Bound states in the continuum of higher-order topological insulators

Alexander Cerjan; Wladimir A. Benalcazar

REVIEW 2 major objections 4 minor 77 references

Bound states in the continuum of higher-order topological insulators

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A two-dimensional higher-order topological lattice hosts four zero-energy corner-bound states inside its gapless bulk continuum, protected by the simultaneous presence of C4v and chiral symmetry.

desk verdict Loss-probe diagnostic is mathematically unsound, so the central BIC claim is not established; the symmetry idea is still worth exploring. read the letter →

arxiv 1908.05687 v4 pith:G5B6D4F6 submitted 2019-08-15 cond-mat.str-el cond-mat.mes-hall

Wladimir A. Benalcazar , Alexander Cerjan This is my paper · ORCID

classification cond-mat.str-elcond-mat.mes-hall

keywords boundstatesinthecontinuumhigher-ordertopologicalinsulatorscornerchiralsymmetryC4vresonancesfillinganomalynon-Hermitianprobing

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a two-dimensional lattice with higher-order topology can pin four zero-energy states to its corners even though the bulk spectrum is gapless at that same energy, making these states genuine bound states in the continuum (BICs). The authors propose a practical diagnostic: add a small uniform loss to every site outside the corner regions, then identify BICs as eigenstates whose complex energy stays real as the system grows. Using this test on a $C_{4v}$- and chiral-symmetric four-site lattice, they find four corner BICs in the topological phase ($|t|<1$) and none in the trivial phase. They argue that the BICs are protected by the simultaneous presence of $C_{4v}$ and chiral symmetry, and that breaking either symmetry converts the BICs into corner-localized resonances. If correct, the result shows that topological boundary states can survive without spectral isolation, widening the search space for crystalline topological phases.

What carries the argument

The central mechanism is the loss-probe diagnostic: the lattice is split into four corner 'system' regions $S$ and the rest as the 'environment' $R$, and a uniform on-site loss $-i\kappa$ is added to $R$ (Eq. 2). Bulk and edge states acquire negative imaginary energies of order $\kappa$, whereas BIC wave functions, being exponentially confined to $S$, have imaginary energies that vanish exponentially with system size. The supporting symmetry mechanism is the representation analysis of the corner states ($A_1\oplus B_2\oplus E$) against the bulk zero-energy states (pure $E$), combined with chiral symmetry, which forces any $E$-type hybridization to remain at zero energy and therefore cannot produce a physical avoided crossing that would destroy the BICs.

What would settle it

Diagonalize the original Hermitian lattice for increasing $n$ and isolate the zero-energy subspace: if the four corner-localized states have bulk weight that decays only algebraically, or if the loss-probe imaginary energies scale as $\kappa$ rather than exponentially with $n$, the BIC claim fails. A direct calculation that exhibits a basis of the degenerate zero-energy subspace in which the four corner states are not exact eigenstates would also falsify the paper's central assertion.

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Extended reading notes

Core claim

For the Bloch Hamiltonian $h(k)$ in Eq. (1), the paper's central claim is that the topological phase with dimerized hopping $|t|<1$ hosts four degenerate zero-energy corner-localized bound states in the continuum, embedded in the zero-energy continuum of the central bulk band, while the trivial phase $|t|>1$ does not. The four corner states are exact eigenstates in the thermodynamic limit whose penetration into the bulk decays exponentially, and they are unaffected by the loss-probe's environmental loss because they have no support in the lossy region. The symmetry argument is that under $C_{4v}$ the four corner states form the representations $A_1\oplus B_2\oplus E$, while all degenerate bulk states at zero energy transform as the two-dimensional $E$ representation; chiral symmetry pins every $E$-type state to zero energy, so a corner-bulk hybrid cannot move off zero energy and split. Thus the simultaneous preservation of $C_{4v}$ and chiral symmetry protects an identifiable BIC subspace, and breaking either symmetry turns the corner states into higher-order topological resonances.

Load-bearing premise

The load-bearing premise is that adding a small uniform loss to the environment and declaring 'BIC' any eigenstate whose imaginary energy vanishes as the lattice grows is a complete and valid way to certify bound states in the continuum of the original lossless lattice.

