<p>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 <InlineEquation ID="IEq1"><EquationSource Format="TEX">${N}_{{\ell}}$</EquationSource><EquationSource Format="MATHML"><math><msub><mrow><mi>N</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></EquationSource></InlineEquation> and the number of topological minigaps <InlineEquation ID="IEq2"><EquationSource Format="TEX">${M}_{{\ell}}$</EquationSource><EquationSource Format="MATHML"><math><msub><mrow><mi>M</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></EquationSource></InlineEquation> grow exponentially with the fractal generation index <InlineEquation ID="IEq3"><EquationSource Format="TEX">${\ell}$</EquationSource><EquationSource Format="MATHML"><math><mi>ℓ</mi></math></EquationSource></InlineEquation>. We find that <InlineEquation ID="IEq4"><EquationSource Format="TEX">${N}_{{\ell}}$</EquationSource><EquationSource Format="MATHML"><math><msub><mrow><mi>N</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></EquationSource></InlineEquation> is an integer multiple of <InlineEquation ID="IEq5"><EquationSource Format="TEX">${M}_{{\ell}}$</EquationSource><EquationSource Format="MATHML"><math><msub><mrow><mi>M</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></EquationSource></InlineEquation>, 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.</p>

Fractal hierarchy enables exponential scaling of topological boundary states

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2026年7月9日

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Limin Song,
Zhichan Hu,
Ziteng Wang,
Domenico Bongiovanni,
Liqin Tang,
Daohong Song,
Roberto Morandotti,
Jingjun Xu,
Hrvoje Buljan,
Zhigang Chen

摘要

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 ${N}_{{\ell}}$ $N_{ℓ}$ and the number of topological minigaps ${M}_{{\ell}}$ $M_{ℓ}$ grow exponentially with the fractal generation index ${\ell}$ $ℓ$ . We find that ${N}_{{\ell}}$ $N_{ℓ}$ is an integer multiple of ${M}_{{\ell}}$ $M_{ℓ}$ , 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.