Scale-Free Topological Boundary States

• Scale-Free Topological Boundary States are boundary phenomena whose energy dispersion and localization remain independent of system size, defined by geometry and boundary conditions.
• They manifest in diverse setups such as gapless phases with flat bands, elastic metamaterials with flexural modes, Weyl waveguides with chiral states, non-Hermitian systems, and fractal lattices with exponential edge-state growth.
• These states enable robust edge transport and novel device functionalities by circumventing conventional finite-size constraints and the need for bulk gap-closing transitions.

Searching arXiv for recent and directly relevant papers on scale-free topological boundary states and related usages. Scale-free topological boundary states are topological boundary phenomena whose existence, quantization, localization, or multiplicity is not constrained in the conventional finite-size manner. In the current literature, the label is used for several distinct settings: dispersionless zero-energy surface flat bands in gapless topological phases (Matsuura et al., 2012); flexural free-edge states in graphene-like elastic metamaterials that survive even in ribbons only two unit-cells wide (Huang et al., 2024); chiral extended states in Weyl metamaterial waveguides whose discrete momenta do not depend on the waveguide thickness (Han et al., 2024); non-Hermitian boundary states with size-dependent localization and loop spectra generated by intrinsic perturbations (Liang et al., 23 Sep 2025); and fractal-inspired lattices in which the number of topological boundary states grows exponentially with the fractal generation index $\ell$ (Song et al., 1 Apr 2026). Taken together, these works show that “scale-free” does not denote a single universal mechanism, but rather a family of boundary effects in which topological behavior is controlled by geometry, boundary conditions, non-normality, or hierarchical structure rather than by the usual requirement of a macroscopically large topological domain.

1. Meanings and scope of the term

The literature attaches the phrase to several technically different boundary phenomena. In gapless topological phases, it refers to strictly flat boundary bands, with $E(k_\parallel)\equiv 0$ across a finite region of surface momentum space. In elastic metamaterials, it refers to edge bands on a free zigzag boundary that remain present when the transverse ribbon width is reduced to two unit cells. In Weyl metamaterial waveguides, it refers to a boundary-induced quantization condition that yields discrete momenta independent of thickness $d$. In non-Hermitian systems, it refers to a regime of “scale-free localization,” in which the inverse localization length scales as $1/L$ and the spectrum forms a point-gap loop. In fractal-like lattices, it refers to exponential scaling of the number of boundary states with generation index $\ell$.

Setting	Scale-free feature	Representative consequence
Gapless topological phases (Matsuura et al., 2012)	Dispersionless boundary bands	Flat-band region inside projection of nodal manifold
Elastic graphene-like metamaterial (Huang et al., 2024)	Edge states not limited by finite size	DB1/DB2 survive in ribbons only two unit-cells wide
Weyl metamaterial waveguide (Han et al., 2024)	Discrete momenta independent of thickness	$p=0$ chiral extended states for any $d$
Non-Hermitian boundary systems (Liang et al., 23 Sep 2025)	Localization length set by system size	$\kappa \approx \kappa_0+\kappa_1/N_x$
Fractal-inspired lattices (Song et al., 1 Apr 2026)	Boundary-state multiplicity grows with $\ell$	$N_\ell$ and $E(k_\parallel)\equiv 0$0 scale exponentially

A common misconception is that scale-free boundary states must be dispersionless. The cited works show otherwise. The gapless flat-band case is dispersionless, but the elastic metamaterial supports an upper edge band DB2 with nonzero slope $E(k_\parallel)\equiv 0$1, and the Weyl waveguide supports chiral extended states with linear dispersion. Another misconception is that scale-free behavior is always a bulk-determined surface effect. The Weyl waveguide work explicitly describes a family of chiral bulk states induced solely by the waveguide boundaries, and the non-Hermitian work identifies a boundary phase transition that does not require a conventional bulk spectral gap closing.

