Fractal hierarchy enables exponential scaling of topological boundary states
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| Main Authors: | , , , , , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915905561165824 |
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| author | Song, Limin Hu, Zhichan Wang, Ziteng Bongiovanni, Domenico Tang, Liqin Song, Daohong Morandotti, Roberto Xu, Jingjun Buljan, Hrvoje Chen, Zhigang |
| author_facet | Song, Limin Hu, Zhichan Wang, Ziteng Bongiovanni, Domenico Tang, Liqin Song, Daohong Morandotti, Roberto Xu, Jingjun Buljan, Hrvoje Chen, Zhigang |
| contents | Exponential growth describes an extremely rapid process ubiquitous across mathematics and 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 Sierpinski-gasket unit cells. We show that, for suitable coupling parameters, both the number of topological boundary states $N_{\ell}$ and the number of topological minigaps $M_{\ell}$ grow exponentially with the fractal generation index $\ell$. We find that $N_{\ell}$ is an integer multiple of $M_{\ell}$, with the integer determined by the underlying symmetry. This hierarchical scaling law is captured by multi-topological-phase theory and confirmed experimentally in laser-written photonic lattices. Our results identify fractal hierarchy as a materials architecture principle for controlling boundary-state multiplicity, revealing an interplay between topology, self-similar geometry, and periodic order. More broadly, this work suggests a route to synthetic materials and integrated photonic platforms in which large numbers of robust boundary modes can be engineered within compact architectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00814 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractal hierarchy enables exponential scaling of topological boundary states Song, Limin Hu, Zhichan Wang, Ziteng Bongiovanni, Domenico Tang, Liqin Song, Daohong Morandotti, Roberto Xu, Jingjun Buljan, Hrvoje Chen, Zhigang Optics Materials Science Applied Physics Exponential growth describes an extremely rapid process ubiquitous across mathematics and 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 Sierpinski-gasket unit cells. We show that, for suitable coupling parameters, both the number of topological boundary states $N_{\ell}$ and the number of topological minigaps $M_{\ell}$ grow exponentially with the fractal generation index $\ell$. We find that $N_{\ell}$ is an integer multiple of $M_{\ell}$, with the integer determined by the underlying symmetry. This hierarchical scaling law is captured by multi-topological-phase theory and confirmed experimentally in laser-written photonic lattices. Our results identify fractal hierarchy as a materials architecture principle for controlling boundary-state multiplicity, revealing an interplay between topology, self-similar geometry, and periodic order. More broadly, this work suggests a route to synthetic materials and integrated photonic platforms in which large numbers of robust boundary modes can be engineered within compact architectures. |
| title | Fractal hierarchy enables exponential scaling of topological boundary states |
| topic | Optics Materials Science Applied Physics |
| url | https://arxiv.org/abs/2604.00814 |