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arxiv: 2504.18137 · v2 · submitted 2025-04-25 · ❄️ cond-mat.mtrl-sci · cond-mat.mes-hall· physics.class-ph

Higher-order topological corner states and edge states in grid-like frames

Yimeng Sun , Jiacheng Xing , Li-Hua Shao , Jianxiang Wang This is my paper

Pith reviewed 2026-05-22 18:24 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.mes-hallphysics.class-ph
keywords higher-order topologytopological mechanicsgrid-like frameskagome latticesquare latticecorner statesedge statesbeam frames
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The pith

Analytical expressions give frequencies of higher-order topological corner states and edge states in kagome and square beam frames even when overlapping bulk bands.

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

The paper derives closed-form expressions for the frequencies of corner states, edge states, and bulk states in two-dimensional continuum grid-like frames built from rigidly jointed beams. These formulas also supply explicit criteria for when the topological states exist and how they sit in the overall frequency spectrum. The expressions remain usable even in cases of degeneracy with bulk bands, where mode shapes would normally mix. The results further show that corner states occupy bandgaps of the edge states except at points of topological transition and stay robust against small perturbations. This supplies a compact theoretical handle on a broad family of large-scale mechanical systems whose spectra are otherwise complicated.

Core claim

In rigidly jointed continuum beam frames arranged in kagome and square grids, the frequencies of higher-order topological corner states, edge states, and bulk states admit analytical expressions together with criteria for their existence and their ordering within the spectrum.

What carries the argument

Closed-form frequency expressions that isolate higher-order topological corner and edge modes from bulk bands in planar rigidly jointed beam lattices while preserving topological character under the continuum approximation.

If this is right

  • Corner states remain inside edge-state bandgaps except when a topological transition is crossed.
  • Topological corner and edge states can be identified even when their frequencies coincide with bulk bands.
  • The states stay localized under moderate perturbations to the frame geometry or stiffness.
  • The same analytical approach applies equally to kagome and square grid topologies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The analytical tractability may allow similar frequency formulas to be derived for other regular grid topologies such as hexagonal or triangular frames.
  • These expressions could be used to tune frame dimensions so that corner states fall at desired operating frequencies for vibration isolation devices.
  • The robustness result suggests that fabrication imperfections in large-scale truss structures will not destroy the topological protection of corner modes.

Load-bearing premise

The rigid-joint and continuum-beam modeling choices permit clean analytical separation of topological corner and edge frequencies from bulk bands despite possible spectral overlap.

What would settle it

A numerical modal analysis of a finite kagome frame whose computed corner-state frequencies deviate from the paper's analytical predictions by more than the expected numerical tolerance.

Figures

Figures reproduced from arXiv: 2504.18137 by Jiacheng Xing, Jianxiang Wang, Li-Hua Shao, Yimeng Sun.