2. Gapless topological phases and dispersionless boundary flat bands

A systematic formulation for protected boundary states in gapless phases is given by the classification of stable Fermi surfaces and nodal lines using K-theory (Matsuura et al., 2012). If the Brillouin-zone dimension is $E(k_\parallel)\equiv 0$2 and the Fermi-surface or nodal-manifold dimension is $E(k_\parallel)\equiv 0$3, the codimension is

$E(k_\parallel)\equiv 0$4

For chiral systems, the Hamiltonian can be written in off-diagonal form,

$E(k_\parallel)\equiv 0$5

and flattened to a unitary block

$E(k_\parallel)\equiv 0$6

Whenever a $E(k_\parallel)\equiv 0$7 classification applies, the topological invariant is the winding number of this off-diagonal block on a small sphere $E(k_\parallel)\equiv 0$8 enclosing the gapless defect.

The generalized bulk–boundary correspondence proceeds by fixing the surface momentum $E(k_\parallel)\equiv 0$9 and viewing the remaining perpendicular problem as effectively lower-dimensional. If the boundary invariant $d$0 is nonzero, the half-space problem hosts zero-energy boundary modes at that $d$1. Deforming the sphere surrounding the bulk nodal object into large hemispheres shows that all $d$2 inside the projection of the nodal manifold inherit nontrivial topology. In class AIII nodal lines in three dimensions, this gives

$d$3

with $d$4 inside the projection and $d$5 outside.

The resulting boundary spectrum is strictly flat. In the half-space Dirac construction, if the bulk invariant at $d$6 is nonzero, there exists a normalizable zero mode satisfying $d$7, and the entire family remains at zero energy for all $d$8 in the flat-band region:

$d$9

The surface density of states therefore acquires a delta-peak,

$1/L$0

which broadens to a Lorentzian only after introducing a small $1/L$1.

The explicit examples in class AIII and class DIII illustrate two-dimensional zero-energy flat bands over the interior of projected nodal rings or nodal loops. In this usage, the boundary states are “scale-free” because the zero-energy manifold fills a finite region of surface momentum space rather than forming an isolated dispersing branch. This suggests a notion of scale freedom tied to codimension and projection geometry: the topological boundary response is fixed by the bulk nodal structure and does not depend on a narrow fine-tuned boundary momentum.

3. Free-edge flexural states in graphene-like elastic metamaterials

In graphene-like elastic metamaterials, scale-free topological boundary states arise in the flexural sector of a plate governed by linear elasticity (Huang et al., 2024). Starting from the three-dimensional displacement vector $1/L$2, the elastic equation is

$1/L$3

For out-of-plane modes $1/L$4, the relevant bands are selected through the out-of-plane polarization ratio

$1/L$5

with $1/L$6 for flexural modes. In the pristine hexagonal lattice with point group $1/L$7, this yields a doubly-degenerate Dirac cone at $1/L$8.

Introducing three identical one-beam resonators per cell lowers the symmetry to $1/L$9 while preserving the Dirac points. Near $\ell$0, the two flexural bands are mapped to an effective honeycomb tight-binding Hamiltonian

$\ell$1

Expanding in the reciprocal-lattice basis gives

$\ell$2

Treating $\ell$3 as a parameter and viewing $\ell$4 as a one-dimensional chain in $\ell$5 yields a Zak phase

$\ell$6

The result is $\ell$7 for $\ell$8 and $\ell$9 otherwise, predicting a pair of counter-propagating flexural edge modes on zigzag terminations between the projections of $p=0$0 and $p=0$1.

For a traction-free boundary, the elastic condition is $p=0$2. In zigzag ribbons, the flexural band structure contains two edge bands, DB1 and DB2, traversing the Dirac frequency $p=0$3. DB1 is nearly flat near $p=0$4, reproducing the zero-group-velocity midgap mode familiar from graphene zigzag edges. DB2, however, has nonzero slope, so $p=0$5 and the group velocity along the boundary is finite. The decisive scale-free statement is that these bands survive even in ribbons only two unit-cells wide: varying the number $p=0$6 of cells in the transverse direction does not remove DB1 or DB2. Here, “scale-free” means that the topological edge states do not require a large finite sample or a domain-wall interface between two distinct phases; they appear at any free zigzag boundary.

