Fractal hierarchy enables exponential scaling of topological boundary states.

Abstract

Exponential growth describes an extremely rapid process ubiquitous in mathematics and across diverse physical, biological, and technological systems. Here, we introduce a class of fractal-inspired lattices that combine long-range periodic order with self-similar hierarchy, establishing a structural motif that enables exponential scaling of topological boundary states. We demonstrate this phenomenon in (i) a quasi-one-dimensional lattice chain constructed from Koch-curve unit cells and (ii) a two-dimensional periodic tiling lattice composed of Sierpiński-gasket unit cells. We show that, for suitable coupling parameters, both the number of topological boundary states <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>N</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math> and the number of topological minigaps <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>M</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math> grow exponentially with the fractal generation index <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ℓ</mi></math>. We find that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>N</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math> is an integer multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>M</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math>, with the integer determined by the underlying symmetry. This hierarchical scaling law is captured by  the multi-topological-phase theory and confirmed experimentally in laser-written photonic lattices. Our results identify fractal hierarchy as a design principle for controlling boundary-state multiplicity, revealing a fundamental interplay between topology, self-similar geometry, and periodic order. More broadly, this work suggests a route toward synthetic materials and integrated photonic platforms in which large numbers of robust boundary modes can be engineered within hierarchically structured architectures.