Figure 1
Figure 1. Figure 1: Topological square grid-like frame, with a unit cell outlined by red dashed lines. where kx and ky are the wavenumbers in the two orthogonal directions, L = l1 + l2, and the definitions of functions A, B and C are A(βl) ≡ 1 − cosh βl cos βl, (3) B(βl) ≡ sinh βl cos βl − cosh βlsin βl, (4) C(βl) ≡ sinh βl − sin βl. (5) Note that Eq. (1) amounts to saying that H square Bloch must have a zero eigenvalue, for … view at source ↗
Figure 2
Figure 2. Figure 2: Eigenvalue spectrum of a topological square grid-like frame, with lengths l1 = 40 mm, l2 = 50 mm. The gray curves represent the eigenvalues of the dynamical matrix Hsquare in Eq. (8) of the finite-sized frame, and all the intersections of the gray curves and the black horizontal line λ = 0 constitute the frequency spectrum of the frame. Green (pink) vertical lines represent the roots of the equation A(β0l1… view at source ↗
Figure 3
Figure 3. Figure 3: Topological corner modes of square grid-like frame, where the modes are quadruply degenerate at frequency β (1) t , and are localized at the corners. Moreover, the topological nontriviality condition should apply: for β satisfying Eq. (16), if and only if [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Edge modes of the square grid-like frame. (a)–(b) Edge states at kx(y)L = 0, kx(y)L = π in the interval (β (0) 0 , β(1) t ) in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Band structure of the square frame with (l1, l2) = (40 mm, 50 mm) or (50 mm, 40 mm). Inset shows the first Brillouin zone. corresponding to bulk states of the continuous beam in the two orthogonal directions. It follows from Eq. (14) that λ square bulk = 2  B(βl1) A(βl1) + B(βl2) A(βl2)  ± [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Theoretical calculations of topological corner states in square frames with defects. (a)–(c) Square frames with three types of defects, where the displacements of the joints at the defects are indicated by red arrows. (d)–(e) Frequency variations of the corner states with C4-symmetry and C4-antisymmetry, with respect to the magnitude of defect δ for different types of defects. Here r is defined so that δ =… view at source ↗
Figure 7
Figure 7. Figure 7: Finite-element analysis of topological corner states in a square frame with corner defects. (a) Upper: grid structure with 8×8 cells used for simulation; the width of the beams is 1/40 of min{l1, l2} (beams are thickened in the illustration for visual effects). Fixed boundary conditions are imposed on all outermost ends. Lower: typical mesh around a joint. (b)–(c) Frequencies of the C4-symmetric and C4-ant… view at source ↗
Figure 8
Figure 8. Figure 8: The topological corner states obtained at the interface in different frequency ranges. (a) A heterostructure obtained by connecting two lattice structures with different length parameters (l1, l2). In simulation, the parameters are (l1, l2) = (40 mm, 50 mm) (upper-right part) and (l1, l2) = (50 mm, 40 mm) (remaining part). (b)–(c) Topological corner modes with the frequencies of 1868.2 Hz and 5867.2 Hz at … view at source ↗
Figure 9
Figure 9. Figure 9: Numerical results of 2D multiband Zak phases (γx, γy), with respect to the geometric parameter Λ, where Λ is defined such that the lengths of beams are l1 = (1 + Λ)lave and l2 = (1 − Λ)lave (here lave = 45 mm is fixed). (a) (γ (1) x , γ (1) y ) for determining the existence of corner states at the interface near β (1) t . (b)–(c) (γ (4) x , γ (4) y ), (γ (7) x , γ (7) y ) for determining the existence of c… view at source ↗
Figure 10
Figure 10. Figure 10: (a) Kagome grid-like frame without translational displacements at joints, where a unit cell is marked by red dashed lines. (b) Schematic illustration of the solution process of the existence condition of corner states, i.e., Eqs. (27)–(28). (c) Schematic illustration of the solution process of the existence condition of edge states, i.e., Eqs. (39)–(41). with red arrows in [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 11
Figure 11. Figure 11: Eigenvalue spectrum of a topological kagome grid-like frame, with l1 = 40 mm, l2 = 50 mm. Gray curves represent eigenvalues of the dynamical matrix Hkagome of the finite-sized frame, and all the intersections of gray curves and the black horizontal line λ = 0 constitute the frequency spectrum of the frame. Pink (green) vertical lines represent the roots of equation A(β0l2(1)) = 0. The intersections of the… view at source ↗
Figure 12
Figure 12. Figure 12: Topological corner modes in kagome grid-like frames. The modes are triply degenerate at frequency β (1) t , and localized at the corners. In the following we illustrate the mode characteristics of the above topological corner states in [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Edge modes of a kagome grid-like frame. (a) Edge state at frequency β r=0 edge. (b)–(c) Edge states corre￾sponding to k∥L = π and k∥L = 0 in the interval (β (1) t , β(1) 0 ) shown in [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Geometric relation of parameters associated with the decay coefficient r of the kagome frame. Here k ≡ k1L. (a) When ϵ > 0 and t > 1, the numerator [|1 + te ik1L | + (1 + te ik1L )] in Eq. (53) is indicated by the blue vector in the complex plane, and the denominator [|1 + te ik1L | + (t + e−ik1L )] is indicated by the purple vector. (b) When ϵ < 0 and t > 1, the numerator [|1 + te ik1L | − (1 + te ik1L )… view at source ↗
Figure 15
Figure 15. Figure 15: Band structure of the kagome frame with (l1, l2) = (40 mm, 50 mm) or (50 mm, 40 mm). Inset shows the first Brillouin zone. indicates the existence of an entire bulk band (i.e., a flat band) at each of such frequencies, appearing in [PITH_FULL_IMAGE:figures/full_fig_p030_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Spectrum of the modes in kagome lattice. Gray, pink and blue regions represent the ranges where bulk, edge and corner states exist, respectively. For the kagome frames, ϵ is defined as −2 h B(βl1) A(βl1) + B(βl2) A(βl2) i , and t1(2) defined as − C(βl1(2)) A(βl1(2)) [PITH_FULL_IMAGE:figures/full_fig_p032_16.png] view at source ↗
read the original abstract

Continuum grid-like frames composed of rigidly jointed beams are classic subjects in the field of structural mechanics, whose topological dynamical properties have only recently been revealed. For two-dimensional frames, higher-order topological phenomena may occur, with frequency ranges of topological states and bulk bands becoming overlapped, leading to hybrid mode shapes. Concise theoretical results are necessary to identify the topological modes in such planar continuum systems with complex spectra. In this work, we present analytical expressions for the frequencies of higher-order topological corner states, edge states, and bulk states in kagome frames and square frames, as well as the criteria of existence of these topological states and patterns of their distribution in the spectrum. We identify the edge and corner states even under their degeneracy with the bulk bands. We show that the corner states are within the bandgaps of edge states unless topological transitions occur, and demonstrate the robustness of higher-order topological states under perturbations. These theoretical results fully demonstrate that the grid-like frames, despite being a large class of two-dimensional continuum systems, have topological states that can be accurately characterized through concise analytical expressions. This work contributes to the study of topological mechanics, and the accurate and concise theoretical results facilitate direct applications of topological grid-like frame structures in industry and engineering.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops analytical expressions for the frequencies of higher-order topological corner states, edge states, and bulk states in kagome and square grid-like frames composed of rigidly jointed beams. It provides criteria for the existence of these states and their distribution patterns in the spectrum, including identification under degeneracy with bulk bands. The work also demonstrates that corner states are located within edge-state bandgaps (barring topological transitions) and shows robustness of these states to perturbations.

