Anomalous Higher-Order Boundary States

• Anomalous higher-order boundary states are defined as novel modes localized on codimension>1 features (e.g., corners, hinges) that escape conventional bulk invariants.
• They encompass diverse regimes—including Floquet dynamics, spectrally embedded states, and non-Hermitian effects—each with distinct localization and symmetry properties.
• This topic refines bulk-boundary correspondences by introducing new diagnostics such as dynamical polarization, symmetry-adapted invariants, and higher-group theoretical classifications.

Searching arXiv for the papers on arXiv to ground the synthesis. Anomalous higher-order boundary states are boundary phenomena on codimension-$>1$ loci—corners, hinges, rotation axes, or boundary defects—whose existence is not exhausted by conventional bulk band topology or by the static bulk invariants usually associated with higher-order topological insulators. In the cited literature, the phrase covers several distinct but mathematically related situations: Floquet corner modes present even when the Floquet bands are topologically trivial, corner-localized bound states embedded in a bulk continuum, non-Hermitian higher-order boundary modes obscured by skin accumulation, codimension-$n$ Dirac cones with parity anomaly, and interacting higher-order boundaries realized as anomalous interfaces between symmetry-enriched topological orders (Huang et al., 2018, Zhang et al., 2020, Benalcazar et al., 2019, Li et al., 2022, Wei et al., 12 May 2025). A persistent theme is that higher-order boundary localization alone is not a sufficient diagnostic: the relevant bulk or boundary invariant depends on whether the anomaly is dynamical, crystalline, non-Hermitian, interaction-enabled, or tied to higher symmetry (Jung et al., 2020).

1. Terminology, diagnostics, and recurring misconceptions

A central correction in the modern literature is that corner-localized states, fractional corner charge, filling anomaly, bulk polarization, and genuine higher-order bulk topology are distinct observables. In particular, localized zero-energy corner states need not be consequences of the bulk invariant that diagnoses corner charge, and conversely a nontrivial corner anomaly need not imply an isolated zero-energy corner eigenstate (Jung et al., 2020). This distinction is not cosmetic: it changes what counts as an anomalous higher-order boundary state and what constitutes a valid bulk-boundary correspondence.

For $C^3$-symmetric breathing Kagome systems, the lowest-band bulk polarization can correctly diagnose the fractional corner anomaly, but the zero-energy corner state is instead controlled by the $\mathbb Z_2$ composite Zak phase of edge-localized bands. In that case the corner mode is an edge-decoration effect, not an intrinsic higher-order bulk state. For the $C^4$-symmetric four-band topological crystalline insulator, the widely used quarter-filled polarization is not the correct invariant for the zero-energy corner mode; the proper correspondence is a half-filled corner charge $Q_c=\frac12$ together with chiral symmetry, which pins the compensating corner state to zero energy (Jung et al., 2020). This separates intrinsic higher-order corner states from edge-decoration-induced corner states and from trivial defect states.

The same caution appears in gapless settings. A nonzero filling anomaly is a robust higher-order crystalline diagnostic because it implies fractional corner charge, but it does not by itself guarantee a spectrally isolated corner eigenstate. In a gapless bulk, the same anomaly can manifest either as an exact corner-localized bound state in the continuum or merely as a resonance, depending on extra symmetry constraints that prevent hybridization with bulk states at the same energy (Benalcazar et al., 2019). This makes “anomalous” ambiguous unless the mechanism is specified: the anomaly may be in the filling, in the spectral embedding, or in the failure of an expected bulk invariant.

A broader anomaly correspondence also appears at the level of symmetry. For bosonic SPTs related by crystalline equivalence, anomalous mirror-protected and time-reversal-protected boundaries can be paired directly, both for surface SETs and for critical boundaries. This does not by itself classify hinge or corner states, but it provides a field-theoretic route for transferring anomaly data between crystalline and internal-symmetry settings (Zhang et al., 2023). This suggests that anomalous higher-order boundary states should be distinguished not only by codimension, but also by whether the anomaly is intrinsic to a full boundary, to a symmetry-fixed submanifold, or to a decorated boundary defect.

