---
title: Anomalous Higher-Order Boundary States
url: https://www.emergentmind.com/topics/anomalous-higher-order-boundary-states
type: topic
---

# Anomalous Higher-Order Boundary States

Searching arXiv for the provided papers 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 [1811.00555] [2010.07945] [1908.05687] [2203.01957] [2505.08820]. 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 [2010.10299].

## 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 [2010.10299]. 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 [2010.10299]. 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 [1908.05687]. 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 [2310.19266]. 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 [1811.00555] [2010.07945]. This is the canonical Floquet meaning of anomaly: static band topology fails, but micromotion remains topologically obstructed.

One formulation uses the periodized evolution
\[
U(\mathbf{k},t)=U_\varepsilon(\mathbf{k},t)\,[U(\mathbf{k},T)]_\varepsilon^{t/T},
\]
where \(U_\varepsilon(\mathbf{k},t)\) is \(T\)-periodic and satisfies
\[
U_\varepsilon(\mathbf{k},0)=U_\varepsilon(\mathbf{k},T)=\mathbb I.
\]
The anomalous information lives in \(U_\varepsilon\), not in the quasienergy bands of \(U(\mathbf{k},T)\). 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
\[
\hat x_{\mathrm{mean}}(t)=\frac{\hat x(t)+\hat x(0)}{2},
\]
whose associated dynamical Wilson loops define first-order branches \(\nu_{x,\mu}(k_y,t)\) and nested second-order branches \(\nu^{(\nu_x)}_{y,\tilde\mu}(k_x,t)\). 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:
\[
\tilde P^{(+\nu_x)}_{xy,\tilde\mu}=\int_0^T dt\, \partial_t\langle \nu^{(+\nu_x)}_{y,\tilde\mu}\rangle(t)\quad (\mathrm{mod}\ 1)=0,1\in\mathbb Z_2.
\]
The nontrivial value diagnoses a higher-order Floquet phase whose corner states can appear in both the \(0\)- and \(\pi\)-gaps while the static quadrupole diagnosis vanishes [1811.00555].

A complementary formulation characterizes intrinsic anomalous Floquet HOTIs by singularities in the phase spectrum of the return map \(\widetilde U(\mathbf{k},t)\). In two dimensions these singularities live in the three-dimensional parameter space \((k_x,k_y,t)\) and take the form of dynamical Weyl pairs or mirror-protected dynamical Dirac points at the principal phase-zone boundary \(\widetilde\phi=\pm\pi\). Their local dispersions are unconventional: they cannot be realized as spectra of any static \(3d\) lattice, but resemble surface states of \(4d\) topological crystalline insulators [2010.07945]. The corresponding higher-order bulk-boundary correspondence is established by phase-band dimensional reduction, which maps the \(2d\) Floquet problem to a \(1d\) 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 \((k_x,k_y,t)\) and dimensional reduction to lower-dimensional anomalous Floquet phases. Both replace static higher-order indices by genuinely dynamical ones [1811.00555] [2010.07945].

## 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 \(C_{4v}\)- and chiral-symmetric square-lattice model with four orbitals per unit cell, the simultaneous presence of chiral symmetry and \(C_{4v}\) forces the bulk spectrum to be gapless at zero energy. Nevertheless, in the topological phase \(|t|<1\), the system supports four corner-localized zero-energy bound states in the continuum [1908.05687].

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 \(1b\), quantized dipole polarization
\[
\mathbf P=\left(\frac e2,\frac e2\right),
\]
and corner filling anomalies quantified by
\[
Q^{(4)}=\frac14,\ \frac12,\ \frac14
\]
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 [1908.05687].

Exact BIC protection requires extra symmetry beyond the topology protecting the filling anomaly. All zero-energy bulk states transform in the two-dimensional irrep \(E\) of \(C_{4v}\), while the four corner states decompose as
\[
A_1\oplus B_2 \oplus E.
\]
The \(A_1\) and \(B_2\) corner states cannot hybridize with bulk \(E\)-states by irrep mismatch. The remaining \(E\) corner doublet is pinned to zero by the combined action of \(C_{4v}\) 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 [1908.05687].

The paper proposes a direct condensed-matter BIC diagnostic by adding a fictitious non-Hermitian absorber on the bulk-like region,
\[
h_{\mathrm{loss}}=-i\kappa \sum_{\mathbf r\in\mathcal R}\sum_{\alpha=1}^4 c^\dagger_{\mathbf r,\alpha}c_{\mathbf r,\alpha},\qquad 0<\kappa\ll1,
\]
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 \(C_{4v}\) is broken [1908.05687].

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 [1908.05687].