Editorial extensions

If this is right

Corner bound states can exist without a bulk gap, so higher-order topological boundary states are not contingent on spectral isolation.
The BICs are protected by $C_{4v}$ and chiral symmetry together; preserving the symmetries that protect the topological phase but breaking either of these two converts them into higher-order topological resonances.
The loss-probe method provides a direct numerical way to identify BICs in closed crystalline lattices, complementing open-system radiative definitions of BICs.
BIC protection holds for nonseparable Hamiltonians, so the separability of $h(k_x,k_y)$ into $k_x$ and $k_y$ parts is not the origin of these BICs.
The corner filling anomaly is a necessary onset condition but not sufficient: additional symmetries decide whether the corner states emerge as BICs or as resonances.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extending the symmetry logic, any chiral- and $C_n$-symmetric higher-order topological insulator whose corner states and bulk states carry incompatible irreps, or whose $E$-type partners are pinned to zero by chirality, should host corner BICs even with a gapless bulk; the paper does not test this general criterion.
The same loss-probe protocol could be applied to hinge modes of three-dimensional HOTIs or to other boundary-localized states, as long as the lossless region fully contains the exponentially confined mode — a straightforward but untested extension.
Because the loss probe maps onto gain-loss contrast, a photonic or acoustic metamaterial realization of this lattice should show four corner modes with near-zero linewidth while the surrounding bulk remains lossy; observing this would confirm the BIC character experimentally.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Here's my read of Benalcazar and Cerjan's BIC-in-HOTI paper. The concept is genuinely interesting: they argue that corner states can survive as exact bound states even when the bulk is gapless at the same energy, and they propose a symmetry protection mechanism (C4v + chiral) that goes beyond the known separability argument. That's a meaningful conceptual step, and the symmetry analysis is plausible. The paper also does a decent job situating itself relative to the acoustic HOTI work of Chen et al. (Ref. [57]).

The problem is in the diagnostic. The method is: add a uniform loss -iκ to the environment R (everything outside the four ns×ns corner squares), diagonalize H = H0 - iκ P_R, and pick states with Im E → 0 as the lattice size n grows. But there is an exact identity: for any eigenstate ψ of H, Im E = -κ ||P_R ψ||^2/||ψ||^2. So a vanishing imaginary part requires the eigenstate to have identically zero support in R. An exponentially localized BIC has a tail that penetrates into R, and for fixed ns that tail's weight in R is independent of n once n exceeds the localization length. So Im E should saturate at a non-zero value of order κ e^{-ns/ξ}, not decay to zero with increasing n. The observed decay in Fig. 2(b) is inconsistent with the stated exponential penetration. Either the selected states are exactly compact (and thus eigenstates of a decoupled cluster, not the infinite lattice), or ns is secretly varying. The paper doesn't resolve this. This is load-bearing: the loss probe is the main evidence that these are BICs rather than resonances.

The symmetry argument is a better foundation, but it's not a proof. It shows that the corner and bulk zero-energy states of the same E irrep cannot generically hybridize into split eigenstates while chiral symmetry holds, but it doesn't guarantee that exact corner-localized eigenstates of H0 exist. The numerics directly on H0 (e.g., spatial decay, participation ratio) would have settled the matter, but those are not presented. Also, there's a minor sign inconsistency: Eq. (2) says 0<κ<<1, but the figures use κ=-5×10^-2, which would be gain, not loss. And there's no code/data deposit.

Verdict: I would not reject the idea. The BIC existence in this model may well be true—the symmetry argument is suggestive—but the current manuscript's main diagnostic is flawed. A serious referee should ask for a direct numerical demonstration on the Hermitian lattice, or a corrected complex-energy analysis that accounts for the identity above. This deserves peer review, but as a major-revision candidate, not acceptance.

Who is this for? Researchers in topological photonics/metamaterials and HOTI theory. It's a think-piece that, if fixed, could change how people look for BICs in closed systems. As written, treat the central claim with caution.