2. Floquet anomaly: micromotion, phase-band singularities, and dynamical multipoles

In periodically driven systems, anomalous higher-order boundary states arise when robust corner modes are protected by the full time evolution $U(\mathbf{k},t)$ rather than by the topology of the one-period Floquet operator $U(\mathbf{k},T)$. A two-dimensional Floquet higher-order topological insulator can therefore host corner modes in the $0$- and $\pi$-quasienergy gaps even when all Floquet bands are topologically trivial (Huang et al., 2018, Zhang et al., 2020). This is the canonical Floquet meaning of anomaly: static band topology fails, but micromotion remains topologically obstructed.

One formulation uses the periodized evolution

$n$0

where $n$1 is $n$2-periodic and satisfies

$n$3

The anomalous information lives in $n$4, not in the quasienergy bands of $n$5. Because no meaningful occupied-band subspace exists for the full micromotion, static nested Wilson loops fail, and the relevant bulk object becomes a dynamical polarization. The key construction is the dynamical mean polarization

$n$6

whose associated dynamical Wilson loops define first-order branches $n$7 and nested second-order branches $n$8. The higher-order Floquet invariant is not an instantaneous quantized quadrupole moment, but the winding of the averaged dynamical quadrupole branches over a drive cycle: $n$9 The nontrivial value diagnoses a higher-order Floquet phase whose corner states can appear in both the $C^3$0- and $C^3$1-gaps while the static quadrupole diagnosis vanishes (Huang et al., 2018).

A complementary formulation characterizes intrinsic anomalous Floquet HOTIs by singularities in the phase spectrum of the return map $C^3$2. In two dimensions these singularities live in the three-dimensional parameter space $C^3$3 and take the form of dynamical Weyl pairs or mirror-protected dynamical Dirac points at the principal phase-zone boundary $C^3$4. Their local dispersions are unconventional: they cannot be realized as spectra of any static $C^3$5 lattice, but resemble surface states of $C^3$6 topological crystalline insulators (Zhang et al., 2020). The corresponding higher-order bulk-boundary correspondence is established by phase-band dimensional reduction, which maps the $C^3$7 Floquet problem to a $C^3$8 class-AIII anomalous Floquet topological insulator whose protected end modes become the original corner states.

These two descriptions are consistent rather than redundant. The dynamical-polarization framework emphasizes nonequilibrium multipole transport and the obstruction of symmetry-protected micromotion at open boundaries, while the phase-band framework emphasizes singularity structure in $C^3$9 and dimensional reduction to lower-dimensional anomalous Floquet phases. Both replace static higher-order indices by genuinely dynamical ones (Huang et al., 2018, Zhang et al., 2020).

3. Spectrally embedded corner states: bound states in the continuum and resonances

A distinct anomalous regime occurs when higher-order boundary states are not in-gap modes at all, but exact corner-localized eigenstates embedded in a bulk continuum. In a $\mathbb Z_2$0- and chiral-symmetric square-lattice model with four orbitals per unit cell, the simultaneous presence of chiral symmetry and $\mathbb Z_2$1 forces the bulk spectrum to be gapless at zero energy. Nevertheless, in the topological phase $\mathbb Z_2$2, the system supports four corner-localized zero-energy bound states in the continuum (Benalcazar et al., 2019).

The higher-order bulk diagnosis in that model is multifold. The topological phase is characterized by symmetry representations at high-symmetry points, Wannier centers at the maximal Wyckoff position $\mathbb Z_2$3, quantized dipole polarization

$\mathbb Z_2$4

and corner filling anomalies quantified by

$\mathbb Z_2$5

for the first, middle, and upper band. Yet the paper emphasizes that the filling anomaly alone predicts only a corner-induced reorganization of state counting and fractional corner charge; it does not guarantee an isolated or exact corner eigenstate (Benalcazar et al., 2019).