The frequency range of the edge states is quantified through an elastic analogue of Shannon entropy. For an eigenmode $p=0$7, the normalized probability density is

$p=0$8

and the Shannon entropy is

$p=0$9

Boundary-localized modes have low $d$0, while bulk-extended modes have high $d$1. Using the criterion $d$2 so that $d$3 remains below the single-cell threshold yields a frequency band

$d$4

Experimentally, edge-state transport is observed only for $d$5, with a $d$6 bandwidth, in correspondence with the entropy prediction.

The numerical validation uses COMSOL Multiphysics Structural Mechanics with tetrahedral mesh, maximum element size $d$7, and curvature factor $d$8. The material is Al-5745 with $d$9, $\kappa \approx \kappa_0+\kappa_1/N_x$0, and $\kappa \approx \kappa_0+\kappa_1/N_x$1. The experiment employs a $\kappa \approx \kappa_0+\kappa_1/N_x$2 aluminum plate, a $\kappa \approx \kappa_0+\kappa_1/N_x$3 PZT transducer driven by a $\kappa \approx \kappa_0+\kappa_1/N_x$4 amplifier, and scanning LDV at $\kappa \approx \kappa_0+\kappa_1/N_x$5 sampling with $\kappa \approx \kappa_0+\kappa_1/N_x$6 records and $\kappa \approx \kappa_0+\kappa_1/N_x$7 resolution over approximately $\kappa \approx \kappa_0+\kappa_1/N_x$8 points. Both simulation and experiment show flexural edge waves that circumvent corners and persist in the presence of missing resonators and glued-in defects, confirming robust boundary transport.

4. Boundary-induced chiral extended states in Weyl metamaterial waveguides

A different form of scale-free boundary phenomenon appears in Weyl metamaterial waveguides, where the boundary induces chiral extended states that propagate through the bulk of the slab (Han et al., 2024). The effective two-band Hamiltonian is

$\kappa \approx \kappa_0+\kappa_1/N_x$9

which hosts four Weyl points at

$\ell$0

with chiral charges $\ell$1. Confinement is introduced by perfect electric-conductor plates at $\ell$2 and $\ell$3, with boundary condition

$\ell$4

and $\ell$5.

Seeking spinor solutions of the form

$\ell$6

one obtains the secular equation

$\ell$7

For real $\ell$8, the bulk-guided modes are

$\ell$9

with dispersion

$N_\ell$0

For imaginary $N_\ell$1, one obtains surface modes localized near $N_\ell$2 or $N_\ell$3.

The scale-free condition occurs at special in-plane wavevectors satisfying $N_\ell$4, which gives $N_\ell$5. For $N_\ell$6, this solution satisfies the boundary condition irrespective of $N_\ell$7. Thus the quantization along the confined direction is not set by waveguide thickness. Restricting to $N_\ell$8 and taking $N_\ell$9 yields the closed-form dispersion

$E(k_\parallel)\equiv 0$00

with spinors proportional to $E(k_\parallel)\equiv 0$01 and $E(k_\parallel)\equiv 0$02. These are the chiral extended states.

At fixed energy, the conventional two-dimensional Fermi arcs are bound to the two surfaces, while the chiral extended states lie at the junctions where those arcs meet. The work characterizes them as “wormhole tunnels” connecting Fermi-arc surface states living in different two-dimensional spaces via the third dimension. The quantity

$E(k_\parallel)\equiv 0$03

distinguishes bottom-surface modes ($E(k_\parallel)\equiv 0$04), top-surface modes ($E(k_\parallel)\equiv 0$05), and states evenly distributed across $E(k_\parallel)\equiv 0$06 ($E(k_\parallel)\equiv 0$07). In the limit $E(k_\parallel)\equiv 0$08, the CES profile is nearly constant in $E(k_\parallel)\equiv 0$09,