Referee Report

2 major / 4 minor

Summary. The paper claims that two-dimensional higher-order topological insulators can host corner-localized bound states in the continuum despite having gapless zero-energy bulk bands. To identify such states, the authors propose a non-Hermitian loss-probe method: a small lossless region S (four corner squares) is kept intact while a large environment R is given uniform local loss, and eigenstates of the resulting non-Hermitian Hamiltonian whose imaginary energies tend to zero as the system grows are declared to be BICs of the original Hermitian lattice. Applying this to a C4v- and chiral-symmetric lattice, they find four zero-energy corner states in the topological phase, argue that they are protected by the simultaneous presence of C4v and chiral symmetries, and show that breaking either symmetry transforms them into corner-localized resonances. The paper also argues that their BICs are not due to Hamiltonian separability and that the filling anomaly alone is insufficient for BIC protection.

Significance. If established, the result would be significant: it would extend the notion of higher-order topological corner states to systems without a bulk gap, broaden the design space for topological metamaterials, and offer a generic diagnostic for BICs in closed crystalline systems. The paper includes direct spatial-density evidence and a symmetry-based protection argument that goes beyond earlier separability-based BIC mechanisms. However, the central diagnostic is formally flawed, and the symmetry argument for two of the four claimed BICs is incomplete. The core claim is therefore not supported by the presented numerics, although the underlying idea may be salvageable with a different analysis.

major comments (2)

[Bound states in the continuum; Eq. (2); Fig. 2(b); Supplemental Fig. S2] The loss-probe criterion is not a valid test for BICs of the original Hermitian Hamiltonian. For H = H0 - iκ P_R, any right eigenvector ψ satisfies Im E = -κ ||P_R ψ||^2 / ||ψ||^2. Thus a vanishing imaginary part requires the eigenstate to have exactly zero support in the lossy environment R. For a fixed lossless region S of size ns and an exponentially localized state with penetration length ξ, the integrated weight in R is dominated by the region just outside S and is independent of the total lattice size n once n exceeds ξ; it is of order e^{-2ns/ξ}, a nonzero constant. The imaginary part should therefore saturate at a nonzero value as n grows, not decay exponentially. The decay shown in Fig. 2(b) and the exactly zero values for fixed ns=4 in Fig. S2(a) imply that the selected states have no support in R, i.e., they are either exactly compact within S or some parameter such as ns or κ is being varied without being reported. This contradicts the paper's stated exponential penetration of the BICs into the bulk. The presented numerics therefore do not establish that the four corner states are eigenstates of the original Hermitian H0 embedded in the continuum.
[Symmetry protection of the BICs] The protection argument for the two E-representation corner states is incomplete. The claim that the hybridized states |ψ1,2⟩ are merely arbitrary choices in the highly degenerate zero-energy subspace does not prove that the localized corner states are eigenstates. A Hermitian Hamiltonian diagonalizes within the degenerate subspace, and generically the true eigenstates are mixtures of corner and bulk components unless an additional symmetry or a vanishing coupling forces a localized eigenstate to exist. To establish that the E corner states are BICs, one must show that there is an eigenstate whose overlap with all bulk states is zero, for example by symmetry or by a direct inverse-participation-ratio calculation in the thermodynamic limit. The loss-probe was intended to supply this evidence, but as argued above it is invalid.

minor comments (4)

[Eq. (2) and figure captions] The loss term is defined with 0<κ≪1 in Eq. (2), but the captions of Figs. 2, 3, and 4 state κ=-5×10^{-2}; a negative κ would produce gain rather than loss, so the sign convention is inconsistent and should be corrected.
[Fig. 2(b)] The caption describes the horizontal axis as 'system size' without stating whether the varied parameter is the total lattice size n or the size ns of the lossless corner regions; this ambiguity is material to the claimed exponential decay and should be clarified.
[BICs as a signature of the topological phase] The statement that indirect gap closings 'start to occur at t=0.5' appears to conflict with the earlier assertion that the topological transition is at |t|=1; please clarify which quantity is being described.
[Supplemental Fig. S2] The insets in Fig. S2 lack labeled axes and scales, making it difficult to verify the claim that the BIC imaginary energies are exactly zero as a function of n; adding labels would improve the presentation.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the corner BIC demonstration rests on direct numerical diagnostics and a symmetry argument, with only a minor non-load-bearing self-citation.