Exact BIC protection requires extra symmetry beyond the topology protecting the filling anomaly. All zero-energy bulk states transform in the two-dimensional irrep $\mathbb Z_2$6 of $\mathbb Z_2$7, while the four corner states decompose as

$\mathbb Z_2$8

The $\mathbb Z_2$9 and $C^4$0 corner states cannot hybridize with bulk $C^4$1-states by irrep mismatch. The remaining $C^4$2 corner doublet is pinned to zero by the combined action of $C^4$3 and chiral symmetry, since the two states are both degenerate and related as chiral partners. This produces exact corner BICs even though many zero-energy bulk states are present at the same real energy (Benalcazar et al., 2019).

The paper proposes a direct condensed-matter BIC diagnostic by adding a fictitious non-Hermitian absorber on the bulk-like region,

$C^4$4

and then identifying nearly real eigenvalues whose imaginary parts vanish exponentially with system size. This distinguishes exact corner BICs from higher-order topological resonances, which retain strong corner weight but acquire finite linewidth when either chiral symmetry or $C^4$5 is broken (Benalcazar et al., 2019).

This regime broadens the spectroscopic meaning of anomalous higher-order boundary states. The anomaly is no longer only that boundary modes evade static higher-order bulk indices, but that the bulk-boundary correspondence can produce corner-localized states that remain exact eigenstates inside a gapless bulk continuum. When extra symmetry is relaxed, the same higher-order crystalline topology persists in weaker form as a resonance rather than a BIC (Benalcazar et al., 2019).

4. Non-Hermitian and extrinsic anomalous boundary correspondence

In non-Hermitian systems, anomalous higher-order boundary states are complicated by the coexistence of topological boundary localization and the non-Hermitian skin effect. A normal-density corner accumulation is therefore not sufficient to identify a genuine higher-order boundary mode, because bulk, edge, and corner sectors can all be driven to the same physical corner by nonreciprocity (Yang et al., 2024). The exact framework developed for nonreciprocal hypercubic lattices separates the problem into two parts: destructive-interference-induced boundary localization in a reciprocal transformed model, and nonreciprocal skin localization in the original model.

The decisive conceptual distinction is between the surface gap and the separation gap. For a codimension-$C^4$6 boundary mode, the surface gap measures the distance to bulk modes at the same nonzero surface momentum, whereas the separation gap measures the minimal complex-energy distance to any bulk mode. For $C^4$7, these quantities generally differ: $C^4$8 As a result, a higher-order boundary mode can lose spectral isolation from the bulk on the complex-energy plane while remaining sharply boundary-localized because its surface gap stays open (Yang et al., 2024). Topological transitions are then tracked by generalized biorthogonal polarization and generalized surface Brillouin zones, while amoeba theory diagnoses bulk spectral embedding. This resolves a higher-dimensional anomalous bulk-boundary correspondence rather than abolishing it.

A second non-Hermitian mechanism arises from the sensitivity of non-normal boundary Hamiltonians to intrinsic bulk-induced perturbations. Writing the full Hamiltonian in block form and integrating out the bulk yields the exact effective boundary Hamiltonian

$C^4$9

Although the induced couplings between opposite topological boundaries are exponentially small in the transverse size, the boundary Hamiltonian is non-normal, with condition number

$Q_c=\frac12$0

Because $Q_c=\frac12$1 can grow exponentially with boundary length, tiny intrinsic perturbations can reorganize the boundary spectrum and produce a transition between hybrid skin-topological states and scale-free topological boundary states (Liang et al., 23 Sep 2025). In the non-Hermitian BBH example this leads to corner states with loop-like spectra and localization length proportional to system size, so that the existence of zero-energy higher-order corner states becomes size dependent.