$E(k_\parallel)\equiv 0$10

The topological content is inherited from the Weyl points. Each CES carries the same monopole charge as the underlying Weyl point, and one may equivalently define a one-dimensional invariant by integrating the Berry connection around a small loop in $E(k_\parallel)\equiv 0$11 around the CES locus. This shows that the CES is not a trivial waveguide mode but a boundary-selected topological channel. The key mechanism is that imposing $E(k_\parallel)\equiv 0$12 at both surfaces projects out one pseudospin sector and selects only the forward-propagating even-parity spinor. This suggests a boundary-controlled mode-selection principle distinct from the more familiar bulk-controlled Fermi-arc correspondence.

The experimental implementation uses a saddle-shaped metallic inclusion in a dielectric with $E(k_\parallel)\equiv 0$13, lattice constants $E(k_\parallel)\equiv 0$14 and $E(k_\parallel)\equiv 0$15, wire thickness $E(k_\parallel)\equiv 0$16, hole radius $E(k_\parallel)\equiv 0$17, frame radius $E(k_\parallel)\equiv 0$18, layer spacing $E(k_\parallel)\equiv 0$19, and copper thickness $E(k_\parallel)\equiv 0$20. A $E(k_\parallel)\equiv 0$21 supercell is sandwiched by PEC plates at $E(k_\parallel)\equiv 0$22 and $E(k_\parallel)\equiv 0$23. Measured field maps and Fourier transforms show a single gapless mode in the pseudo-bandgap at each in-plane momentum, four discrete branches in each quadrant of the isofrequency contour, and scattering-suppressed unidirectional bulk propagation past large obstacles.

5. Non-Hermitian scale-free topology and anomalous higher-order boundary states

In non-Hermitian systems, scale-free topological boundary states are formulated through an effective boundary Hamiltonian obtained by integrating out the bulk (Liang et al., 23 Sep 2025). With the full Hamiltonian written as

$E(k_\parallel)\equiv 0$24

the effective boundary model is

$E(k_\parallel)\equiv 0$25

The term

$E(k_\parallel)\equiv 0$26

is called an intrinsic perturbation because it is generated by the bulk Green’s function and feeds the bulk influence back into the boundary sector.

When the boundary consists of two physical edges $E(k_\parallel)\equiv 0$27 and $E(k_\parallel)\equiv 0$28,

$E(k_\parallel)\equiv 0$29

where $E(k_\parallel)\equiv 0$30 are induced on-site shifts and $E(k_\parallel)\equiv 0$31 are inter-edge couplings. The central mechanism is the sensitivity of a non-normal boundary Hamiltonian to these perturbations. By the Bauer–Fike theorem, strong spectral rearrangement occurs when

$E(k_\parallel)\equiv 0$32

and in a one-dimensional topological boundary such as an SSH chain the non-normality grows exponentially with length,

$E(k_\parallel)\equiv 0$33

At the same time, the induced inter-edge couplings scale as

$E(k_\parallel)\equiv 0$34

The resulting criterion for scale-free localization is expressed by

$E(k_\parallel)\equiv 0$35

where $E(k_\parallel)\equiv 0$36 denote the sign of the spectral winding on each edge. If this inequality holds, the open-boundary spectrum forms a loop or area, i.e. a point-gap, and eigenstates exhibit scale-free localization. If it is violated, the spectrum is line-like and the system is in the hybrid skin–topological regime. The boundary phase transition can also be characterized by the aspect ratio

$E(k_\parallel)\equiv 0$37

with hybrid skin–topological states for $E(k_\parallel)\equiv 0$38 and scale-free topological states for $E(k_\parallel)\equiv 0$39. At criticality, the point-gap radius follows square-root scaling,

$E(k_\parallel)\equiv 0$40

which is characteristic of an exceptional-point bifurcation.

