Observation of fractality-induced topology in photonic crystals

Technical Summary: Observation of Fractality-Induced Topology in Photonic Crystals

Problem Statement
The discovery of symmetry-protected topological phases has reshaped the understanding of quantum matter and inspired analogous states in classical wave systems. While traditional topological phases rely on mechanisms such as magnetic fields (breaking time-reversal symmetry), spin-orbit coupling, or specific lattice modulations (e.g., breathing modes), recent theoretical work suggested that fractal geometry itself could induce nontrivial topology. Specifically, L. Eek et al. predicted that the self-similar nature of fractal structures, such as the Sierpiński gasket, could lift degeneracies and generate topological states in systems that would otherwise be trivial. However, despite successful implementations of pre-existing topological models on fractal lattices, the experimental observation of topology induced solely by fractality—without external driving mechanisms like magnetic fields or staggered hopping—remained elusive.

Methodology
The authors addressed this challenge by designing and fabricating a tight-binding-like photonic crystal based on a Kagome lattice with Sierpiński gasket fractal geometry.

• Theoretical Framework: The study began with a standard Kagome lattice where intra-cell and inter-cell couplings were equal ($w = v = 1$), resulting in a gapless band structure with a degenerate Dirac cone at the K point. By introducing Sierpiński fractal geometry, the authors utilized the concept of isospectral reduction (ISR). This mathematical technique partitioned the fractal lattice sites into subsets, effectively reducing the original Hamiltonian to an equivalent "breathing" Kagome lattice Hamiltonian. In this reduced model, the fractal geometry induces energy-dependent intra-cell couplings ($v(E)$) and onsite potentials ($a(E)$), creating a condition where $v(E) < w$, which is known to support higher-order topological insulator (HOTI) phases.
• Photonic Implementation: To realize this in a physical system, the authors constructed a scalar-wave-like photonic crystal using dielectric rods (sites) and metallic rods (to confine Mie resonance and satisfy the tight-binding approximation). They fabricated three samples: a standard Kagome lattice, a first-generation Sierpiński fractal Kagome lattice, and a second-generation fractal lattice.
• Experimental Characterization: The samples were fabricated using perforated air foam embedded with copper and dielectric rods. Measurements were conducted using a vector network analyzer (Keysight E5080) with monopole antennas acting as a point source and a probe. The team measured transmission spectra and mapped the electric field ($E_z$) distributions. Fast Fourier transforms of the field distributions were used to extract experimental photonic band structures, which were compared against COMSOL Multiphysics simulations.

Key Contributions and Results
The paper reports the first experimental observation of fractality-induced topology in a photonic system.

1. Lifting of Degeneracy and Bandgap Opening: Experimental and simulated results confirmed that while the standard Kagome lattice exhibits a symmetry-protected Dirac point, the introduction of Sierpiński fractal geometry lifts this degeneracy. This lifting opens photonic bandgaps solely due to the fractal structure, without the need for external symmetry-breaking fields or engineered coupling variations.
2. Identification of Topological vs. Trivial Gaps: The authors distinguished between topological and trivial bandgaps using rotational invariants ($\chi^{(3)}$) derived from symmetry eigenvalues at high-symmetry points. The fractal lattices were shown to possess topological bandgaps characterized by nontrivial rotational invariants and non-zero nominal corner charges.
3. Observation of Topological Corner States: Within the fractality-induced topological bandgaps, the authors directly observed localized topological corner states.
   • For the first-generation fractal, three degenerate topological corner states were observed in the topological bandgaps, with field distributions tightly localized at the three corners of the Sierpiński structure.
   • For the second-generation fractal, a greater number of topological bandgaps and corresponding corner states were observed, demonstrating that higher-generation fractals can generate more complex topological features.
4. Experimental Validation: The measured bulk band structures (color maps) matched the simulated white solid lines with high fidelity. Transmission spectra showed distinct peaks in the topological bandgaps when the probe was placed at the corner, whereas no such peaks appeared in the trivial bandgaps. The measured electric field distributions confirmed the corner-localized nature of these states, aligning with theoretical predictions.

Significance
The authors claim that this work experimentally establishes fractality itself as a novel mechanism for generating nontrivial topological states. By demonstrating that self-similarity and non-integer dimensions can induce higher-order topological phases in an otherwise trivial system, the study expands the fundamental frontier of topological physics. The results suggest a new avenue for designing robust topological photonic devices without relying on traditional driving mechanisms. The authors note that the underlying mechanism is general and could potentially be extended to other physical platforms, including condensed-matter systems, acoustics, mechanics, electric circuits, and cold-atom setups, though they do not detail specific future applications beyond this general potential.