A third, conceptually parallel, development concerns quantum walks. There, a large class of Floquet-anomalous boundary states can be interpreted as extrinsic rather than intrinsic: the topology resides in a lower-dimensional boundary unitary $Q_c=\frac12$2, not in the bulk effective Hamiltonian alone (Bessho et al., 2021). The resulting boundary counting relation is modified to

$Q_c=\frac12$3

while a boundary decoration with extrinsic invariant $Q_c=\frac12$4 shifts the individual $Q_c=\frac12$5- and $Q_c=\frac12$6-boundary charges simultaneously. This is formally analogous to extrinsic higher-order topology in static HOTIs, except that in quantum walks the extrinsic mechanism already governs first-order boundary states (Bessho et al., 2021).

Taken together, these non-Hermitian and quantum-walk results enlarge the anomaly concept from “boundary mode without a bulk invariant” to “boundary mode whose nature is obscured by non-Bloch localization, non-normal sensitivity, or boundary-decoration topology.” In all cases, a bulk-only Bloch classification is insufficient (Yang et al., 2024, Liang et al., 23 Sep 2025, Bessho et al., 2021).

5. Interacting realizations: anomalous gapped boundaries and topological order

Interactions can transmute higher-order boundary anomalies rather than eliminate them. For inversion-protected $Q_c=\frac12$7 higher-order topological insulators and superconductors, free-fermion hinge or corner modes can be symmetrically gapped by covering inversion-conjugate surface regions with non-Abelian surface topological orders $Q_c=\frac12$8 and $Q_c=\frac12$9. The higher-order anomaly then survives not as a propagating codimension-2 free mode, but as an anomalous interface inside the surface: the common boundary between the two inversion-related STOs cannot be realized as a strictly $U(\mathbf{k},t)$0 inversion-symmetric interface with the same symmetry action (Li et al., 2022).

For class A and AII higher-order topological insulators, the relevant STO is the $U(\mathbf{k},t)$1 $U(\mathbf{k},t)$2-Pfaffian,

$U(\mathbf{k},t)$3

while for class D and DIII higher-order superconductors the relevant order is $U(\mathbf{k},t)$4, obtained through the conformal embedding

$U(\mathbf{k},t)$5

The interface can be fully gapped by symmetry-preserving interactions only when the original higher-order bulk contribution is included. Thus the interacting manifestation of higher-order topology is an anomalous gapped boundary between inversion-related STOs, or, for third-order phases, anomalous inversion-related point defects on such an interface (Li et al., 2022).

A complementary interaction-driven mechanism appears in boundaries with pure global anomaly. A $U(\mathbf{k},t)$6 chiral $U(\mathbf{k},t)$7 doublet carrying Witten’s anomaly can be driven into a fully gapped $U(\mathbf{k},t)$8 topological order by breaking symmetry, gapping the defect cores, and condensing bound states of vortex-loop flavors. The topological order remains anomalous because the link of the two vison-loop flavors is a Hopf soliton, and the Hopf soliton is fermionic (You et al., 2015). The same structure persists one dimension higher: on the $U(\mathbf{k},t)$9 boundary of a $U(\mathbf{k},T)$0 topological superconductor, linked flux membranes realize the higher-dimensional analog of the anomaly and likewise obstruct symmetric confinement (You et al., 2015).

This distinction between perturbative and global anomaly is crucial. The paper argues that if the boundary anomaly is perturbative, a fully gapped symmetry-preserving boundary topological order is obstructed; if the anomaly is purely global, a symmetric gapped topological order may exist for spatial dimension $U(\mathbf{k},T)$1, but it cannot be further confined to a trivial symmetric phase because special defect links remain anomalous (You et al., 2015). For higher-order systems, this implies that anomalous hinge or corner modes need not survive as explicit gapless excitations under interactions; they may reappear as anomalous defect sectors of a gapped surface topological order (Li et al., 2022, You et al., 2015).