In the BBH example, two classes of corner states appear. The first are scale-free corner states whose eigenvalues form a loop of radius $E(k_\parallel)\equiv 0$41 that slowly grows with $E(k_\parallel)\equiv 0$42, and whose wave functions satisfy

$E(k_\parallel)\equiv 0$43

Thus the inverse localization length scales as $E(k_\parallel)\equiv 0$44. The rescaled profiles collapse when plotted against $E(k_\parallel)\equiv 0$45. The second are zero-energy corner states inherited from the Hermitian BBH model, pinned at $E(k_\parallel)\equiv 0$46 for all sizes and having fractal dimension $E(k_\parallel)\equiv 0$47.

A critical point in this formulation is that the phase transition is a non-Hermitian boundary transition. No conventional bulk spectral gap closes; rather, the transition occurs in the effective boundary model generated by intrinsic perturbations. This directly counters the assumption that topology-changing transitions must be accompanied by a bulk gap closing in the full system. A plausible implication is that geometry and boundary non-normality can serve as control parameters for higher-order boundary states even when the bulk spectrum remains qualitatively unchanged.

6. Fractal hierarchy and exponential multiplicity of boundary states

A further use of “scale-free” appears in fractal-inspired lattices that combine self-similar hierarchy with translational order and thereby produce exponential proliferation of topological boundary modes (Song et al., 1 Apr 2026). Two concrete realizations are given. In the quasi-one-dimensional Koch-curve chain, generation $E(k_\parallel)\equiv 0$48 is the two-site SSH model, generation $E(k_\parallel)\equiv 0$49 replaces each bond by the five-site Koch element $E(k_\parallel)\equiv 0$50–$E(k_\parallel)\equiv 0$51–$E(k_\parallel)\equiv 0$52–$E(k_\parallel)\equiv 0$53–$E(k_\parallel)\equiv 0$54, and in general generation $E(k_\parallel)\equiv 0$55 has $E(k_\parallel)\equiv 0$56 sublattices per unit cell. In the two-dimensional Sierpiński-gasket tiling, generation $E(k_\parallel)\equiv 0$57 is the breathing Kagome lattice, generation $E(k_\parallel)\equiv 0$58 replaces each triangle by the first-generation Sierpiński gasket, and in general the number of sublattices is

$E(k_\parallel)\equiv 0$59

The key quantities are the total number of topological minigaps $E(k_\parallel)\equiv 0$60 and the total number of topological boundary states $E(k_\parallel)\equiv 0$61 across all minigaps. For the Koch-chain lattices,

$E(k_\parallel)\equiv 0$62

For the Sierpiński-tiling lattices with $E(k_\parallel)\equiv 0$63,

$E(k_\parallel)\equiv 0$64

The integer ratio $E(k_\parallel)\equiv 0$65 is fixed by symmetry: $E(k_\parallel)\equiv 0$66 in one dimension and $E(k_\parallel)\equiv 0$67 in two dimensions, enforced respectively by inversion ($E(k_\parallel)\equiv 0$68) or threefold ($E(k_\parallel)\equiv 0$69) symmetry.

The underlying framework is multi-topological-phase theory. The Bloch Hamiltonian is decomposed as

$E(k_\parallel)\equiv 0$70

after splitting the unit-cell degrees of freedom into intra-sites and inter-sites. Each intra-site index $E(k_\parallel)\equiv 0$71 defines an independent winding number

$E(k_\parallel)\equiv 0$72

which is either $E(k_\parallel)\equiv 0$73 or $E(k_\parallel)\equiv 0$74. By construction, the set $E(k_\parallel)\equiv 0$75 gives one invariant per minigap, and the exact count becomes

$E(k_\parallel)\equiv 0$76

with $E(k_\parallel)\equiv 0$77 and $E(k_\parallel)\equiv 0$78 in the one-dimensional Koch chain, and $E(k_\parallel)\equiv 0$79 and $E(k_\parallel)\equiv 0$80 in the two-dimensional Sierpiński tiling.

