Observation of fractality-induced topology in photonic crystals

Abstract: Fractal topology--achieved by integrating nontrivial topology into fractal geometries with self-similarity and non-integer dimensions--has opened new avenues for exploring topological phases of matter. Recent theoretical advances revealed a counterintuitive fractal topology: fractality itself can induce nontrivial topology in an otherwise trivial system. Here, we report the first experimental observation of fractality-induced topology in a tight-binding-like photonic crystal, without relying on traditional driving mechanisms such as magnetic fields, staggered hopping, or spin-orbit coupling. We demonstrate that fractality alone is sufficient to lift the degeneracy of Kagome lattice band structure and induce topological corner states within the bandgap of the resulting fractal Kagome photonic crystal, which is a photonic higher-order topological insulator. This work experimentally reveals a novel mechanism for realizing nontrivial topological states, expanding both the fundamental frontier and potential application of topological physics.

• The paper establishes that fractal geometry, applied to a Kagome lattice, exclusively induces topological bandgaps and protected corner states without traditional symmetry-breaking.
• The paper employs microwave-scale experiments and near-field Fourier analysis to map the fractalized lattice to an effective ‘breathing’ Kagome model, validating theoretical predictions.
• The paper reveals that higher fractal generations yield multiple degenerate corner modes, highlighting the scalability of fractality for robust topological photonic devices.

Fractality-Induced Topology in Photonic Crystals: Experimental Realization and Mechanistic Insights

Introduction

This work establishes the experimental realization of nontrivial topology arising solely from fractal geometry in photonic crystals. Traditional approaches to topological phases in condensed matter and classical wave systems rely on symmetry breaking mechanisms—such as magnetic fields, staggered hopping, or spin-orbit coupling—to induce nontrivial band topology and support topologically protected states. Recent theoretical work, however, suggests fractal architectures can themselves serve as primary drivers of topological phenomena, independent of these conventional mechanisms. This study provides experimental confirmation of these theoretical predictions by observing higher-order topological insulator (HOTI) corner states induced by Sierpiński-type fractality in tight-binding-like Kagome photonic crystals.

Theoretical Framework and Mechanism

The foundational system is a Kagome lattice photonic crystal with equal inter- and intra-cell couplings ($w = v = 1$), supporting a gapless bulk spectrum with Dirac cone degeneracy at the $K$ point. Introducing the Sierpiński gasket fractal geometry lifts this degeneracy and opens a photonic bandgap. Through isospectral reduction (ISR), the fractalized Kagome Hamiltonian is mapped to an effective "breathing" Kagome lattice, in which energy-dependent intra-cell couplings $v(E)$ and onsite potentials $a(E)$ enact the necessary asymmetries to support higher-order topology. Notably, in the absence of any explicit symmetry-breaking perturbation or modulation of the coupling constants, fractality alone accounts for the emergence of localized topological corner states at specific eigenenergies ($E = -0.414$ for the first-generation Sierpiński fractal). This nonperturbative mechanism constitutes a marked departure from standard HOTI implementations and asserts fractality as a sufficient origin of band topology.

Experimental Methods

Experiments employ microwave-scale tight-binding-like Kagome photonic crystals constructed with dielectric and metallic rods embedded in triangulated air foam. Sierpiński fractalization is introduced up to second-generation, systematically controlling the self-similar geometric hierarchy. The photonic band structure is experimentally accessed via near-field measurements: excitation and probing are performed with monopole antennas interfaced to a vector network analyzer. The measured field distributions are Fourier-transformed to recover experimental band structures that can be compared with full-wave numerical simulations.

Key Results

Three principal findings characterize the results:

1. Fractality-Induced Bandgaps: Introduction of Sierpiński fractality opens complete photonic bandgaps at the formerly degenerate Dirac points. For first- and second-generation fractal crystals, three and seven topological bandgaps, respectively, are observed experimentally, interleaved with trivial gaps. The bulk gaps are characterized using rotational symmetry eigenvalues, yielding nontrivial topological invariants for the bandgaps.
2. Corner State Localization: Within each topological bandgap, highly localized corner modes are experimentally observed—manifesting as sharp peaks in transmission spectra when measured with probes positioned at the lattice corners, and absent when probed in the bulk. The spatial distribution of these states, reconstructed from measured $E_z$ field maps, confirms strong localization to corners of the Sierpiński fractal region and concordance with simulated eigenmode profiles.
3. Higher-Order Topology Robustness and Multiplicity: Second-generation fractal crystals support multiple (≥3) degenerate corner states per topological bandgap, distributed at distinct geometric corners. This scaling, tied directly to fractal generation, establishes the geometric self-similarity as a multiplier of topological mode count, not merely a single-mode effect.

These findings exclusively attribute the emergence of photonic topological corner states to fractality—a claim validated by both numerical simulation and experimental observation, absent engineered symmetry-breaking terms or modulated coupling networks.

Implications and Future Directions

The demonstration that fractality alone induces nontrivial topology in photonic crystals has several profound implications:

• Topological Design in Non-integer Dimensions: The ability to design topological states via fractal geometry provides a pathway to photonic devices that exploit non-integer spatial dimensionality for robust light localization and transport. This expands the topological design space beyond traditional Bravais lattice frameworks.
• Mechanism Generalization: The fractality-induced mechanism is basis-agnostic and can, in principle, be implemented in various classical and quantum platforms, including acoustics, mechanics, electronic circuits, and cold atom lattices.
• Scalability and Mode Multiplicity: The direct relationship between fractal generation order and the number of corner states enables scalable architectures for multi-mode topological devices, such as multi-channel routers, robust delay lines, and topological quantum memory elements.

Further investigation is warranted into the potential realization of higher-dimensional (e.g., three-dimensional Sierpiński constructs) fractality-induced topological phases, including surface and hinge state generalizations. Additionally, the observed mechanism may support explorations of dynamically reconfigurable topological metamaterials and systems that leverage fractal-global symmetries for protected transport and encoding.

Conclusion

This work constitutes the first experimental observation of fractality-induced photonic topology in tight-binding-like Sierpiński fractal Kagome crystals, with robust evidence of topological bandgap formation and emergent corner states arising solely from geometric self-similarity. The results expand the taxonomy of topological phases accessible to engineered photonic and classical wave systems, positioning fractal geometry as a standalone generator of nontrivial band topology. These findings open opportunities for the design and application of fractal-based topological matter in photonics and other engineered platforms.