full rationale

The central claim—that the two-dimensional HOTI lattice of Eq. (1) hosts zero-energy corner-localized bound states in the continuum—is supported by an explicit numerical probe and by a symmetry analysis, not by fitting a parameter to the claimed outcome. The non-Hermitian loss term in Eq. (2) is used as a diagnostic: eigenstates whose imaginary energies approach zero with system size are identified as BICs, and Fig. 2(c)-(d) independently shows the selected states are corner-localized and embedded in the zero-energy bulk continuum. No parameter is fitted to force the four near-real eigenvalues; they emerge from the spectrum of H0 - iκP_R. The symmetry-protection section additionally provides a non-numerical argument: under C4v and chiral symmetry, the corner-state E-representation partners |C+> and |C-> are pinned to zero energy, and any C4v-allowed hybridization with bulk E states remains in the zero-energy subspace rather than producing a physical resonance. This argument does not reduce to the paper's conclusion by construction. The only notable self-citation is Ref. [11], used for Wannier-center positions and filling-anomaly indices. That prior work is published, external to the present model's BIC diagnosis, and is explicitly described as only the 'onset' for the states; the paper's own diagnostic and symmetry argument carry the load. The loss-probe criterion could be questioned on physical grounds (e.g., whether fixed loss region R gives a true exponential-in-n decay), but that is a validation/correctness concern, not circularity. Overall, the derivation is self-contained and the central claim is not equivalent to its inputs.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper's claim rests on the specific tight-binding model, a non-Hermitian detection criterion proposed here, and previously derived filling-anomaly invariants. No new physical entities are introduced; the term higher-order topological resonance is descriptive, not a new object.

free parameters (3)

t (hopping ratio) = 0.25 (Figs. 2, 4; 0.15 in Fig. 1)
Ratio between dimerized hoppings; sets the lattice in the topological phase |t|<1. The qualitative result is the same across the phase, so t is a model parameter rather than a fitted value.
k (environment loss strength) = 0.05, though captions write k=-5e-2
Strength of the imaginary on-site potential in the environment. The diagnostic requires small loss; the limit behavior is tested by system-size scaling rather than by fitting.
n_s (lossless corner region size) = 3 (Figs. 2-4) and 4 (Fig. S2)
Size of each corner region left lossless. Must exceed the BIC localization length; the paper states results are robust for sufficiently large n_s.

assumptions (6)

domain assumption The tight-binding Bloch Hamiltonian in Eq. (1) with four sites per unit cell is the physical model, and finite open-boundary lattices approximate the thermodynamic limit.
All numerical evidence and symmetry analysis are for this specific lattice; no proof is given that the conclusions extend beyond it.
ad hoc to paper The loss-probe criterion identifies BICs: eigenstates of the non-Hermitian Hamiltonian in Eq. (2) with imaginary parts tending to zero as system size grows correspond to BICs of the original Hermitian Hamiltonian.
This is the paper's proposed detection method. It is validated numerically by spatial profiles and scaling, but not proved.
domain assumption All zero-energy bulk states transform as the E irreducible representation of C4v, and the four corner states transform as A1+B2+E.
This representation assignment comes from the specific model and from Table S1 in the supplement; it is used in the hybridization and protection argument.
domain assumption The corner-induced filling anomaly and symmetry-indicator invariants from Ref. [11] apply to this lattice, so a fractional corner charge exists in the topological phase.
The paper imports these prior results, largely from the same group, to connect the BICs to the filling anomaly.
standard math Chiral symmetry forces zero-energy states to come in chiral pairs or to be eigenstates of the chiral operator.
This is standard linear algebra following from {h, Pi}=0 and is stated in Supplement C.
domain assumption The random symmetry-preserving perturbations described in Supplement D are generic, so no accidental symmetry beyond those stated protects the BICs.
The constraints ensure the desired symmetries, but genericity of the random matrices is assumed rather than proven.