## 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 [2405.03750]. 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-\((d-n)\) 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 \(n\ge1\), these quantities generally differ:
\[
|\Delta E_{\mathrm{Surf.}}|\neq |\Delta E_{\mathrm{Sep.}}|.
\]
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 [2405.03750]. 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
\[
H_{\mathrm{eff}}=H_{\mathrm{boundary}}+X_1(E_{\mathrm{boundary}}-H_{\mathrm{bulk}})^{-1}X_2
=H_{\mathrm{boundary}}+H_{\mathrm{perturb}}.
\]
Although the induced couplings between opposite topological boundaries are exponentially small in the transverse size, the boundary Hamiltonian is non-normal, with condition number
\[
\kappa(V)=\frac{\sigma_{\max}(V)}{\sigma_{\min}(V)}.
\]
Because \(\kappa(V)\) 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 [2509.18952]. 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 \(U_{\mathrm{BDQW}}(\mathbf{k}_\parallel)\), not in the bulk effective Hamiltonian alone [2112.03167]. The resulting boundary counting relation is modified to
\[
\sum_{\epsilon_\alpha=0}\nu_\alpha^0-(-1)^d\sum_{\epsilon_\alpha=\pi}\nu_\alpha^\pi=n_{\mathrm{bulk}},
\]
while a boundary decoration with extrinsic invariant \(n\) shifts the individual \(0\)- and \(\pi\)-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 [2112.03167].

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 [2405.03750] [2509.18952] [2112.03167].

## 5. Interacting realizations: anomalous gapped boundaries and topological order

Interactions can transmute higher-order boundary anomalies rather than eliminate them. For inversion-protected \(3d\) 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 \(\mathcal A_{\mathrm N}\) and \(\mathcal A_{\mathrm S}=\overline{\mathcal A_{\mathrm N}}\). 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 \(2d\) inversion-symmetric interface with the same symmetry action [2203.01957].

For class A and AII higher-order topological insulators, the relevant STO is the \(2d\) \(\mathcal T\)-Pfaffian,
\[
\mathrm{T\!-\!Pf}\equiv [\mathrm{Ising}\times U(1)_{-8}]/\mathbb Z_2,
\]
while for class D and DIII higher-order superconductors the relevant order is \(\mathsf{SO}(3)_3\), obtained through the conformal embedding
\[
\mathfrak{so}(9)_1 \supset \mathfrak{so}(3)_3^{(1)}\times \mathfrak{so}(3)_3^{(2)}.
\]
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 [2203.01957].

A complementary interaction-driven mechanism appears in boundaries with pure global anomaly. A \((3+1)d\) chiral \(SU(2)\) doublet carrying Witten’s anomaly can be driven into a fully gapped \(\mathbb Z_{2N}\) 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 [1502.07752]. The same structure persists one dimension higher: on the \((4+1)d\) boundary of a \((5+1)d\) topological superconductor, linked flux membranes realize the higher-dimensional analog of the anomaly and likewise obstruct symmetric confinement [1502.07752].

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 \(d\ge2\), but it cannot be further confined to a trivial symmetric phase because special defect links remain anomalous [1502.07752]. 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 [2203.01957] [1502.07752].

## 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 \(d\)-dimensional boundary can be regulated as the symmetric boundary of a higher SPT protected by a \(d\)-group symmetry [1511.02929]. Generalized cobordism theory further classifies higher-SPTs and their boundary fermionic or bosonic anomalies through bordism groups \(\Omega_d^H(B\mathbb G)\), where \(H\) encodes the tangential spacetime structure and \(B\mathbb G\) is a higher-group classifying space [1812.11967]. 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 \(3d\) 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 [2310.19266]. 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 \((3+1)d\) boundary of a \((4+1)d\) fermionic SPT with \(C_N\times \mathbb Z_2^{\mathrm F}\) symmetry. Under crystalline correspondence, the anomalous \(\mathbb Z_{2N}^{\mathrm F}\) symmetry of \(\nu\) same-chirality Weyl fermions is reinterpreted as a rotational crystalline anomaly concentrated on lower-dimensional boundary structures: a rotation axis and \(N\) decorated half-planes meeting at that axis [2411.05786]. For \(\nu=N\), the paper constructs a symmetry-preserving gapped \((3+1)d\) boundary whose low-energy theory is a \(\mathbb Z_4\) gauge theory. For \(\nu=N/2\), the construction yields a non-TQFT symmetric gapped boundary formed by stacking lower-dimensional \((2+1)d\) topological orders inhomogeneously around the axis [2411.05786]. This is an explicitly higher-order, defect-network realization of an anomalous boundary.

High-dimensional synthetic lattices provide another codimension-\(n\) extension. An \(n\)-th order topological insulator in a \((2j+n)\)-dimensional synthetic lattice can support anomalous \((2j)\)-dimensional Dirac cones on codimension-\(n\) boundaries, thereby realizing the parity anomaly in \(2j+1\) space-time dimensions [2505.08820]. In the explicit \(4d\) second-order model,
\[
H_{\mathrm{corner}}(k_x,k_z)=s_x\sin k_x+s_y\sin k_z,
\]
each corner hosts a single \(2d\) Dirac cone. After adding a time-reversal-breaking mass \(H_{\mathrm M}=-Ms_z\), each corner contributes a half-integer Hall response,
\[
j_x=\frac12\frac{e^2}{h}E_z.
\]
The bulk diagnosis combines nested Wilson loops with time-reversal polarization to produce \(Z_2\) invariants \(\nu_{yw}^{G_xG_z}\) that determine both the number and the momentum-space location of higher-order boundary Dirac cones [2505.08820]. 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 a synthesis 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-\(>1\) 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-\(n\) Dirac theories [1511.02929] [1812.11967] [2310.19266] [2411.05786] [2505.08820].

Source: https://www.emergentmind.com/topics/anomalous-higher-order-boundary-states