The experiments use CW-laser writing in nonlinear SBN crystals to create single-mode photonic waveguides with tunable couplings $E(k_\parallel)\equiv 0$81 and $E(k_\parallel)\equiv 0$82. In the one-dimensional Koch-chain experiment, $E(k_\parallel)\equiv 0$83 and $E(k_\parallel)\equiv 0$84 lattices of ten unit cells are studied. In the nontrivial regime $E(k_\parallel)\equiv 0$85, light remains confined to the edge waveguides; in the trivial regime, it diffracts into the bulk. The observed number of edge modes is $E(k_\parallel)\equiv 0$86 for $E(k_\parallel)\equiv 0$87 and $E(k_\parallel)\equiv 0$88 for $E(k_\parallel)\equiv 0$89, matching $E(k_\parallel)\equiv 0$90 and $E(k_\parallel)\equiv 0$91. In the two-dimensional Sierpiński-tiling experiment with $E(k_\parallel)\equiv 0$92 layers, there are $E(k_\parallel)\equiv 0$93 corner modes in each gap for $E(k_\parallel)\equiv 0$94 and $E(k_\parallel)\equiv 0$95 corner modes distributed across three gaps for $E(k_\parallel)\equiv 0$96. Fourier-space imaging resolves exactly $E(k_\parallel)\equiv 0$97 minigaps, and the states within each gap exhibit near-zero group velocity.

This version of scale-free topology differs from the earlier ones: the principal invariant is not a width-independent edge branch or a size-dependent localization length, but an exponential law for topological multiplicity embedded within a compact unit-cell architecture. The data suggest a design principle in which self-similar hierarchy controls not only whether boundary states exist, but how many of them appear.

7. Recurring mechanisms, distinctions, and open interpretations

Across these settings, scale-free topological boundary states are generated by markedly different mechanisms. In gapless topological phases, the driver is generalized bulk–boundary correspondence for stable Fermi surfaces or nodal lines, producing zero-energy flat bands over projected momentum regions (Matsuura et al., 2012). In the elastic metamaterial, the driver is the combination of free zigzag boundary conditions, flexural Dirac physics, and a nontrivial Zak phase, yielding DB1 and DB2 that survive in ultra-narrow ribbons (Huang et al., 2024). In the Weyl waveguide, the mechanism is the PEC spinor projector $E(k_\parallel)\equiv 0$98, which locks the $E(k_\parallel)\equiv 0$99 condition independently of thickness (Han et al., 2024). In the non-Hermitian case, the mechanism is exponential amplification of intrinsic perturbations by the large condition number of non-normal boundary Hamiltonians (Liang et al., 23 Sep 2025). In fractal-inspired lattices, the mechanism is the growth of independent winding invariants with fractal generation index $d$00 (Song et al., 1 Apr 2026).

Several distinctions are therefore essential. First, scale-free boundary states are not uniformly boundary-localized in the same sense. The gapless flat bands are zero-energy surface states; the elastic states are flexural free-edge modes with either nearly flat or finite-slope dispersion; the Weyl chiral extended states are bulk-spanning in the confined direction and connect Fermi arcs; the non-Hermitian states may have localization lengths set by system size; and the fractal hierarchy work concerns the total count of edge or corner states. Second, scale-free behavior need not imply zero group velocity. The elastic DB2 branch has finite group velocity, and the Weyl CES is a chiral transport channel. Third, conventional bulk gap closing is not a universal diagnostic. The non-Hermitian boundary transition occurs without a conventional bulk spectral gap closing.

These distinctions help prevent overgeneralization. The shared feature is not a single band-theoretic signature, but the weakening of the usual finite-size constraint on boundary topology. Depending on the platform, this appears as width-independent existence, thickness-independent quantization, localization-length scaling with system size, or exponential boundary-state multiplicity. A plausible synthesis is that scale-free topology identifies boundary phenomena whose decisive control variable is not simply system extent, but the structure of the boundary problem itself—whether encoded in projected nodal topology, free-edge elasticity, boundary spinor projectors, intrinsic non-Hermitian perturbations, or hierarchical unit-cell design.