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Cite this review

Pith. "Pith review of Bound states in the continuum of higher-order topological insulators." pith.science (2026). https://pith.science/paper/G5B6D4F6

@misc{pith2026190805687,
  author       = {Pith},
  title        = {Pith review of: Bound states in the continuum of higher-order topological insulators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G5B6D4F6}},
  note         = {Machine review of arXiv:1908.05687}
}

read the original abstract

We show that lattices with higher-order topology can support corner-localized bound states in the continuum (BICs). We propose a method for the direct identification of BICs in condensed matter settings and use it to demonstrate the existence of BICs in a concrete lattice model. Although the onset for these states is given by corner-induced filling anomalies in certain topological crystalline phases, additional symmetries are required to protect the BICs from hybridizing with their degenerate bulk states. We demonstrate the protection mechanism for BICs in this model and show how breaking this mechanism transforms the BICs into higher-order topological resonances. Our work shows that topological states arising from the bulk-boundary correspondence in topological phases are more robust than previously expected, expanding the search space for crystalline topological phases to include those with boundary-localized BICs or resonances.

Figures

Figures reproduced from arXiv: 1908.05687 by the authors.

**Figure 2.** FIG. 2. Probing the existence of bound states in the contin [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Complex energies in the lattice as a function of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Breaking the symmetries that protect the BICs. En [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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See Supplemental Material for a detailed description of the topological phases and invariants in this model, the implementation of the symmetry-breaking terms added to the pristine Hamiltonian for Fig. 4, and scaling argu- ments for the BICs. This material includes Refs. [65-67]
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A case in point is the lattice in Fig

have a corner-induced ﬁlling anomaly. A case in point is the lattice in Fig. 2(e) in Ref. 4. The symmetry indicator invariants and their topolog- ical indices for polarization and corner ﬁlling anomalies for the bands in our model are shown in Table S4 and S5. C. Constraints o...
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For chiral symmetry Under chiral symmetry, Πh(k)Π = −h(k), all T ma- trices must obey {T, Π} = 0. (S12)
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To ﬁrst satisfy C2 symmetry, ˆr2h(kx,ky)ˆr† 2 = h(−kx,−ky), we require [Tx1, ˆr2] = 0, {Tx2, ˆr2} = 0

For C4 symmetry Let us ﬁrst focus on the nearest neighbor T ma- trices. To ﬁrst satisfy C2 symmetry, ˆr2h(kx,ky)ˆr† 2 = h(−kx,−ky), we require [Tx1, ˆr2] = 0, {Tx2, ˆr2} = 0. (S13) Now, to satisfyC4 symmetry, ˆr4h(kx,ky)ˆr† 4 =h(ky,−kx), we additionally require Ty1 = ˆr4Tx1ˆr†...
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(S14) The two next nearest neighborT matrices are odd under C2 symmetry. Take ﬁrst T1 to obey {T1, ˆr2} = 0, (S15) and then determine T2 via the constraint due to C4 sym- metry, T2 =−ˆr4T2ˆr†
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For reﬂection symmetry Under reﬂection symmetry along x, ˆMxh(kx,ky) ˆM † x = h(−kx,ky), four T matrices are even under Mx and two are odd, [Tx1, ˆMx] = 0 , [Ty1, ˆMx] = 0, [Ty2, ˆMx] = 0, [T1, ˆMx] = 0, (S17) {Tx2, ˆMx} = 0, {T2, ˆMx} = 0. If more than one symmetry is to be p...
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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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These quantized dipole moments are accompanied by two edge energy bands (i.e., bands with edge-localized states) spectrally isolated from the bulk energy bands [Fig

in the ﬁrst and fourth bands of the lattice [11]. These quantized dipole moments are accompanied by two edge energy bands (i.e., bands with edge-localized states) spectrally isolated from the bulk energy bands [Fig. 1(c)]. In addition to the dipole moments, the topological phase has a corner-induced ﬁlling anomaly [11] [62] with secondary topological indi...

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