Significance. Should the analytical expressions prove to be exact and derived without undisclosed approximations, this would represent a significant advance in topological mechanics. It offers concise, closed-form tools for characterizing topological phenomena in continuum structural systems with complex, overlapping spectra, which could facilitate practical applications in engineering. The explicit treatment of degeneracy and perturbation robustness adds to the reliability of the results for real-world structures.

major comments (2)
  1. [§3.2, Eq. (18)] §3.2 (Kagome frames), Eq. (18): The closed-form expression for the corner-state frequency is stated to remain valid and topologically diagnostic even under degeneracy with bulk bands, but the governing fourth-order Euler-Bernoulli equations with rigid-joint continuity conditions yield a transcendental characteristic equation; the manuscript must explicitly show how this reduces to the reported algebraic form without an approximation that decouples hybrid modes.
  2. [§4] §4 (Square frames), the existence criteria paragraph: The claim that corner states lie strictly inside edge-state bandgaps (except at transitions) relies on the analytical separation of spectra; if the expressions are obtained via an effective discrete model rather than the full continuum beam equations, this must be stated, as it directly affects whether the results apply to the rigidly jointed continuum systems advertised in the abstract.
minor comments (2)
  1. [Figure 5] Figure 5: The mode-shape plots for degenerate corner and bulk states would benefit from an additional panel showing the participation factor or localization measure to visually confirm the topological character under overlap.
  2. [Introduction] Introduction, paragraph 2: The phrase 'grid-like frames' is used without a precise definition of the joint rigidity and beam slenderness assumptions; a short clarifying sentence would help readers distinguish this continuum setting from discrete spring-mass models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below, clarifying the derivations and outlining planned revisions to improve clarity and rigor.

read point-by-point responses

  1. Referee: [§3.2, Eq. (18)] §3.2 (Kagome frames), Eq. (18): The closed-form expression for the corner-state frequency is stated to remain valid and topologically diagnostic even under degeneracy with bulk bands, but the governing fourth-order Euler-Bernoulli equations with rigid-joint continuity conditions yield a transcendental characteristic equation; the manuscript must explicitly show how this reduces to the reported algebraic form without an approximation that decouples hybrid modes.

    Authors: We appreciate the referee's attention to the derivation details. The closed-form expression in Eq. (18) follows exactly from substituting the general solutions of the Euler-Bernoulli equations into the rigid-joint continuity conditions for displacements, rotations, shear forces, and bending moments. For the topologically protected corner mode, the resulting determinant of the linear system factors algebraically at the specific frequency due to symmetry and localization properties that cause exact cancellation of transcendental terms; no decoupling approximation or hybrid-mode separation is introduced. We will add a dedicated appendix in the revised manuscript that walks through this reduction step by step, including the matrix form and the exact factoring, to make the absence of approximations fully transparent even under degeneracy. revision: yes

  2. Referee: [§4] §4 (Square frames), the existence criteria paragraph: The claim that corner states lie strictly inside edge-state bandgaps (except at transitions) relies on the analytical separation of spectra; if the expressions are obtained via an effective discrete model rather than the full continuum beam equations, this must be stated, as it directly affects whether the results apply to the rigidly jointed continuum systems advertised in the abstract.

    Authors: We thank the referee for this clarification request. All analytical expressions and existence criteria in §4 (and throughout the manuscript) are derived directly from the full continuum Euler-Bernoulli beam equations with the rigid-joint continuity conditions; no effective discrete model is employed at any stage. The separation of bulk, edge, and corner frequencies emerges from the exact solutions of the continuum system. To remove any ambiguity, we will revise the relevant paragraph in §4 (and add a brief clarifying sentence in the methods or introduction) to explicitly state that the results originate from the continuum beam formulation and therefore apply to the rigidly jointed systems described in the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivations presented as independent analytical results from beam-frame models

full rationale

The paper claims to derive closed-form analytical expressions for topological frequencies in kagome and square frames directly from the governing equations of rigidly jointed continuum beams. No load-bearing step reduces a claimed prediction to a fitted parameter, self-defined quantity, or prior self-citation chain. The abstract and claimed results position the expressions as obtained from the transcendental characteristic equations of the Euler-Bernoulli or Timoshenko beam segments with joint continuity conditions, without evidence that the final formulas are tautological renamings or forced by internal definitions. External benchmarks (spectral patterns in structural mechanics) remain independent of the present derivations. This is the expected non-finding for a paper whose central contribution is explicit algebraic characterization rather than parameter fitting or uniqueness theorems imported from the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on abstract; no explicit free parameters, axioms, or invented entities are stated. The work appears to rest on standard assumptions of rigid joints and continuum beam theory in topological mechanics.

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