rebbi mechanism generically guarantees the existence and quantization of codimension-$n$ zero modes at the boundaries where $n$ masses simultaneously reverse sign.

3. Topological Invariants and Momentum-Space Characterizations

Conventional HOTP invariants derive from Abelian quantizations (e.g., Chern, $\mathbb{Z}_2$, winding numbers), but a diverse toolkit has emerged:

• Multipole moments: Quadrupole and, in higher dimensions, octupole moments act as robust bulk-to-corner topological indices (Niu et al., 2020, Hua et al., 2024). The 2D quadrupole moment $Q_{xy}$ distinguishes higher-order phases ($Q_{xy}=0.5$ for nontrivial SOTIs).
• Nested Wilson loops: HOTP invariants in momentum space, defined recursively (e.g., projecting onto Wannier bands then forming additional Wilson loops) capture the hierarchy of edge, hinge, and corner-state topology (Hua et al., 2024).
• Polarized topological charges: For chiral HOTPs, the net polarized topological charge $$W_d = (-2)^{-d} \sum_{n}\mathcal{C}_n\,\mathcal{P}_n$$, with $\mathcal{C}_n$ the node’s Jacobian sign and $\mathcal{P}_n$ its polarization, unifies the classification across dimensions (Jia et al., 2024).
• Non-Abelian invariants: In multi-band, PT-symmetric systems, invariants can take values in non-commutative groups like the quaternion group $Q_8$. In such "non-Abelian HOTPs," the topological phase is labeled by vectors like $(Q_x, w)\in Q_8 \times \mathbb{Z}$, and the presence of protected corner states requires both components to be nontrivial (Pan et al., 24 Dec 2025).

Phase transitions between HOTPs may occur via either bulk gap-closing (type-I) or by closing of boundary gaps (type-II), captured quantitatively in the momentum-space evolution of topological charges across band-inversion surfaces and monitored directly in quench dynamics (Jia et al., 2022).

4. Realizations in Model Systems and Synthetic Architectures

HOTPs have been theoretically and experimentally realized in a variety of settings:

• Electronic solids and moiré systems: Twisted bilayer graphene (TBG), WTe$_2$, Bi, and Bi$_4$Br$_4$ host SOTIs and 1D helical hinge modes, detectable via STM or transport (Hua et al., 2024).
• Photonic crystals and acoustic/phononic metamaterials: Chiral-symmetric and $C_3$-symmetric photonic/kagome lattices exhibit fractional corner charges and multiple robust corner modes controlled via geometry or long-range coupling (Wang et al., 2021, Wang et al., 2023).
• Fractal and amorphous networks: Fractal lattices such as the Sierpiński carpet or amorphous systems, provided their outer boundaries retain appropriate symmetry, are predicted and observed to host second-order zero modes, with bulk quantized multipole moments persisting well beyond the deep crystalline regime (Agarwala et al., 2019, Manna et al., 2021).
• Topoelectric circuits and network models: Synthetic Majorana or photonic networks with $C_4$, particle–hole, and phase rotation symmetries host higher-order HOTPs with a high multiplicity of protected corner states, robust to disorder and symmetry-preserving deformations (Liu et al., 2020, Liu et al., 2024).
• Floquet-driven and time-periodic systems: Periodically driven (Floquet) HOTPs support boundary modes at both 0 and $\pi$ quasienergies, and can host protected corner states even at gapless critical points (Zhou, 2020, Zhou et al., 14 Jan 2025, Nag et al., 2019).

Table 2 highlights key experimental HOTP platforms:

Platform	HOTP Type	Protection	Detection
Moiré TBG	SOTI	Mirror/rotation	STM, gapless corners
Breathing kagome photonics	SOTI	$C_3$, crystal symmetry	Fractional corner “charge”
Acoustic inversion-bridge	Multiple TCM	Chiral (AIII), LRC vs. NNC	Frequency scan, spatial probes
Network model (rings)	HOTP/STP	$C_4$, phase rotation, PH	LDOS peaks, four-terminal conductance
Floquet lattices	gapped/gapless	Chiral, time-frame, driving	Quasienergy spectroscopy
Fractals	SOTI/SOTSC	U(1)/particle–hole	LDOS, quadrupole invariants

5. Interacting and Non-crystalline Generalizations

Recent expansions of HOTP theory have revealed:

• Subsystem- and interaction-protected HOTPs: HOTSPT states arise in strongly interacting models and do not require global crystalline symmetry. Their existence can be certified by generalized Lieb–Schultz–Mattis theorems forbidding trivial gapped hinge or surface phases under subsystem or translational symmetry (You, 2019).
• Amorphous and fractal HOTPs: HOTPs are robust against disorder provided protecting symmetries persist on the system’s boundary. Quantized corner modes and invariant multipole moments persist in amorphous domains provided the boundary symmetry is preserved, and HOTPs can even arise in lattices of non-integer Hausdorff dimension (Manna et al., 2021, Agarwala et al., 2019).
• Superconducting HOTPs: HOTSCs feature Majorana hinge or corner zero modes, with classes of $\mathcal{C}$ or inversion-protected 3D HOTSCs supporting hinge flat bands, connecting arcs, and unique nodal line phases (Nakai et al., 2023, Roy, 2019).

6. Current Directions and Open Challenges

Research frontiers in HOTPs include:

• Non-Abelian HOTPs and braiding: The demonstration of quaternion or more general non-Abelian invariants opens routes to HOTPs where bulk–boundary correspondence is enriched beyond Abelian physics, offering potential for non-Abelian statistics and quantum information protocols (Pan et al., 24 Dec 2025, Hua et al., 2024).
• Disorder, non-Hermiticity, and Floquet engineering: The role of disorder, gain/loss, and time-periodic modulation are central in stabilizing, tuning, or destroying HOTPs, motivating both theoretical generalizations and experimental advances (Hua et al., 2024).
• High-order phase transitions and dynamical probes: Unified frameworks in momentum space, based on quench or spectroscopic protocols, can identify type-I and type-II HOTPTs using dynamical probes in ultracold atomic quantum simulators or solid-state devices (Jia et al., 2024, Jia et al., 2022).
• Device applications: Exploitation of multiple degenerate modes for lasing, high-density sensing, and robust wave manipulation is under active exploration (Wang et al., 2023).

7. Conclusion

HOTPs establish a flexible and unifying paradigm for robust, symmetry-protected boundary phenomena in both electronic and classical systems, fundamentally expanding the topology of matter beyond the tenfold way. Essential features are the codimension-$n$ bulk–boundary correspondence, a generalized set of topological invariants (Abelian and non-Abelian), and a crucial dependence on symmetry for the protection, classification, and practical realization of boundary modes. Advances now extend to amorphous, fractal, Floquet, interacting, and noncrystalline systems, with experimental confirmation of higher-order corner, hinge, and defect-localized modes in a wide array of physical platforms (Pan et al., 24 Dec 2025, Jia et al., 2024, Hua et al., 2024, Agarwala et al., 2019, You, 2019, Wang et al., 2023, Zhou, 2020).