6. Higher symmetries, crystalline correspondence, and high-dimensional generalizations

A more abstract extension replaces ordinary symmetry by higher symmetry and recasts boundary anomalies as inflow from higher-SPTs. Some anomalies are too severe to be cancelled by an ordinary SPT bulk with only 0-form symmetry, but can be cancelled after extending the symmetry to a higher group and introducing emergent higher-form gauge fields. In this framework, a severe anomaly on a $U(\mathbf{k},T)$2-dimensional boundary can be regulated as the symmetric boundary of a higher SPT protected by a $U(\mathbf{k},T)$3-group symmetry (Thorngren et al., 2015). Generalized cobordism theory further classifies higher-SPTs and their boundary fermionic or bosonic anomalies through bordism groups $U(\mathbf{k},T)$4, where $U(\mathbf{k},T)$5 encodes the tangential spacetime structure and $U(\mathbf{k},T)$6 is a higher-group classifying space (Wan et al., 2018). These works do not directly classify hinge or corner modes, but they furnish a bulk anomaly-inflow language for boundary theories with higher symmetries.

Crystalline correspondence provides an intermediate layer between these field-theoretic constructions and condensed-matter higher-order topology. Anomalous mirror-protected and time-reversal-protected boundaries of $U(\mathbf{k},T)$7 bosonic SPTs can be mapped directly to one another, both for surface SETs and for critical boundaries, and the mirror-line theory on a surface can be obtained as a symmetry-respecting domain wall inside the time-reversal-anomalous surface theory (Zhang et al., 2023). This is already a codimension-2 anomalous boundary structure, even though the paper does not frame it as a complete hinge/corner classification.

A fermionic crystalline realization appears on the $U(\mathbf{k},T)$8 boundary of a $U(\mathbf{k},T)$9 fermionic SPT with $0$0 symmetry. Under crystalline correspondence, the anomalous $0$1 symmetry of $0$2 same-chirality Weyl fermions is reinterpreted as a rotational crystalline anomaly concentrated on lower-dimensional boundary structures: a rotation axis and $0$3 decorated half-planes meeting at that axis (Cheng et al., 2024). For $0$4, the paper constructs a symmetry-preserving gapped $0$5 boundary whose low-energy theory is a $0$6 gauge theory. For $0$7, the construction yields a non-TQFT symmetric gapped boundary formed by stacking lower-dimensional $0$8 topological orders inhomogeneously around the axis (Cheng et al., 2024). This is an explicitly higher-order, defect-network realization of an anomalous boundary.

High-dimensional synthetic lattices provide another codimension-$0$9 extension. An $\pi$0-th order topological insulator in a $\pi$1-dimensional synthetic lattice can support anomalous $\pi$2-dimensional Dirac cones on codimension-$\pi$3 boundaries, thereby realizing the parity anomaly in $\pi$4 space-time dimensions (Wei et al., 12 May 2025). In the explicit $\pi$5 second-order model,

$\pi$6

each corner hosts a single $\pi$7 Dirac cone. After adding a time-reversal-breaking mass $\pi$8, each corner contributes a half-integer Hall response,

$\pi$9

The bulk diagnosis combines nested Wilson loops with time-reversal polarization to produce $n$00 invariants $n$01 that determine both the number and the momentum-space location of higher-order boundary Dirac cones (Wei et al., 12 May 2025). Here the anomalous higher-order boundary state is not a corner zero mode but a codimension-2 Dirac theory whose odd multiplicity is itself anomalous.

These developments suggest a broad unifying picture, although such an overview goes beyond any single paper. Across Floquet, crystalline, non-Hermitian, interacting, and high-dimensional settings, anomalous higher-order boundary states are best understood as codimension-$n$02 boundary manifestations of an obstruction: static band topology, ordinary Bloch theory, or purely lower-dimensional symmetry implementation is insufficient. The obstruction may instead be encoded in micromotion, in defect statistics, in higher-group inflow, in symmetry-fixed submanifolds, or in codimension-$n$03 Dirac theories (Thorngren et al., 2015, Wan et al., 2018, Zhang et al., 2023, Cheng et al., 2024, Wei et al., 12 May 2025).